{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-05-17T12:31:17+00:00"},"categoryId":33728,"data":{"title":"Statistics","slug":"statistics","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"parentCategory":{"categoryId":33720,"title":"Math","slug":"math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"}},"childCategories":[],"description":"Ace your stats class, analyze data for work, or play the odds at the slot machines. Everything you need is in here.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=33728&offset=0&size=5"}},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":218,"items":[{"headers":{"creationTime":"2016-03-26T15:38:44+00:00","modifiedTime":"2022-03-15T16:31:04+00:00","timestamp":"2022-03-15T18:01:08+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Calculate a Confidence Interval for a Population Mean When You Know Its Standard Deviation","strippedTitle":"how to calculate a confidence interval for a population mean when you know its standard deviation","slug":"how-to-calculate-a-confidence-interval-for-a-population-mean-when-you-know-its-standard-deviation","canonicalUrl":"","seo":{"metaDescription":"You've got the standard deviation. Now you want to figure out a confidence interval for the average of a population. Find out how.","noIndex":0,"noFollow":0},"content":"If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is <i>numerical,</i> most people want to estimate the mean (average) value for the population. You estimate the population mean, <em>μ</em>, by using a sample mean,<em> x̄</em>, plus or minus a margin of error. The result is called a <i>confidence interval for the population mean, μ.</i>\r\n\r\nWhen the population standard deviation is known, the formula for a confidence interval (CI) for a population mean is <em>x̄ ± z* σ/√n, </em>where <em>x̄ </em>is the sample mean, <em>σ </em>is the population standard deviation, <em>n</em> is the sample size, and <i>z*</i> represents the appropriate <i>z</i>*-value from the standard normal distribution for your desired confidence level.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><i>z</i>*<i>-</i>values for Various Confidence Levels</td>\r\n</tr>\r\n<tr>\r\n<th>Confidence Level</th>\r\n<th>z*-value</th>\r\n</tr>\r\n<tr>\r\n<td>80%</td>\r\n<td>1.28</td>\r\n</tr>\r\n<tr>\r\n<td>90%</td>\r\n<td>1.645 (by convention)</td>\r\n</tr>\r\n<tr>\r\n<td>95%</td>\r\n<td>1.96</td>\r\n</tr>\r\n<tr>\r\n<td>98%</td>\r\n<td>2.33</td>\r\n</tr>\r\n<tr>\r\n<td>99%</td>\r\n<td>2.58</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nThe above table shows values of <i>z*</i> for the given confidence levels. Note that these values are taken from the standard normal (Z-) distribution. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. Hence this chart can be expanded to other confidence percentages as well. The chart shows only the confidence percentages most commonly used.\r\n<p class=\"Warning\">In this case, the data either have to come from a normal distribution, or if not, then <i>n </i>has to be large enough (at least 30 or so) in order for the Central Limit Theorem to be applied, allowing you to use <i>z*-</i>values in the formula.</p>\r\nTo calculate a CI for the population mean (average), under these conditions, do the following:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Determine the confidence level and find the appropriate <i>z*</i>-value.</p>\r\n<p class=\"child-para\">Refer to the above table.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the sample mean (<em>x̄</em>) for the sample size (<i>n</i>).</p>\r\n<p class=\"child-para\"><b><i>Note:</i></b> The <a href=\"https://www.dummies.com/education/math/statistics/how-population-standard-deviation-affects-standard-error/\" target=\"_blank\" rel=\"noopener\">population standard deviation</a> is assumed to be a known value, <em>σ.</em></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply <i>z*</i> times <em>σ </em>and divide that by the square root of <i>n</i>.</p>\r\n<p class=\"child-para\">This calculation gives you the margin of error.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Take <em>x̄ </em>plus or minus the margin of error to obtain the CI.</p>\r\n<p class=\"child-para\">The lower end of the CI is <em>x̄ </em>minus the margin of error, whereas the upper end of the CI is <em>x̄ </em>plus the margin of error.</p>\r\n</li>\r\n</ol>\r\nFor example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Because you want a 95 percent confidence interval, your <i>z*</i>-value is 1.96.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Suppose you take a random sample of 100 fingerlings and determine that the average length is 7.5 inches; assume the population standard deviation is 2.3 inches. This means <em>x̄ </em>= 7.5, <em>σ </em>= 2.3, and<em> n </em>= 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply 1.96 times 2.3 divided by the square root of 100 (which is 10). The margin of error is, therefore, <em>± </em>1.96(2.3/10) = 1.96*0.23 = 0.45 inches.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Your 95 percent confidence interval for the mean length of walleye fingerlings in this fish hatchery pond is 7.5 inches <em>± </em>0.45 inches.</p>\r\n<p class=\"child-para\">(The lower end of the interval is 7.5 – 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95 inches.)</p>\r\n</li>\r\n</ol>\r\n<p class=\"Remember\">After you calculate a <a href=\"https://www.dummies.com/education/math/statistics/surveying-statistical-confidence-intervals/\" target=\"_blank\" rel=\"noopener\">confidence interval</a>, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.”</p>\r\nIn this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 7.05 and 7.95 inches, based on my sample data.” (Always be sure to include appropriate units.)","description":"If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is <i>numerical,</i> most people want to estimate the mean (average) value for the population. You estimate the population mean, <em>μ</em>, by using a sample mean,<em> x̄</em>, plus or minus a margin of error. The result is called a <i>confidence interval for the population mean, μ.</i>\r\n\r\nWhen the population standard deviation is known, the formula for a confidence interval (CI) for a population mean is <em>x̄ ± z* σ/√n, </em>where <em>x̄ </em>is the sample mean, <em>σ </em>is the population standard deviation, <em>n</em> is the sample size, and <i>z*</i> represents the appropriate <i>z</i>*-value from the standard normal distribution for your desired confidence level.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><i>z</i>*<i>-</i>values for Various Confidence Levels</td>\r\n</tr>\r\n<tr>\r\n<th>Confidence Level</th>\r\n<th>z*-value</th>\r\n</tr>\r\n<tr>\r\n<td>80%</td>\r\n<td>1.28</td>\r\n</tr>\r\n<tr>\r\n<td>90%</td>\r\n<td>1.645 (by convention)</td>\r\n</tr>\r\n<tr>\r\n<td>95%</td>\r\n<td>1.96</td>\r\n</tr>\r\n<tr>\r\n<td>98%</td>\r\n<td>2.33</td>\r\n</tr>\r\n<tr>\r\n<td>99%</td>\r\n<td>2.58</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nThe above table shows values of <i>z*</i> for the given confidence levels. Note that these values are taken from the standard normal (Z-) distribution. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. Hence this chart can be expanded to other confidence percentages as well. The chart shows only the confidence percentages most commonly used.\r\n<p class=\"Warning\">In this case, the data either have to come from a normal distribution, or if not, then <i>n </i>has to be large enough (at least 30 or so) in order for the Central Limit Theorem to be applied, allowing you to use <i>z*-</i>values in the formula.</p>\r\nTo calculate a CI for the population mean (average), under these conditions, do the following:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Determine the confidence level and find the appropriate <i>z*</i>-value.</p>\r\n<p class=\"child-para\">Refer to the above table.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the sample mean (<em>x̄</em>) for the sample size (<i>n</i>).</p>\r\n<p class=\"child-para\"><b><i>Note:</i></b> The <a href=\"https://www.dummies.com/education/math/statistics/how-population-standard-deviation-affects-standard-error/\" target=\"_blank\" rel=\"noopener\">population standard deviation</a> is assumed to be a known value, <em>σ.</em></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply <i>z*</i> times <em>σ </em>and divide that by the square root of <i>n</i>.</p>\r\n<p class=\"child-para\">This calculation gives you the margin of error.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Take <em>x̄ </em>plus or minus the margin of error to obtain the CI.</p>\r\n<p class=\"child-para\">The lower end of the CI is <em>x̄ </em>minus the margin of error, whereas the upper end of the CI is <em>x̄ </em>plus the margin of error.</p>\r\n</li>\r\n</ol>\r\nFor example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Because you want a 95 percent confidence interval, your <i>z*</i>-value is 1.96.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Suppose you take a random sample of 100 fingerlings and determine that the average length is 7.5 inches; assume the population standard deviation is 2.3 inches. This means <em>x̄ </em>= 7.5, <em>σ </em>= 2.3, and<em> n </em>= 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply 1.96 times 2.3 divided by the square root of 100 (which is 10). The margin of error is, therefore, <em>± </em>1.96(2.3/10) = 1.96*0.23 = 0.45 inches.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Your 95 percent confidence interval for the mean length of walleye fingerlings in this fish hatchery pond is 7.5 inches <em>± </em>0.45 inches.</p>\r\n<p class=\"child-para\">(The lower end of the interval is 7.5 – 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95 inches.)</p>\r\n</li>\r\n</ol>\r\n<p class=\"Remember\">After you calculate a <a href=\"https://www.dummies.com/education/math/statistics/surveying-statistical-confidence-intervals/\" target=\"_blank\" rel=\"noopener\">confidence interval</a>, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.”</p>\r\nIn this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 7.05 and 7.95 inches, based on my sample data.” (Always be sure to include appropriate units.)","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208650,"title":"Statistics For Dummies Cheat Sheet","slug":"statistics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208650"}},{"articleId":188342,"title":"Checking Out Statistical Confidence Interval Critical Values","slug":"checking-out-statistical-confidence-interval-critical-values","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188342"}},{"articleId":188341,"title":"Handling Statistical Hypothesis Tests","slug":"handling-statistical-hypothesis-tests","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188341"}},{"articleId":188343,"title":"Statistically Figuring Sample Size","slug":"statistically-figuring-sample-size","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188343"}},{"articleId":188336,"title":"Surveying Statistical Confidence Intervals","slug":"surveying-statistical-confidence-intervals","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188336"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282603,"slug":"statistics-for-dummies-2nd-edition","isbn":"9781119293521","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119293529-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-for-dummies-2nd-edition-cover-9781119293521-203x255.jpg","width":203,"height":255},"title":"Statistics For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p>Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. 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Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. 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Rumsey, PhD</b> is a longtime statistics professor at The Ohio State University specializing in statistics education. She authored <i>Statistics For Dummies, Statistics II For Dummies,</i> and <i>Probability For Dummies.</i> </p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119547518&quot;]}]\" id=\"du-slot-62196dc1b1bb5\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119547518&quot;]}]\" id=\"du-slot-62196dc1b2547\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":259570,"title":"Basic Statistics Formulas You Need","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/259570"}},{"articleId":259574,"title":"Important Probability Definitions and Rules","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/259574"}},{"articleId":259578,"title":"Correlation and Regression Formulas","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/259578"}},{"articleId":259582,"title":"10 Ways to Spot Common Statistical Mistakes","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/259582"}}],"content":[{"title":"Basic statistics formulas you need","thumb":null,"image":null,"content":"<p>In statistics the formulas add up until at the end you have quite a group. The following table presents statistics formulas organize based on when they are typically found in a statistics class, from to bottom. It starts with the sample mean, median, and standard deviation, moving on to Z-values, confidence intervals and hypothesis tests for a mean and a proportion, and the difference of two means and two proportions, plus the formula to use for the normal approximation to the binomial. You can use it to help you determine and remember which formula to use when.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-259571\" src=\"https://www.dummies.com/wp-content/uploads/Stats-z-values.jpg\" alt=\"Stats-z-values\" width=\"529\" height=\"400\" /></p>\n"},{"title":"Important probability definitions and rules","thumb":null,"image":null,"content":"<p>The following table presents the most important probability rules (the addition rule, the multiplication rule, and the complement rule) as well as the well-loved definition of independence and conditional probability. Lots of notation in these rules and definitions.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-259575\" src=\"https://www.dummies.com/wp-content/uploads/stats-probabilities.jpg\" alt=\"stats-probabilities\" width=\"535\" height=\"189\" /></p>\n"},{"title":"Correlation and regression formulas","thumb":null,"image":null,"content":"<p>Correlation and regression are super important — and useful. Here you find formulas for correlation and the best fitting regression line, as well as the individual formulas for the slope and <em>y</em>-intercept of the best fitting line.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-259579\" src=\"https://www.dummies.com/wp-content/uploads/stats-correlation.jpg\" alt=\"stats-correlation\" width=\"535\" height=\"215\" /></p>\n"},{"title":"10 ways to spot common statistical mistakes","thumb":null,"image":null,"content":"<p>Use these everyday reminders in and out of class to help spot common statistical mistakes, either in your own work or in the work of others. Focus on the ways that things can go wrong in statistics: look for misleading graphs, search for bias, ask for the sample size and margin of error, know how the sample was selected, check for confounding variables, think about the value of the correlation, and last but not least, <em>check the math.</em> Avoid selective reporting, and don’t put much stock in anecdotes.</p>\n<ol>\n<li><strong>Scrutinize graphs.</strong><br />\nWatch the scale, the number of bars/slices, the sample size, and the source.</li>\n<li><strong>Search for and specify bias.</strong><br />\nBias is systematic favoritism and can occur in sample selection, data collection, and in graphs and analyses.</li>\n<li><strong>Mark the margin of error.</strong><br />\nEvery poll has a margin of error; it’s important to look for theirs and report yours.</li>\n<li><strong>Scan for sample size.</strong><br />\nSample size plays a big role in how precise the data is, if all goes well, so knowing its value is important</li>\n<li><strong>Study sample selection — it’s gotta be random.</strong><br />\nA random sample gives each group of the same size the same chance to be selected. Without this chance, bias can quickly occur.</li>\n<li><strong>Check for confounding variables.</strong><br />\nConfounding variables are ones that were not included in a study but can influence the results. They are a major source of bias.</li>\n<li><strong>Consider correlation.</strong><br />\nFinding the correlation is step 1; interpreting it is step 2. Make sure a correlation is at least beyond the weak range before believing the results; that is, beyond –0.3 and +0.3. The closer to plus or minus 1, the better.</li>\n<li><strong>Do the math.</strong><br />\nMath mistakes are possible; double-check your work and the work of others before diving into the interpretation of the results.</li>\n<li><strong>Detect selective reporting.</strong><br />\nSelective reporting is where folks leave out the stuff that didn’t work out and only point your attention to the exciting results that may not represent the big picture.</li>\n<li><strong>Avoid the anecdote.</strong><br />\nAnecdotes are stories, which represent samples of size 1. We need more data than that before putting credence in results.</li>\n</ol>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-25T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":208845},{"headers":{"creationTime":"2016-03-27T16:57:44+00:00","modifiedTime":"2022-02-23T15:03:22+00:00","timestamp":"2022-02-24T17:07:36+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"Statistics II For Dummies Cheat Sheet","strippedTitle":"statistics ii for dummies cheat sheet","slug":"statistics-ii-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Build on your foundational statistics knowledge by learning about multiple regression, analysis of variance (ANOVA), Chi-square tests, and more.","noIndex":0,"noFollow":0},"content":"Statistics II elaborates on Statistics I and moves into new territories, including multiple regression, analysis of variance (ANOVA), Chi-square tests, nonparametric procedures, and other key topics. Knowing which data analysis to use and why is important, as is familiarity with computer output if you want your numbers to give you dependable results.","description":"Statistics II elaborates on Statistics I and moves into new territories, including multiple regression, analysis of variance (ANOVA), Chi-square tests, nonparametric procedures, and other key topics. Knowing which data analysis to use and why is important, as is familiarity with computer output if you want your numbers to give you dependable results.","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}},{"articleId":208845,"title":"Statistics Workbook For Dummies Cheat Sheet","slug":"statistics-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208845"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":288190,"slug":"statistics-ii-for-dummies-2nd-edition","isbn":"9781119827399","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119827396/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119827396/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119827396-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119827396/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119827396/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-ii-for-dummies-2nd-edtion-cover-9781119827399-203x255.jpg","width":203,"height":255},"title":"Statistics II For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p>Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.</p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119827399&quot;]}]\" id=\"du-slot-6217bb58d0569\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119827399&quot;]}]\" id=\"du-slot-6217bb58d0ee9\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":194603,"title":"How to Determine Which Data Analysis to Use in Statistics II","slug":"how-to-determine-which-data-analysis-to-use-in-statistics-ii","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194603"}},{"articleId":194597,"title":"Computer Output for Statistics II","slug":"computer-output-for-statistics-ii","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194597"}}],"content":[{"title":"How to determine which data analysis to use in Statistics II","thumb":null,"image":null,"content":"<p class=\"SortTitle\">Statistics II is often about data analysis, and the trick is to know when to use which analysis method. The following table helps you compare, contrast, and decide what data analysis to use and when. Use it for an easy reference and to review for exams.</p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">\n<tbody>\n<tr>\n<th>Analysis</th>\n<th>Purpose</th>\n<th>When It’s Used</th>\n</tr>\n<tr>\n<td>Simple linear regression</td>\n<td>Use <i>x</i> to estimate <i>y,</i> using a line</td>\n<td>Response variable <i></i><i>y</i> quantitative; constant<br />\nvariance across <i>x,</i> which is quantitative</td>\n</tr>\n<tr>\n<td>Multiple regression</td>\n<td>Use multiple <i>x</i> variables (<i>x,</i> i = 1 . . . ,<br />\n<i>k</i>) to estimate y using a plane</td>\n<td><i>y</i> is quantitative; normal distribution for each<br />\n<i>xi</i> combination with constant variance</td>\n</tr>\n<tr>\n<td>Nonlinear regression</td>\n<td>Use <i>x</i> to estimate <i>y</i> using a curve</td>\n<td><i>y</i> is quantitative; normal distribution; constant<br />\nvariance across <i>x</i></td>\n</tr>\n<tr>\n<td>Logistic regression</td>\n<td>Use <i>x</i> to estimate <i>p</i> = probability of success of<br />\n<i>y</i></td>\n<td><i>y</i> is a yes/no variable with success <i>p</i></td>\n</tr>\n<tr>\n<td>One-way ANOVA</td>\n<td>Compare two population means using one factor</td>\n<td><i>y</i> is quantitative; factor is <i>x</i></td>\n</tr>\n<tr>\n<td>Tukey’s test</td>\n<td>Multiple comparisons</td>\n<td>Confidence intervals for all pairs of means; keeps error rates<br />\nlow</td>\n</tr>\n<tr>\n<td>Fisher’s LSD test</td>\n<td>Multiple comparisons</td>\n<td>Confidence intervals for all pairs of means; overall error rate<br />\nhigher than Tukey’s</td>\n</tr>\n<tr>\n<td>Scheffe’s method</td>\n<td>Multiple comparisons</td>\n<td>Looks at linear combinations of means, not just pairs</td>\n</tr>\n<tr>\n<td>Bonferroni adjustment</td>\n<td>Multiple comparisons</td>\n<td>All pairs of <i>t-</i>tests adjusted for number of tests</td>\n</tr>\n<tr>\n<td>Dunnetts’s test</td>\n<td>Multiple comparisons</td>\n<td>Experiments; compares treatment versus control only</td>\n</tr>\n<tr>\n<td>Student Newman-Keuls test (SNK)</td>\n<td>Multiple comparisons</td>\n<td>Stepwise approach, comparing pairs ordered from smallest to<br />\nlargest</td>\n</tr>\n<tr>\n<td>Duncan’s multiple range test (MRT)</td>\n<td>Multiple comparisons</td>\n<td>Adjusts SNK test for more power</td>\n</tr>\n<tr>\n<td>Two-way ANOVA</td>\n<td>Compare more than two population means, using two factors plus<br />\ninteraction</td>\n<td><i>y</i> is quantitative; factors are <i>(x</i><sub>1</sub><i>,<br />\nx</i><sub>2</sub><i>)</i></td>\n</tr>\n<tr>\n<td>Chi-square tests</td>\n<td>Test independence of two variables or goodness-of-fit for one<br />\nqualitative variable</td>\n<td>All variables qualitative</td>\n</tr>\n<tr>\n<td>Sign/Signed rank tests</td>\n<td>Test one population median</td>\n<td><i>y</i> is quantitative or ordinal (based on ranks)</td>\n</tr>\n</tbody>\n</table>\n"},{"title":"Computer output for Statistics II","thumb":null,"image":null,"content":"<p class=\"SortTitle\">If you’re taking Statistics II, you’re likely to face questions on computer output for multiple regression and ANOVA. Professors like to give output on exams and ask you to interpret it. Sometimes they leave empty spaces and ask you to fill them in using the info given. Reviewing the following output analyses can help.</p>\n<p><img loading=\"lazy\" class=\"\" src=\"https://www.dummies.com/wp-content/uploads/161224.image0.jpg\" alt=\"image0.jpg\" width=\"597\" height=\"636\" /></p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2021-09-23T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209320},{"headers":{"creationTime":"2016-03-27T16:57:33+00:00","modifiedTime":"2022-02-14T17:21:49+00:00","timestamp":"2022-02-24T17:07:31+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"SPSS For Dummies Cheat Sheet","strippedTitle":"spss for dummies cheat sheet","slug":"spss-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Learn how to use the Syntax command language that performs statistical analysis on data through the SPSS application.","noIndex":0,"noFollow":0},"content":"SPSS is an application that performs statistical analysis on data. Entering and manipulating information in the application can be done by using SPSS’s proprietary language, which is known as the Syntax command language, or more commonly, as Syntax. The language is quite like other programming languages, and it allows you to define variables (or use predefined ones), and to use them within statements, or to evaluate them with relational or logical operators. Good programmers always know to make their code accessible through the use of comments. Syntax can also be used in conjunction with Basic and Python.","description":"SPSS is an application that performs statistical analysis on data. Entering and manipulating information in the application can be done by using SPSS’s proprietary language, which is known as the Syntax command language, or more commonly, as Syntax. The language is quite like other programming languages, and it allows you to define variables (or use predefined ones), and to use them within statements, or to evaluate them with relational or logical operators. Good programmers always know to make their code accessible through the use of comments. Syntax can also be used in conjunction with Basic and Python.","blurb":"","authors":[{"authorId":10512,"name":"Arthur Griffith","slug":"arthur-griffith","description":"","_links":{"self":"https://dummies-api.dummies.com/v2/authors/10512"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":272533,"title":"How to Run an Analysis in SPSS Statistics","slug":"how-to-run-an-analysis-in-spss-statistics","categoryList":["technology","software","other-software"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/272533"}},{"articleId":272519,"title":"10 SPSS Statistics Gotchas","slug":"10-spss-statistics-gotchas","categoryList":["technology","software","other-software"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/272519"}},{"articleId":272514,"title":"4 SPSS Statistics Licensing Options","slug":"4-spss-statistics-licensing-options","categoryList":["technology","software","other-software"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/272514"}},{"articleId":207521,"title":"SPSS Statistics For Dummies Cheat Sheet","slug":"spss-statistics-for-dummies-cheat-sheet","categoryList":["technology","software","other-software"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207521"}},{"articleId":143753,"title":"SPSS Statistics Commonly Used Analyze Menus","slug":"spss-statistics-commonly-used-analyze-menus","categoryList":["technology","software","other-software"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/143753"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":208845,"title":"Statistics Workbook For Dummies Cheat Sheet","slug":"statistics-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208845"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281869,"slug":"spss-statistics-for-dummies-4th-edition","isbn":"9781119560838","categoryList":["technology","software","other-software"],"amazon":{"default":"https://www.amazon.com/gp/product/1119560837/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119560837/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119560837-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119560837/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119560837/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/spss-statistics-for-dummies-4th-edition-cover-9781119560838-203x255.jpg","width":203,"height":255},"title":"SPSS Statistics For Dummies, 4th Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"9107\">Jesus Salcedo</b> is an independent statistical and data-mining consultant who has been using SPSS products for more than 25 years. He has written numerous SPSS courses and trained thousands of users. <b data-author-id=\"9106\">Keith McCormick</b> has been all over the world training and consulting in all things SPSS, statistics, and data mining. He now authors courses on the LinkedIn Learning platform and coaches executives on how to effectively manage their analytics teams.</p>","authors":[{"authorId":9107,"name":"Jesus Salcedo","slug":"jesus-salcedo","description":"Jesus Salcedo is an independent statistical and data-mining consultant who has been using SPSS products for more than 25 years. He has written numerous SPSS courses and trained thousands of users.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9107"}},{"authorId":9106,"name":"Keith McCormick","slug":"keith-mccormick","description":"Keith McCormick has been all over the world training and consulting in all things SPSS, statistics, and data mining. He now authors courses on the LinkedIn Learning platform and coaches executives on how to effectively manage their analytics teams.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9106"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119560838&quot;]}]\" id=\"du-slot-6217bb5381e48\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119560838&quot;]}]\" id=\"du-slot-6217bb53827e8\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":194410,"title":"SPSS Syntax Language Variable Definitions","slug":"spss-syntax-language-variable-definitions","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194410"}},{"articleId":194412,"title":"SPSS Syntax Language Statements","slug":"spss-syntax-language-statements","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194412"}},{"articleId":194411,"title":"SPSS Syntax Language Predefined Variables","slug":"spss-syntax-language-predefined-variables","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194411"}},{"articleId":194399,"title":"SPSS Syntax Language Comments","slug":"spss-syntax-language-comments","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194399"}},{"articleId":194409,"title":"SSPS Syntax Language Relational Operators","slug":"ssps-syntax-language-relational-operators","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194409"}},{"articleId":194413,"title":"SPSS Syntax Language Logical Operators","slug":"spss-syntax-language-logical-operators","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194413"}}],"content":[{"title":"SPSS Syntax language variable definitions","thumb":null,"image":null,"content":"<p>In Syntax, you can define several different characteristics for each of your variables. Here is the full collection of possibilities among SPSS Syntax language variables:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Name: </b>Short form of the variable name</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Type: </b>Numeric, comma, dot, scientific notation, date, dollar, custom currency, or string</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Width: </b>Maximum number of characters used to display the data</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Decimals: </b>Number of digits to the right of the decimal point</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Label: </b>Long form of the variable name</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Values: </b>Names assigned to specific values</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Missing: </b>Value, or values, to represent missing values</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Columns: </b>Number of spaces into which the value is displayed</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Align: </b>Right, left, or center</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Measure: </b>Scale, ordinal, or nominal</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Role:</b> Input, target, both, none, partition, or split</p>\n</li>\n</ul>\n"},{"title":"SPSS Syntax language statements","thumb":null,"image":null,"content":"<p>A single Syntax language instruction can be very simple, or it can be complex enough to serve as an entire program. A single instruction consists of a command followed by arguments to modify or expand the actions of the command, as follows:</p>\n<pre class=\"code\"><i>command [/option=value]terminator</i></pre>\n<p><b>command:</b> Every statement begins with a command.</p>\n<p><b>option:</b> Each command has a specific set of options.</p>\n<p><b>value:</b> The value, or values, for the option.</p>\n<p><b>terminator:</b> Every statement ends with a period as a terminator.</p>\n"},{"title":"SPSS Syntax language predefined variables","thumb":null,"image":null,"content":"<p>Most of the values used in Syntax are from the variables in the data set you currently have loaded and displayed in SPSS. You simply use your variable names in your program, and SPSS knows where to go and get the values for it.</p>\n<p>Some other variables are already defined, and you can use them anywhere in a program. Predefined variables, which are called <i>system variables,</i> all begin with a dollar sign ($) and already contain values.</p>\n<p>The system variables are listed in the table below:</p>\n<table>\n<tbody>\n<tr>\n<th>Variable Name</th>\n<th>What It Is</th>\n</tr>\n<tr>\n<td><span class=\"code\">$CASENUM</span></td>\n<td>Current case number. It is the count of cases from the<br />\nbeginning to the current one.</td>\n</tr>\n<tr>\n<td><span class=\"code\">$DATE</span></td>\n<td>Current date in international format with two-digit year.</td>\n</tr>\n<tr>\n<td><span class=\"code\">$DATE11</span></td>\n<td>Current date in international format with four-digit year.</td>\n</tr>\n<tr>\n<td><span class=\"code\">$JDATE</span></td>\n<td>Count of the number of days since October 14, 1582 (the first<br />\nday of the Gregorian calendar).</td>\n</tr>\n<tr>\n<td><span class=\"code\">$LENGTH</span></td>\n<td>Current page length.</td>\n</tr>\n<tr>\n<td><span class=\"code\">$SYSMIS</span></td>\n<td>System missing value. This prints as a period (.) or whatever<br />\nis defined as the decimal point.</td>\n</tr>\n<tr>\n<td><span class=\"code\">$TIME</span></td>\n<td>Number of seconds since midnight October 14, 1582 (the first<br />\nday of the Gregorian calendar).</td>\n</tr>\n<tr>\n<td><span class=\"code\">$WIDTH</span></td>\n<td>Current page width.</td>\n</tr>\n</tbody>\n</table>\n"},{"title":"SPSS Syntax language comments","thumb":null,"image":null,"content":"<p>You can insert descriptive text, called a <i>comment</i>, into your program. This text doesn&#8217;t do anything except help make things clear when you read (or somebody else reads) your code. You start a comment the same way you start any other command: on its own line by using the keyword <span class=\"code\">COMMENT</span> or an asterisk or an asterisk-slash. The comment is terminated by a period. For example:</p>\n<pre class=\"code\">COMMENT This is a comment and will not be executed.\r\n* This is a comment and will continue to be\r\n a comment until the terminating period.\r\n/* This is a comment and will continue to be\r\n a comment until the terminating asterisk-slash */</pre>\n"},{"title":"SPSS Syntax language relational operators","thumb":null,"image":null,"content":"<p>Syntax offers conditional statements that are executed only if conditions are right. Usually those conditions are determined by evaluating the contents of a variable with a logical or relational operator. The following table lists the relational operators you can use to compare numbers.</p>\n<table>\n<tbody>\n<tr>\n<th>Symbol</th>\n<th>Alpha</th>\n<th>What It Is</th>\n</tr>\n<tr>\n<td><span class=\"code\">=</span></td>\n<td><span class=\"code\">EQ</span></td>\n<td>Is equal to</td>\n</tr>\n<tr>\n<td><span class=\"code\">&lt;</span></td>\n<td><span class=\"code\">LT</span></td>\n<td>Is less than</td>\n</tr>\n<tr>\n<td><span class=\"code\">&gt;</span></td>\n<td><span class=\"code\">GT</span></td>\n<td>Is greater than</td>\n</tr>\n<tr>\n<td><span class=\"code\">&lt;&gt;</span></td>\n<td><span class=\"code\">NE</span></td>\n<td>Is not equal to</td>\n</tr>\n<tr>\n<td><span class=\"code\">&lt;=</span></td>\n<td><span class=\"code\">LE</span></td>\n<td>Is less than or equal to</td>\n</tr>\n<tr>\n<td><span class=\"code\">&gt;=</span></td>\n<td><span class=\"code\">GE</span></td>\n<td>Is greater than or equal to</td>\n</tr>\n</tbody>\n</table>\n"},{"title":"SPSS Syntax language logical operators","thumb":null,"image":null,"content":"<p>Syntax offers conditional statements that are executed only if conditions are right. Usually those conditions are determined by evaluating the contents of a variable with a logical or relational operator. The following table lists the logical operators you can use for longer, complex comparisons.</p>\n<table>\n<tbody>\n<tr>\n<th>Symbol</th>\n<th>Alpha</th>\n<th>Definition</th>\n</tr>\n<tr>\n<td><span class=\"code\">&amp;</span></td>\n<td><span class=\"code\">AND</span></td>\n<td>Both relational operators must be true</td>\n</tr>\n<tr>\n<td><span class=\"code\">|</span></td>\n<td><span class=\"code\">OR</span></td>\n<td>Either relational operator can be true</td>\n</tr>\n<tr>\n<td><span class=\"code\">~</span></td>\n<td><span class=\"code\">NOT</span></td>\n<td>Reverses the result of a relational operator</td>\n</tr>\n</tbody>\n</table>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209293},{"headers":{"creationTime":"2016-03-27T16:47:58+00:00","modifiedTime":"2022-01-28T18:54:34+00:00","timestamp":"2022-02-24T17:07:27+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"Statistics: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"statistics: 1001 practice problems for dummies cheat sheet","slug":"1001-statistics-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Keep this handy Cheat Sheet nearby as you're learning statistics. It includes terminology, symbols, formulas, and sound strategies.","noIndex":0,"noFollow":0},"content":"There are many types of statistics problems, including the use of pie charts, bar graphs, means, standard deviation to correlation, regression, confidence intervals, and hypothesis tests.\r\n\r\nTo be successful, you need to be able to make connections between statistical <i>ideas</i> and statistical <i>formulas</i>. Through practice, you see what type of technique is required for a problem and why, as well as how to set up the problem, work it out, and make proper conclusions.\r\n\r\nMost statistics problems you encounter likely involve terminology, symbols, and formulas. No worries! This Cheat Sheet gives you tips for success.","description":"There are many types of statistics problems, including the use of pie charts, bar graphs, means, standard deviation to correlation, regression, confidence intervals, and hypothesis tests.\r\n\r\nTo be successful, you need to be able to make connections between statistical <i>ideas</i> and statistical <i>formulas</i>. Through practice, you see what type of technique is required for a problem and why, as well as how to set up the problem, work it out, and make proper conclusions.\r\n\r\nMost statistics problems you encounter likely involve terminology, symbols, and formulas. No worries! This Cheat Sheet gives you tips for success.","blurb":"","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":151951,"title":"Checking Out Statistical Symbols","slug":"checking-out-statistical-symbols","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151951"}},{"articleId":151950,"title":"Terminology Used in Statistics","slug":"terminology-used-in-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151950"}},{"articleId":151947,"title":"Breaking Down Statistical Formulas","slug":"breaking-down-statistical-formulas","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151947"}},{"articleId":151934,"title":"Sticking to a Strategy When You Solve Statistics Problems","slug":"sticking-to-a-strategy-when-you-solve-statistics-problems","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151934"}},{"articleId":147353,"title":"How to Measure Relative Standing 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Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282605,"slug":"statistics-1001-practice-problems-for-dummies","isbn":"9781118776049","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118776046-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-1001-practice-problems-for-dummies-cover-9781118776049-204x255.jpg","width":204,"height":255},"title":"Statistics: 1,001 Practice Problems For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8947\">The Experts at Dummies</b> are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.</p>","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = 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Statistics","slug":"terminology-used-in-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151950"}},{"articleId":151947,"title":"Breaking Down Statistical Formulas","slug":"breaking-down-statistical-formulas","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151947"}},{"articleId":151951,"title":"Checking Out Statistical Symbols","slug":"checking-out-statistical-symbols","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151951"}},{"articleId":151934,"title":"Sticking to a Strategy When You Solve Statistics Problems","slug":"sticking-to-a-strategy-when-you-solve-statistics-problems","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151934"}}],"content":[{"title":"Terminology used in statistics","thumb":null,"image":null,"content":"<p>Like every subject, statistics has its own language. The language is what helps you know what a problem is asking for, what results are needed, and how to describe and evaluate the results in a statistically correct manner. Here&#8217;s an overview of the types of statistical terminology:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Four big terms in statistics are population, sample, parameter, and statistic:</p>\n<ul class=\"level-two\">\n<li>\n<p class=\"first-para\">A <i>population </i>is the entire group of individuals you want to study, and a <i>sample </i>is a subset of that group.</p>\n</li>\n<li>\n<p class=\"first-para\">A <i>parameter</i> is a quantitative characteristic of the population that you&#8217;re interested in estimating or testing (such as a population mean or proportion).</p>\n</li>\n<li>\n<p class=\"first-para\">A <i>statistic </i>is a quantitative characteristic of a sample that often helps estimate or test the population parameter (such as a sample mean or proportion).</p>\n</li>\n</ul>\n</li>\n<li>\n<p class=\"first-para\"><i>Descriptive statistics</i> are single results you get when you analyze a set of data — for example, the sample mean, median, standard deviation, correlation, regression line, margin of error, and test statistic.</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Statistical inference</i> refers to using your data (and its descriptive statistics) to make conclusions about the population. Major types of inference include regression, confidence intervals, and hypothesis tests.</p>\n</li>\n</ul>\n"},{"title":"Breaking down statistical formulas","thumb":null,"image":null,"content":"<p>Formulas abound in statistics problems — there&#8217;s just no getting around them. However, there&#8217;s typically a method to the madness if you can break the formulas into pieces. Here are some helpful tips:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Formulas for descriptive statistics basically take the values in the data set and apply arithmetic operations. Often, the formulas look worse than the process itself. The key: If you can explain to your friend how to calculate a standard deviation, for example, the formula is more of an afterthought.</p>\n</li>\n<li>\n<p class=\"first-para\">Formulas for the regression line have a basis in algebra. Instead of the typical <em>y</em> = <em>mx</em> + <em>b</em> format everyone learns in school, statisticians use <em>y</em> = <em>a</em> + <em>bx</em>.</p>\n<ul class=\"level-two\">\n<li>\n<p class=\"first-para\">The slope, <em>b,</em> is the coefficient of the <em>x </em>variable.</p>\n</li>\n<li>\n<p class=\"first-para\">The <em>y-</em>intercept, <em>a,</em> is where the regression line crosses the <em>y-</em>axis.</p>\n</li>\n</ul>\n<p class=\"child-para\">The formulas for finding <em>a</em> and <em>b</em> involve five statistics: the mean of the <em>x-</em>values, the mean of the <em>y-</em>values, the standard deviations for the <em>x</em>&#8216;s, the standard deviations for the <em>y</em>&#8216;s, and the correlation.</p>\n</li>\n<li>\n<p class=\"first-para\">All the various confidence interval formulas, when made into a list, can look like a hodge-podge of notation. However, they all have the same structure: a descriptive statistic (from your sample) plus or minus a margin of error. The margin of error involves a <em>z*</em>-value (from the <em>Z-</em>distribution) or <em>t*-</em>value (from the <em>t-</em>distribution) times the standard error. The parts you need for standard error are generally provided in the problem, and the <em>z*-</em> or <em>t*-</em>values come from tables.</p>\n</li>\n<li>\n<p class=\"first-para\">Hypothesis tests also have a common structure. Although each one involves a series of steps to carry out, they all boil down to one thing: the test statistic. A <em>test statistic</em> measures how far your data is from what the population supposedly looks like. It takes the difference between your sample statistic and the (claimed) population parameter and standardizes it so you can look it up on a common table and make a decision.</p>\n</li>\n</ul>\n"},{"title":"Symbols used in statistics","thumb":null,"image":null,"content":"<p>Symbols (or notation) found in statistics problems fall into three categories: math symbols, symbols referring to a population, and symbols referring to a sample. Math symbols are easy enough to decipher with a simple review of algebra; they involve items such as square root signs, equations of a line, and combinations of math operations. The other two categories are a bit more challenging, and knowing the difference between them is critical.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/430900.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"135\" /></p>\n"},{"title":"Stick to a strategy when you solve statistics problems","thumb":null,"image":null,"content":"<p>Solving statistics problems is always about having a strategy. You can&#8217;t just read a problem over and over and expect to come up with an answer — all you&#8217;ll get is anxiety! Although not all strategies work for everyone, here&#8217;s a three-step strategy that has proven its worth:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">Label everything the problem gives you.</p>\n<p class=\"child-para\">For example, if the problem says &#8220;<i>X</i> has a normal distribution with a mean of 10 and a standard deviation of 2,&#8221; leap into action: Circle the 10 and write <i>μ,</i> and circle the 2 and write <i>σ.</i> That way you don&#8217;t have to hunt later to find the numbers you need.</p>\n</li>\n<li>\n<p class=\"first-para\">Write down what you&#8217;re asked to find in a statistical manner.</p>\n<p class=\"child-para\">Hint: Questions typically tell you what they want in the last line of the problem. For example, if you&#8217;re asked to find the probability that more than 10 people come to the party, write &#8220;Find <i>P</i>(<i>X</i> &gt; 10).&#8221;</p>\n</li>\n<li>\n<p class=\"first-para\">Use a formula, a process, or an example you&#8217;ve seen to connect what you&#8217;re asked to find with what the problem gives you.</p>\n<p class=\"child-para\">For example, suppose you&#8217;re told that <i>X</i> has a normal distribution with a mean of 80 and a standard deviation of 5, and you want the probability that <i>X</i> is less than 90. Label what you&#8217;re given: &#8220;<i>X</i> normal with <i>μ </i>= 80 and <i>σ </i>= 5.&#8221; Next, write what you need to find, using symbols: &#8220;Find <i>P</i>(<i>X</i> &lt; 90).&#8221; Because <i>X</i> has a normal distribution and you want a probability, the connection is the <i>Z-</i>formula: <i>Z</i> = (<i>X</i> – <i>μ</i>)/<i>σ</i>. You have a good idea that this is the right formula because it includes everything you have: <i>μ,</i> <i>σ,</i> and the value of <i>X</i> (which is 90). Find <i>P</i>(<i>X</i> &lt; 90) = <i>P</i>[<i>Z</i> &lt; (90 – 80)/5] = <i>P</i>(<i>Z</i> &lt; 2) = 0.9772. Voilà!</p>\n</li>\n</ol>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207668},{"headers":{"creationTime":"2016-03-26T15:35:49+00:00","modifiedTime":"2021-12-28T15:03:05+00:00","timestamp":"2022-02-24T17:07:20+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Find Probabilities for a Sample Mean","strippedTitle":"how to find probabilities for a sample mean","slug":"how-to-find-probabilities-for-a-sample-mean","canonicalUrl":"","seo":{"metaDescription":"Learn how to find probabilities in statistics for a sample mean when its distribution is normal, not normal, or unknown.","noIndex":0,"noFollow":0},"content":"In statistics, you can easily find probabilities for a <a href=\"https://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-mean/\" target=\"_blank\" rel=\"noopener\">sample mean</a> if it has a normal distribution. Even if it doesn’t have a normal distribution, or the distribution is not known, you can find probabilities if the sample size, <i>n</i>, is large enough.\r\n\r\nThe normal distribution is a very friendly distribution that has a table for finding probabilities and anything else you need. For example, you can find probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361017.image0.png\" alt=\"image0.png\" width=\"17\" height=\"20\" />\r\n\r\nby converting the<i> </i>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361018.image1.png\" alt=\"image1.png\" width=\"52\" height=\"20\" />\r\n\r\nto a <i>z</i>-value and finding probabilities using the <i>Z</i>-table (see below).\r\n\r\nThe general conversion formula from\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361019.image2.png\" alt=\"image2.png\" width=\"151\" height=\"69\" />\r\n\r\nSubstituting the appropriate values of the mean and standard error of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361020.image3.png\" alt=\"image3.png\" width=\"17\" height=\"20\" />\r\n\r\nthe conversion formula becomes:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361021.image4.png\" alt=\"image4.png\" width=\"73\" height=\"61\" />\r\n<p class=\"Tip\">Don’t forget to divide by the square root of<i> n </i>in the denominator of <i>z</i>. Always divide by the square root of<i> n </i>when the question refers to the <i>average</i> of the<i> x-</i>values.</p>\r\nFor example, suppose<i> X </i>is the time it takes a randomly chosen clerical worker in an office to type and send a standard letter of recommendation. Suppose<i> X </i>has a normal distribution, and assume the mean is 10.5 minutes and the standard deviation 3 minutes. You take a random sample of 50 clerical workers and measure their times. What is the chance that their average time is less than 9.5 minutes?\r\n\r\nThis question translates to finding\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361022.image5.png\" alt=\"image5.png\" width=\"79\" height=\"29\" />\r\n\r\nAs<i> X </i>has a normal distribution to start with, you know\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361023.image6.png\" alt=\"image6.png\" width=\"17\" height=\"20\" />\r\n\r\nalso has an exact (not approximate) normal distribution. Converting to <i>z,</i> you get:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361024.image7.png\" alt=\"image7.png\" width=\"204\" height=\"61\" />\r\n\r\nSo you want P(<i>Z</i> < –2.36).\r\n\r\n<img class=\"alignnone size-full wp-image-287021\" src=\"https://www.dummies.com/wp-content/uploads/z-score-table-1.png\" alt=\"z-score table 1\" width=\"535\" height=\"933\" />\r\n\r\n<img class=\"alignnone size-full wp-image-287022\" src=\"https://www.dummies.com/wp-content/uploads/z-score-table-2.png\" alt=\"z-score table 2\" width=\"535\" height=\"912\" />\r\n\r\nUsing the above <i>Z</i>-table, you find that P(<i>Z</i> < –2.36)=0.0091. So the probability that a random sample of 50 clerical workers average less than 9.5 minutes to complete this task is 0.91% (very small).\r\n\r\nHow do you find probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361027.image10.png\" alt=\"image10.png\" width=\"17\" height=\"20\" />\r\n\r\nif<i> X </i>is<i> not </i>normal, or unknown? As a result of the Central Limit Theorem (CLT), the distribution of<i> X </i>can be non-normal or even unknown and as long as<i> n </i>is large enough, you can still find <i>approximate </i>probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361028.image11.png\" alt=\"image11.png\" width=\"17\" height=\"20\" />\r\n\r\nusing the standard normal (<i>Z</i>-)distribution and the process described above. That is, convert to a <i>z</i>-value and find approximate probabilities using the <i>Z</i>-table.\r\n<p class=\"Remember\">When you use the CLT to find a probability for</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361029.image12.png\" alt=\"image12.png\" width=\"17\" height=\"20\" />\r\n\r\n(that is, when the distribution of <i>X</i> is<i> not </i>normal or is unknown), be sure to say that your answer is an <i>approximation.</i> You also want to say the approximate answer should be close because you’ve got a large enough<i> n </i>to use the CLT. (If<i> n </i>is not large enough for the CLT, you can use the<i> t</i>-distribution in many cases.)","description":"In statistics, you can easily find probabilities for a <a href=\"https://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-mean/\" target=\"_blank\" rel=\"noopener\">sample mean</a> if it has a normal distribution. Even if it doesn’t have a normal distribution, or the distribution is not known, you can find probabilities if the sample size, <i>n</i>, is large enough.\r\n\r\nThe normal distribution is a very friendly distribution that has a table for finding probabilities and anything else you need. For example, you can find probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361017.image0.png\" alt=\"image0.png\" width=\"17\" height=\"20\" />\r\n\r\nby converting the<i> </i>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361018.image1.png\" alt=\"image1.png\" width=\"52\" height=\"20\" />\r\n\r\nto a <i>z</i>-value and finding probabilities using the <i>Z</i>-table (see below).\r\n\r\nThe general conversion formula from\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361019.image2.png\" alt=\"image2.png\" width=\"151\" height=\"69\" />\r\n\r\nSubstituting the appropriate values of the mean and standard error of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361020.image3.png\" alt=\"image3.png\" width=\"17\" height=\"20\" />\r\n\r\nthe conversion formula becomes:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361021.image4.png\" alt=\"image4.png\" width=\"73\" height=\"61\" />\r\n<p class=\"Tip\">Don’t forget to divide by the square root of<i> n </i>in the denominator of <i>z</i>. Always divide by the square root of<i> n </i>when the question refers to the <i>average</i> of the<i> x-</i>values.</p>\r\nFor example, suppose<i> X </i>is the time it takes a randomly chosen clerical worker in an office to type and send a standard letter of recommendation. Suppose<i> X </i>has a normal distribution, and assume the mean is 10.5 minutes and the standard deviation 3 minutes. You take a random sample of 50 clerical workers and measure their times. What is the chance that their average time is less than 9.5 minutes?\r\n\r\nThis question translates to finding\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361022.image5.png\" alt=\"image5.png\" width=\"79\" height=\"29\" />\r\n\r\nAs<i> X </i>has a normal distribution to start with, you know\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361023.image6.png\" alt=\"image6.png\" width=\"17\" height=\"20\" />\r\n\r\nalso has an exact (not approximate) normal distribution. Converting to <i>z,</i> you get:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361024.image7.png\" alt=\"image7.png\" width=\"204\" height=\"61\" />\r\n\r\nSo you want P(<i>Z</i> < –2.36).\r\n\r\n<img class=\"alignnone size-full wp-image-287021\" src=\"https://www.dummies.com/wp-content/uploads/z-score-table-1.png\" alt=\"z-score table 1\" width=\"535\" height=\"933\" />\r\n\r\n<img class=\"alignnone size-full wp-image-287022\" src=\"https://www.dummies.com/wp-content/uploads/z-score-table-2.png\" alt=\"z-score table 2\" width=\"535\" height=\"912\" />\r\n\r\nUsing the above <i>Z</i>-table, you find that P(<i>Z</i> < –2.36)=0.0091. So the probability that a random sample of 50 clerical workers average less than 9.5 minutes to complete this task is 0.91% (very small).\r\n\r\nHow do you find probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361027.image10.png\" alt=\"image10.png\" width=\"17\" height=\"20\" />\r\n\r\nif<i> X </i>is<i> not </i>normal, or unknown? As a result of the Central Limit Theorem (CLT), the distribution of<i> X </i>can be non-normal or even unknown and as long as<i> n </i>is large enough, you can still find <i>approximate </i>probabilities for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361028.image11.png\" alt=\"image11.png\" width=\"17\" height=\"20\" />\r\n\r\nusing the standard normal (<i>Z</i>-)distribution and the process described above. That is, convert to a <i>z</i>-value and find approximate probabilities using the <i>Z</i>-table.\r\n<p class=\"Remember\">When you use the CLT to find a probability for</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/361029.image12.png\" alt=\"image12.png\" width=\"17\" height=\"20\" />\r\n\r\n(that is, when the distribution of <i>X</i> is<i> not </i>normal or is unknown), be sure to say that your answer is an <i>approximation.</i> You also want to say the approximate answer should be close because you’ve got a large enough<i> n </i>to use the CLT. (If<i> n </i>is not large enough for the CLT, you can use the<i> t</i>-distribution in many cases.)","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208650,"title":"Statistics For Dummies Cheat Sheet","slug":"statistics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208650"}},{"articleId":188342,"title":"Checking Out Statistical Confidence Interval Critical Values","slug":"checking-out-statistical-confidence-interval-critical-values","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188342"}},{"articleId":188341,"title":"Handling Statistical Hypothesis Tests","slug":"handling-statistical-hypothesis-tests","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188341"}},{"articleId":188343,"title":"Statistically Figuring Sample Size","slug":"statistically-figuring-sample-size","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188343"}},{"articleId":188336,"title":"Surveying Statistical Confidence Intervals","slug":"surveying-statistical-confidence-intervals","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188336"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282603,"slug":"statistics-for-dummies-2nd-edition","isbn":"9781119293521","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119293529-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-for-dummies-2nd-edition-cover-9781119293521-203x255.jpg","width":203,"height":255},"title":"Statistics For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p>Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.</p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb48868c6\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb4887250\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":169362},{"headers":{"creationTime":"2016-03-26T15:38:40+00:00","modifiedTime":"2021-12-21T20:36:48+00:00","timestamp":"2022-02-24T17:07:18+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"Using Linear Regression to Predict an Outcome","strippedTitle":"using linear regression to predict an outcome","slug":"using-linear-regression-to-predict-an-outcome","canonicalUrl":"","seo":{"metaDescription":"Linear regression is a commonly used way to predict the value of a variable when you know the value of other variables.","noIndex":0,"noFollow":0},"content":"Statistical researchers often use a linear relationship to predict the (average) numerical value of<i> Y </i>for a given value of<i> X </i>using a straight line (called the <i>regression line</i>)<i>.</i>\r\n\r\nIf you know the slope and the <i>y</i>-intercept of that regression line, then you can plug in a value for<i> X </i>and predict the average value for <i>Y.</i> In other words, you predict (the average)<i> Y </i>from <i>X.</i>\r\n\r\nIf you establish at least a moderate correlation between <i>X </i>and <i>Y</i> through both a correlation coefficient and a scatterplot, then you know they have some type of linear relationship.\r\n<p class=\"Warning\">Never do a regression analysis unless you have already found at least a moderately strong correlation between the two variables. (A good rule of thumb is it should be at or beyond either positive or negative 0.50.) If the data don’t resemble a line to begin with, you shouldn’t try to use a line to fit the data and make predictions (but people still try).</p>\r\nBefore moving forward to find the equation for your regression line, you have to identify which of your two variables is<i> X </i>and which is<i> Y</i>. When doing correlations, the choice of which variable is<i> X</i><i> </i>and which is<i> Y </i>doesn’t matter, as long as you’re consistent for all the data. But when fitting lines and making predictions, the choice of<i> X </i>and<i> Y </i>does make a difference.\r\n\r\nSo how do you determine which variable is which? In general,<i> Y </i>is the variable that you want to predict, and <i>X</i> is the variable you are using to make that prediction. For example, say you are using the number of times a population of crickets chirp to predict the temperature. In this case you would make the variable <i>Y</i> the temperature, and the variable <i>X</i> the number of chirps. Hence<i> Y </i>can be predicted by<i> X </i>using the equation of a line if a strong enough linear relationship exists.\r\n\r\nStatisticians call the <i>X</i>-variable (cricket chirps in this example) the <i>explanatory variable,</i> because if<i> X </i>changes, the slope tells you (or explains) how much<i> Y </i>is expected to change in response. Therefore, the <i>Y</i> variable is called the <i>response variable.</i> Other names for <i>X</i> and <i>Y</i> include the <i>independent</i> and <i>dependent</i> variables, respectively.\r\n\r\nIn the case of two numerical variables, you can come up with a line that enables you to predict<i> Y </i>from <i>X,</i> if (and only if) the following two conditions are met:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The scatterplot must form a linear pattern.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The correlation, <i>r,</i> is moderate to strong (typically beyond 0.50 or –0.50).</p>\r\n</li>\r\n</ul>\r\nSome researchers actually don’t check these conditions before making predictions. Their claims are not valid unless the two conditions are met.\r\n\r\nBut suppose the correlation is high; do you still need to look at the scatterplot? Yes. In some situations the data have a somewhat curved shape, yet the correlation is still strong; in these cases making predictions using a straight line is still invalid. Predictions in these cases need to be made based on other methods that use a curve instead.","description":"Statistical researchers often use a linear relationship to predict the (average) numerical value of<i> Y </i>for a given value of<i> X </i>using a straight line (called the <i>regression line</i>)<i>.</i>\r\n\r\nIf you know the slope and the <i>y</i>-intercept of that regression line, then you can plug in a value for<i> X </i>and predict the average value for <i>Y.</i> In other words, you predict (the average)<i> Y </i>from <i>X.</i>\r\n\r\nIf you establish at least a moderate correlation between <i>X </i>and <i>Y</i> through both a correlation coefficient and a scatterplot, then you know they have some type of linear relationship.\r\n<p class=\"Warning\">Never do a regression analysis unless you have already found at least a moderately strong correlation between the two variables. (A good rule of thumb is it should be at or beyond either positive or negative 0.50.) If the data don’t resemble a line to begin with, you shouldn’t try to use a line to fit the data and make predictions (but people still try).</p>\r\nBefore moving forward to find the equation for your regression line, you have to identify which of your two variables is<i> X </i>and which is<i> Y</i>. When doing correlations, the choice of which variable is<i> X</i><i> </i>and which is<i> Y </i>doesn’t matter, as long as you’re consistent for all the data. But when fitting lines and making predictions, the choice of<i> X </i>and<i> Y </i>does make a difference.\r\n\r\nSo how do you determine which variable is which? In general,<i> Y </i>is the variable that you want to predict, and <i>X</i> is the variable you are using to make that prediction. For example, say you are using the number of times a population of crickets chirp to predict the temperature. In this case you would make the variable <i>Y</i> the temperature, and the variable <i>X</i> the number of chirps. Hence<i> Y </i>can be predicted by<i> X </i>using the equation of a line if a strong enough linear relationship exists.\r\n\r\nStatisticians call the <i>X</i>-variable (cricket chirps in this example) the <i>explanatory variable,</i> because if<i> X </i>changes, the slope tells you (or explains) how much<i> Y </i>is expected to change in response. Therefore, the <i>Y</i> variable is called the <i>response variable.</i> Other names for <i>X</i> and <i>Y</i> include the <i>independent</i> and <i>dependent</i> variables, respectively.\r\n\r\nIn the case of two numerical variables, you can come up with a line that enables you to predict<i> Y </i>from <i>X,</i> if (and only if) the following two conditions are met:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The scatterplot must form a linear pattern.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The correlation, <i>r,</i> is moderate to strong (typically beyond 0.50 or –0.50).</p>\r\n</li>\r\n</ul>\r\nSome researchers actually don’t check these conditions before making predictions. Their claims are not valid unless the two conditions are met.\r\n\r\nBut suppose the correlation is high; do you still need to look at the scatterplot? Yes. In some situations the data have a somewhat curved shape, yet the correlation is still strong; in these cases making predictions using a straight line is still invalid. Predictions in these cases need to be made based on other methods that use a curve instead.","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208650,"title":"Statistics For Dummies Cheat Sheet","slug":"statistics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208650"}},{"articleId":188342,"title":"Checking Out Statistical Confidence Interval Critical Values","slug":"checking-out-statistical-confidence-interval-critical-values","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188342"}},{"articleId":188341,"title":"Handling Statistical Hypothesis Tests","slug":"handling-statistical-hypothesis-tests","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188341"}},{"articleId":188343,"title":"Statistically Figuring Sample Size","slug":"statistically-figuring-sample-size","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188343"}},{"articleId":188336,"title":"Surveying Statistical Confidence Intervals","slug":"surveying-statistical-confidence-intervals","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188336"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282603,"slug":"statistics-for-dummies-2nd-edition","isbn":"9781119293521","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119293529-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-for-dummies-2nd-edition-cover-9781119293521-203x255.jpg","width":203,"height":255},"title":"Statistics For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p>Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.</p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb4701085\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb4701a14\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":169714},{"headers":{"creationTime":"2016-03-26T15:32:10+00:00","modifiedTime":"2021-12-21T20:20:50+00:00","timestamp":"2022-02-24T17:07:18+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Interpret the Shape of Statistical Data in a Histogram","strippedTitle":"how to interpret the shape of statistical data in a histogram","slug":"how-to-interpret-the-shape-of-statistical-data-in-a-histogram","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"One of the features that a histogram can show you is the <i>shape </i>of the statistical data — in other words, the manner in which the data fall into groups. For example, all the data may be exactly the same, in which case the histogram is just one tall bar; or the data might have an equal number in each group, in which case the shape is flat.\r\n\r\nSome data sets have a distinct shape. Here are three shapes that stand out:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Symmetric.</b><i> </i>A histogram is symmetric if you cut it down the middle and the left-hand and right-hand sides resemble mirror images of each other:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/362516.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"277\" /> The above graph shows a symmetric data set; it represents the amount of time each of 50 survey participants took to fill out a certain survey. You see that the histogram is close to symmetric.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Skewed right. </b>A skewed right histogram looks like a lopsided mound, with a tail going off to the right:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"https://www.dummies.com/wp-content/uploads/362517.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"359\" /> This graph, which shows the ages of the Best Actress Academy Award winners, is skewed right. You see on the right side there are a few actresses whose ages are older than the rest. Most of the actresses were between 20 and 50 years of age when they won. A few actresses were between 60–65 years of age when they won their Oscars, and a handful were 70 years or older. The last three bars are what make the data have a shape that is skewed right.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Skewed left.</b><i> </i>If a histogram is skewed left, it looks like a lopsided mound with a tail going off to the left:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/362518.image2.jpg\" alt=\"image2.jpg\" width=\"400\" height=\"282\" /> This graph shows a histogram of 17 exam scores. The shape is skewed left; you see a few students who scored lower than everyone else.[/caption]</li>\r\n</ul>\r\nFollowing, are some particulars about classifying the shape of a data set:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Don't expect symmetric data to have an exact and perfect shape.</b> Data hardly ever fall into perfect patterns, so you have to decide whether the data shape is close enough to be called symmetric.</p>\r\n<p class=\"child-para\">If the differences aren't significant enough, you can classify it as symmetric or roughly symmetric. Otherwise, you classify the data as non-symmetric.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Don't assume that data are skewed if the shape is non-symmetric.</b> Data sets come in all shapes and sizes, and many of them don't have a distinct shape at all. Skewness is mentioned here because it's one of the more common non-symmetric shapes, and it's one of the shapes included in a standard introductory statistics course.</p>\r\n<p class=\"child-para\">If a data set does turn out to be skewed (or close to it), make sure to denote the direction of the skewness (left or right).</p>\r\n</li>\r\n</ul>","description":"One of the features that a histogram can show you is the <i>shape </i>of the statistical data — in other words, the manner in which the data fall into groups. For example, all the data may be exactly the same, in which case the histogram is just one tall bar; or the data might have an equal number in each group, in which case the shape is flat.\r\n\r\nSome data sets have a distinct shape. Here are three shapes that stand out:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Symmetric.</b><i> </i>A histogram is symmetric if you cut it down the middle and the left-hand and right-hand sides resemble mirror images of each other:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/362516.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"277\" /> The above graph shows a symmetric data set; it represents the amount of time each of 50 survey participants took to fill out a certain survey. You see that the histogram is close to symmetric.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Skewed right. </b>A skewed right histogram looks like a lopsided mound, with a tail going off to the right:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"https://www.dummies.com/wp-content/uploads/362517.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"359\" /> This graph, which shows the ages of the Best Actress Academy Award winners, is skewed right. You see on the right side there are a few actresses whose ages are older than the rest. Most of the actresses were between 20 and 50 years of age when they won. A few actresses were between 60–65 years of age when they won their Oscars, and a handful were 70 years or older. The last three bars are what make the data have a shape that is skewed right.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Skewed left.</b><i> </i>If a histogram is skewed left, it looks like a lopsided mound with a tail going off to the left:</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/362518.image2.jpg\" alt=\"image2.jpg\" width=\"400\" height=\"282\" /> This graph shows a histogram of 17 exam scores. The shape is skewed left; you see a few students who scored lower than everyone else.[/caption]</li>\r\n</ul>\r\nFollowing, are some particulars about classifying the shape of a data set:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Don't expect symmetric data to have an exact and perfect shape.</b> Data hardly ever fall into perfect patterns, so you have to decide whether the data shape is close enough to be called symmetric.</p>\r\n<p class=\"child-para\">If the differences aren't significant enough, you can classify it as symmetric or roughly symmetric. Otherwise, you classify the data as non-symmetric.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Don't assume that data are skewed if the shape is non-symmetric.</b> Data sets come in all shapes and sizes, and many of them don't have a distinct shape at all. Skewness is mentioned here because it's one of the more common non-symmetric shapes, and it's one of the shapes included in a standard introductory statistics course.</p>\r\n<p class=\"child-para\">If a data set does turn out to be skewed (or close to it), make sure to denote the direction of the skewness (left or right).</p>\r\n</li>\r\n</ul>","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208650,"title":"Statistics For Dummies Cheat Sheet","slug":"statistics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208650"}},{"articleId":188342,"title":"Checking Out Statistical Confidence Interval Critical Values","slug":"checking-out-statistical-confidence-interval-critical-values","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188342"}},{"articleId":188341,"title":"Handling Statistical Hypothesis Tests","slug":"handling-statistical-hypothesis-tests","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188341"}},{"articleId":188343,"title":"Statistically Figuring Sample Size","slug":"statistically-figuring-sample-size","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188343"}},{"articleId":188336,"title":"Surveying Statistical Confidence Intervals","slug":"surveying-statistical-confidence-intervals","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188336"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282603,"slug":"statistics-for-dummies-2nd-edition","isbn":"9781119293521","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119293529-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-for-dummies-2nd-edition-cover-9781119293521-203x255.jpg","width":203,"height":255},"title":"Statistics For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p>Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.</p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb46e482d\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-6217bb46e51b9\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-12-21T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":169003},{"headers":{"creationTime":"2016-03-26T08:26:22+00:00","modifiedTime":"2021-10-27T16:09:32+00:00","timestamp":"2022-02-24T17:07:05+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Use the T-table to Solve Statistics Problems","strippedTitle":"how to use the t-table to solve statistics problems","slug":"how-to-use-the-t-table-to-solve-statistics-problems","canonicalUrl":"","seo":{"metaDescription":"How exactly does a t-table differ from a z-table? Learn about all the important statistical differences here.","noIndex":0,"noFollow":0},"content":"The t<i>-</i>table (for the t<i>-</i>distribution) is different from the <i>z-</i>table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t<i>-</i>table, is a valuable statistics skill. Use the t-table as necessary to solve the following sample problems below.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451675.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"796\" />\r\n<h2 id=\"tab1\" >Sample questions</h2>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 17</p>\r\n<p class=\"child-para\">The study involving one population and a sample size of 18 has <i>n</i> – 1 = 18 – 1 = 17 degrees of freedom.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">For a study involving a paired design with a total of 44 observations, with the results assuming a t<i>-</i>distribution, what row of the table will you use to find the probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 21</p>\r\n<p class=\"child-para\">A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: <i>n</i> – 1 = 22 – 1 = 21.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">A <i>t-</i>value of 2.35, from a <i>t-</i>distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table?</p>\r\n<p class=\"child-para\"><b>Answer: </b>0.025 and 0.01</p>\r\n<p class=\"child-para\">Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.</p>\r\n<p class=\"child-para\">Hence, the upper-tail probability for a <i>t-</i>value of 2.35 must lie between 0.025 and 0.01.</p>\r\n</li>\r\n</ol>","description":"The t<i>-</i>table (for the t<i>-</i>distribution) is different from the <i>z-</i>table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t<i>-</i>table, is a valuable statistics skill. Use the t-table as necessary to solve the following sample problems below.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451675.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"796\" />\r\n<h2 id=\"tab1\" >Sample questions</h2>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 17</p>\r\n<p class=\"child-para\">The study involving one population and a sample size of 18 has <i>n</i> – 1 = 18 – 1 = 17 degrees of freedom.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">For a study involving a paired design with a total of 44 observations, with the results assuming a t<i>-</i>distribution, what row of the table will you use to find the probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 21</p>\r\n<p class=\"child-para\">A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: <i>n</i> – 1 = 22 – 1 = 21.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">A <i>t-</i>value of 2.35, from a <i>t-</i>distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table?</p>\r\n<p class=\"child-para\"><b>Answer: </b>0.025 and 0.01</p>\r\n<p class=\"child-para\">Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.</p>\r\n<p class=\"child-para\">Hence, the upper-tail probability for a <i>t-</i>value of 2.35 must lie between 0.025 and 0.01.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Sample questions","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":207668,"title":"Statistics: 1001 Practice Problems For Dummies Cheat Sheet","slug":"1001-statistics-practice-problems-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207668"}},{"articleId":151951,"title":"Checking Out Statistical Symbols","slug":"checking-out-statistical-symbols","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151951"}},{"articleId":151950,"title":"Terminology Used in Statistics","slug":"terminology-used-in-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151950"}},{"articleId":151947,"title":"Breaking Down Statistical Formulas","slug":"breaking-down-statistical-formulas","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151947"}},{"articleId":151934,"title":"Sticking to a Strategy When You Solve Statistics Problems","slug":"sticking-to-a-strategy-when-you-solve-statistics-problems","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151934"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282605,"slug":"statistics-1001-practice-problems-for-dummies","isbn":"9781118776049","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118776046-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118776046/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-1001-practice-problems-for-dummies-cover-9781118776049-204x255.jpg","width":204,"height":255},"title":"Statistics: 1,001 Practice Problems For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8947\">The Experts at Dummies</b> are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.</p>","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118776049&quot;]}]\" id=\"du-slot-6217bb399bfce\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118776049&quot;]}]\" id=\"du-slot-6217bb399c92a\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-09-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":147282},{"headers":{"creationTime":"2016-03-26T08:26:10+00:00","modifiedTime":"2021-10-21T19:37:50+00:00","timestamp":"2022-02-24T17:07:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Use the Z-Table","strippedTitle":"how to use the z-table","slug":"how-to-use-the-z-table","canonicalUrl":"","seo":{"metaDescription":"You can use the Z-score table to find a full set of \"less-than\" probabilities for a wide range of z-values using the z-score formula.","noIndex":0,"noFollow":0},"content":"You can use the Z-score table to find a full set of \"less-than\" probabilities for a wide range of z-values using the z-score formula. Below you will find both the positive z-score and negative z-score table. In figuring out statistics problems, make sure you understand how to use the <i>Z-</i>table to find the probabilities you want.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451654.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"933\" />\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451655.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"912\" />\r\n<h2 id=\"tab1\" >Z Score Table Sample Problems</h2>\r\nUse these sample z-score math problems to help you learn the z-score formula.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (<em>Z</em> ≤ 1.5) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.9332</p>\r\n<p class=\"child-para\">To find the answer using the <i>Z-</i>table, find where the row for 1.5 intersects with the column for 0.00; this value is 0.9332. The <i>Z-</i>table shows only \"less than\" probabilities so it gives you exactly what you need for this question. <i>Note:</i> No probability is exactly at one single point, so:</p>\r\n<em>P </em>(<em>Z</em> ≤ 1.5) = <em>P</em> (<em>Z</em> < 1.5)</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (<em>Z</em> ≥ 1.5) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.0668</p>\r\n<p class=\"child-para\">Use the <i>Z-</i>table to find where the row for 1.5 intersects with the column for 0.00, which is 0.9332. Because the <i>Z-</i>table gives you only \"less than\" probabilities, subtract <i>P</i>(<i>Z</i> < 1.5) from 1 (remember that the total probability for the normal distribution is 1.00, or 100%):</p>\r\n<em>P</em> (<em>Z</em> ≥ 1.5) = 1 – <em>P</em> (<em>Z</em> < 1.5)\r\n\r\n= 1 – 0.9332 = 0.0668</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (–0.5 ≤ <em>Z</em> ≤ 1.0) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.5328</p>\r\n<p class=\"child-para\">To find the probability that <i>Z</i> is between two values, use the <i>Z-</i>table to find the probabilities corresponding to each <i>z-</i>value, and then find the difference between the probabilities.</p>\r\n<p class=\"child-para\">Here, you want the probability that <i>Z</i> is between –0.5 and 1.0. First, use the <i>Z-</i>table to find the value where the row for –0.5 intersects with the column for 0.00, which is 0.3085. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413.</p>\r\n<p class=\"child-para\">Because the <i>Z-</i>table gives you only \"less than\" probabilities, find the difference between the probability less than 1.0 and the probability less than –0.5:</p>\r\n<em>P</em> (–0.5 ≤ <em>Z</em> ≤ 1.0) = <em>P</em> (<em>Z</em> ≤ 1.0) – <em>P</em> (<em>Z </em>≤ –0.50)\r\n\r\n= 0.8413 – 0.3085 = 0.5328</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (–1.0 ≤ <em>Z</em> ≤ 1.0) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.6826</p>\r\n<p class=\"child-para\">To find the probability that <i>Z</i> is between two values, use the <i>Z-</i>table to find the probabilities corresponding to each <i>z-</i>value, and then find the difference between the probabilities.</p>\r\n<p class=\"child-para\">Here, you want the probability that <i>Z</i> is between –1.0 and 1.0. First, use the <i>Z-</i>table to find the value where the row for –1.0 intersects with 0.00, which is 0.1587. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413.</p>\r\n<p class=\"child-para\">Because the <i>Z-</i>table gives you only \"less than\" probabilities, find the difference between probability less than 1.0 and the probability less than –1.0:</p>\r\n<em>P</em> (–1.0 ≤ <em>Z</em> ≤ 1.0) = <em>P</em> (<em>Z</em> ≤ 1.0) – <em>P</em> (<em>Z </em>≤ –1.0)\r\n\r\n= 0.8413 – 0.1587 = 0.6826</li>\r\n</ol>","description":"You can use the Z-score table to find a full set of \"less-than\" probabilities for a wide range of z-values using the z-score formula. Below you will find both the positive z-score and negative z-score table. In figuring out statistics problems, make sure you understand how to use the <i>Z-</i>table to find the probabilities you want.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451654.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"933\" />\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451655.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"912\" />\r\n<h2 id=\"tab1\" >Z Score Table Sample Problems</h2>\r\nUse these sample z-score math problems to help you learn the z-score formula.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (<em>Z</em> ≤ 1.5) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.9332</p>\r\n<p class=\"child-para\">To find the answer using the <i>Z-</i>table, find where the row for 1.5 intersects with the column for 0.00; this value is 0.9332. The <i>Z-</i>table shows only \"less than\" probabilities so it gives you exactly what you need for this question. <i>Note:</i> No probability is exactly at one single point, so:</p>\r\n<em>P </em>(<em>Z</em> ≤ 1.5) = <em>P</em> (<em>Z</em> < 1.5)</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (<em>Z</em> ≥ 1.5) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.0668</p>\r\n<p class=\"child-para\">Use the <i>Z-</i>table to find where the row for 1.5 intersects with the column for 0.00, which is 0.9332. Because the <i>Z-</i>table gives you only \"less than\" probabilities, subtract <i>P</i>(<i>Z</i> < 1.5) from 1 (remember that the total probability for the normal distribution is 1.00, or 100%):</p>\r\n<em>P</em> (<em>Z</em> ≥ 1.5) = 1 – <em>P</em> (<em>Z</em> < 1.5)\r\n\r\n= 1 – 0.9332 = 0.0668</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (–0.5 ≤ <em>Z</em> ≤ 1.0) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.5328</p>\r\n<p class=\"child-para\">To find the probability that <i>Z</i> is between two values, use the <i>Z-</i>table to find the probabilities corresponding to each <i>z-</i>value, and then find the difference between the probabilities.</p>\r\n<p class=\"child-para\">Here, you want the probability that <i>Z</i> is between –0.5 and 1.0. First, use the <i>Z-</i>table to find the value where the row for –0.5 intersects with the column for 0.00, which is 0.3085. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413.</p>\r\n<p class=\"child-para\">Because the <i>Z-</i>table gives you only \"less than\" probabilities, find the difference between the probability less than 1.0 and the probability less than –0.5:</p>\r\n<em>P</em> (–0.5 ≤ <em>Z</em> ≤ 1.0) = <em>P</em> (<em>Z</em> ≤ 1.0) – <em>P</em> (<em>Z </em>≤ –0.50)\r\n\r\n= 0.8413 – 0.3085 = 0.5328</li>\r\n \t<li>\r\n<p class=\"first-para\">What is <em>P</em> (–1.0 ≤ <em>Z</em> ≤ 1.0) ?</p>\r\n<p class=\"child-para\"><b>Answer:</b> 0.6826</p>\r\n<p class=\"child-para\">To find the probability that <i>Z</i> is between two values, use the <i>Z-</i>table to find the probabilities corresponding to each <i>z-</i>value, and then find the difference between the probabilities.</p>\r\n<p class=\"child-para\">Here, you want the probability that <i>Z</i> is between –1.0 and 1.0. First, use the <i>Z-</i>table to find the value where the row for –1.0 intersects with 0.00, which is 0.1587. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413.</p>\r\n<p class=\"child-para\">Because the <i>Z-</i>table gives you only \"less than\" probabilities, find the difference between probability less than 1.0 and the probability less than –1.0:</p>\r\n<em>P</em> (–1.0 ≤ <em>Z</em> ≤ 1.0) = <em>P</em> (<em>Z</em> ≤ 1.0) – <em>P</em> (<em>Z </em>≤ –1.0)\r\n\r\n= 0.8413 – 0.1587 = 0.6826</li>\r\n</ol>","blurb":"","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Z Score Table Sample Problems","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":207668,"title":"Statistics: 1001 Practice Problems For Dummies Cheat Sheet","slug":"1001-statistics-practice-problems-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207668"}},{"articleId":151951,"title":"Checking Out Statistical 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Dummies Cheat 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1,001 Practice Problems For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8947\">The Experts at Dummies</b> are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.</p>","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = 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Statistics Articles

Ace your stats class, analyze data for work, or play the odds at the slot machines. Everything you need is in here.

Articles From Statistics

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218 results
Statistics How to Calculate a Confidence Interval for a Population Mean When You Know Its Standard Deviation

Article / Updated 03-15-2022

If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is numerical, most people want to estimate the mean (average) value for the population. You estimate the population mean, μ, by using a sample mean, x̄, plus or minus a margin of error. The result is called a confidence interval for the population mean, μ. When the population standard deviation is known, the formula for a confidence interval (CI) for a population mean is x̄ ± z* σ/√n, where x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and z* represents the appropriate z*-value from the standard normal distribution for your desired confidence level. z*-values for Various Confidence Levels Confidence Level z*-value 80% 1.28 90% 1.645 (by convention) 95% 1.96 98% 2.33 99% 2.58 The above table shows values of z* for the given confidence levels. Note that these values are taken from the standard normal (Z-) distribution. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. Hence this chart can be expanded to other confidence percentages as well. The chart shows only the confidence percentages most commonly used. In this case, the data either have to come from a normal distribution, or if not, then n has to be large enough (at least 30 or so) in order for the Central Limit Theorem to be applied, allowing you to use z*-values in the formula. To calculate a CI for the population mean (average), under these conditions, do the following: Determine the confidence level and find the appropriate z*-value. Refer to the above table. Find the sample mean (x̄) for the sample size (n). Note: The population standard deviation is assumed to be a known value, σ. Multiply z* times σ and divide that by the square root of n. This calculation gives you the margin of error. Take x̄ plus or minus the margin of error to obtain the CI. The lower end of the CI is x̄ minus the margin of error, whereas the upper end of the CI is x̄ plus the margin of error. For example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond. Because you want a 95 percent confidence interval, your z*-value is 1.96. Suppose you take a random sample of 100 fingerlings and determine that the average length is 7.5 inches; assume the population standard deviation is 2.3 inches. This means x̄ = 7.5, σ = 2.3, and n = 100. Multiply 1.96 times 2.3 divided by the square root of 100 (which is 10). The margin of error is, therefore, ± 1.96(2.3/10) = 1.96*0.23 = 0.45 inches. Your 95 percent confidence interval for the mean length of walleye fingerlings in this fish hatchery pond is 7.5 inches ± 0.45 inches. (The lower end of the interval is 7.5 – 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95 inches.) After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.” In this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 7.05 and 7.95 inches, based on my sample data.” (Always be sure to include appropriate units.)

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Statistics Statistics Workbook For Dummies Cheat Sheet

Cheat Sheet / Updated 02-25-2022

This cheat sheet is for you to use as a quick resource for finding important basic statistical formulas such as mean, standard deviation, and Z-values; important and always useful probability definitions such as independence and rules such as the multiplication rule and the addition rule; and 10 quick ways to spot statistical mistakes either in your own work, or out there in the media as a consumer of statistical information.

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Statistics Statistics II For Dummies Cheat Sheet

Cheat Sheet / Updated 02-23-2022

Statistics II elaborates on Statistics I and moves into new territories, including multiple regression, analysis of variance (ANOVA), Chi-square tests, nonparametric procedures, and other key topics. Knowing which data analysis to use and why is important, as is familiarity with computer output if you want your numbers to give you dependable results.

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Statistics SPSS For Dummies Cheat Sheet

Cheat Sheet / Updated 02-14-2022

SPSS is an application that performs statistical analysis on data. Entering and manipulating information in the application can be done by using SPSS’s proprietary language, which is known as the Syntax command language, or more commonly, as Syntax. The language is quite like other programming languages, and it allows you to define variables (or use predefined ones), and to use them within statements, or to evaluate them with relational or logical operators. Good programmers always know to make their code accessible through the use of comments. Syntax can also be used in conjunction with Basic and Python.

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Statistics Statistics: 1001 Practice Problems For Dummies Cheat Sheet

Cheat Sheet / Updated 01-28-2022

There are many types of statistics problems, including the use of pie charts, bar graphs, means, standard deviation to correlation, regression, confidence intervals, and hypothesis tests. To be successful, you need to be able to make connections between statistical ideas and statistical formulas. Through practice, you see what type of technique is required for a problem and why, as well as how to set up the problem, work it out, and make proper conclusions. Most statistics problems you encounter likely involve terminology, symbols, and formulas. No worries! This Cheat Sheet gives you tips for success.

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Statistics How to Find Probabilities for a Sample Mean

Article / Updated 12-28-2021

In statistics, you can easily find probabilities for a sample mean if it has a normal distribution. Even if it doesn’t have a normal distribution, or the distribution is not known, you can find probabilities if the sample size, n, is large enough. The normal distribution is a very friendly distribution that has a table for finding probabilities and anything else you need. For example, you can find probabilities for by converting the to a z-value and finding probabilities using the Z-table (see below). The general conversion formula from Substituting the appropriate values of the mean and standard error of the conversion formula becomes: Don’t forget to divide by the square root of n in the denominator of z. Always divide by the square root of n when the question refers to the average of the x-values. For example, suppose X is the time it takes a randomly chosen clerical worker in an office to type and send a standard letter of recommendation. Suppose X has a normal distribution, and assume the mean is 10.5 minutes and the standard deviation 3 minutes. You take a random sample of 50 clerical workers and measure their times. What is the chance that their average time is less than 9.5 minutes? This question translates to finding As X has a normal distribution to start with, you know also has an exact (not approximate) normal distribution. Converting to z, you get: So you want P(Z < –2.36). Using the above Z-table, you find that P(Z < –2.36)=0.0091. So the probability that a random sample of 50 clerical workers average less than 9.5 minutes to complete this task is 0.91% (very small). How do you find probabilities for if X is not normal, or unknown? As a result of the Central Limit Theorem (CLT), the distribution of X can be non-normal or even unknown and as long as n is large enough, you can still find approximate probabilities for using the standard normal (Z-)distribution and the process described above. That is, convert to a z-value and find approximate probabilities using the Z-table. When you use the CLT to find a probability for (that is, when the distribution of X is not normal or is unknown), be sure to say that your answer is an approximation. You also want to say the approximate answer should be close because you’ve got a large enough n to use the CLT. (If n is not large enough for the CLT, you can use the t-distribution in many cases.)

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Statistics Using Linear Regression to Predict an Outcome

Article / Updated 12-21-2021

Statistical researchers often use a linear relationship to predict the (average) numerical value of Y for a given value of X using a straight line (called the regression line). If you know the slope and the y-intercept of that regression line, then you can plug in a value for X and predict the average value for Y. In other words, you predict (the average) Y from X. If you establish at least a moderate correlation between X and Y through both a correlation coefficient and a scatterplot, then you know they have some type of linear relationship. Never do a regression analysis unless you have already found at least a moderately strong correlation between the two variables. (A good rule of thumb is it should be at or beyond either positive or negative 0.50.) If the data don’t resemble a line to begin with, you shouldn’t try to use a line to fit the data and make predictions (but people still try). Before moving forward to find the equation for your regression line, you have to identify which of your two variables is X and which is Y. When doing correlations, the choice of which variable is X and which is Y doesn’t matter, as long as you’re consistent for all the data. But when fitting lines and making predictions, the choice of X and Y does make a difference. So how do you determine which variable is which? In general, Y is the variable that you want to predict, and X is the variable you are using to make that prediction. For example, say you are using the number of times a population of crickets chirp to predict the temperature. In this case you would make the variable Y the temperature, and the variable X the number of chirps. Hence Y can be predicted by X using the equation of a line if a strong enough linear relationship exists. Statisticians call the X-variable (cricket chirps in this example) the explanatory variable, because if X changes, the slope tells you (or explains) how much Y is expected to change in response. Therefore, the Y variable is called the response variable. Other names for X and Y include the independent and dependent variables, respectively. In the case of two numerical variables, you can come up with a line that enables you to predict Y from X, if (and only if) the following two conditions are met: The scatterplot must form a linear pattern. The correlation, r, is moderate to strong (typically beyond 0.50 or –0.50). Some researchers actually don’t check these conditions before making predictions. Their claims are not valid unless the two conditions are met. But suppose the correlation is high; do you still need to look at the scatterplot? Yes. In some situations the data have a somewhat curved shape, yet the correlation is still strong; in these cases making predictions using a straight line is still invalid. Predictions in these cases need to be made based on other methods that use a curve instead.

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Statistics How to Interpret the Shape of Statistical Data in a Histogram

Article / Updated 12-21-2021

One of the features that a histogram can show you is the shape of the statistical data — in other words, the manner in which the data fall into groups. For example, all the data may be exactly the same, in which case the histogram is just one tall bar; or the data might have an equal number in each group, in which case the shape is flat. Some data sets have a distinct shape. Here are three shapes that stand out: Symmetric. A histogram is symmetric if you cut it down the middle and the left-hand and right-hand sides resemble mirror images of each other: Skewed right. A skewed right histogram looks like a lopsided mound, with a tail going off to the right: Skewed left. If a histogram is skewed left, it looks like a lopsided mound with a tail going off to the left: Following, are some particulars about classifying the shape of a data set: Don't expect symmetric data to have an exact and perfect shape. Data hardly ever fall into perfect patterns, so you have to decide whether the data shape is close enough to be called symmetric. If the differences aren't significant enough, you can classify it as symmetric or roughly symmetric. Otherwise, you classify the data as non-symmetric. Don't assume that data are skewed if the shape is non-symmetric. Data sets come in all shapes and sizes, and many of them don't have a distinct shape at all. Skewness is mentioned here because it's one of the more common non-symmetric shapes, and it's one of the shapes included in a standard introductory statistics course. If a data set does turn out to be skewed (or close to it), make sure to denote the direction of the skewness (left or right).

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Statistics How to Use the T-table to Solve Statistics Problems

Article / Updated 10-27-2021

The t-table (for the t-distribution) is different from the z-table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t-table, is a valuable statistics skill. Use the t-table as necessary to solve the following sample problems below. Sample questions For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results? Answer: df = 17 The study involving one population and a sample size of 18 has n – 1 = 18 – 1 = 17 degrees of freedom. For a study involving a paired design with a total of 44 observations, with the results assuming a t-distribution, what row of the table will you use to find the probability affiliated with the study results? Answer: df = 21 A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: n – 1 = 22 – 1 = 21. A t-value of 2.35, from a t-distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table? Answer: 0.025 and 0.01 Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01. Hence, the upper-tail probability for a t-value of 2.35 must lie between 0.025 and 0.01.

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Statistics How to Use the Z-Table

Article / Updated 10-21-2021

You can use the Z-score table to find a full set of "less-than" probabilities for a wide range of z-values using the z-score formula. Below you will find both the positive z-score and negative z-score table. In figuring out statistics problems, make sure you understand how to use the Z-table to find the probabilities you want. Z Score Table Sample Problems Use these sample z-score math problems to help you learn the z-score formula. What is P (Z ≤ 1.5) ? Answer: 0.9332 To find the answer using the Z-table, find where the row for 1.5 intersects with the column for 0.00; this value is 0.9332. The Z-table shows only "less than" probabilities so it gives you exactly what you need for this question. Note: No probability is exactly at one single point, so: P (Z ≤ 1.5) = P (Z < 1.5) What is P (Z ≥ 1.5) ? Answer: 0.0668 Use the Z-table to find where the row for 1.5 intersects with the column for 0.00, which is 0.9332. Because the Z-table gives you only "less than" probabilities, subtract P(Z < 1.5) from 1 (remember that the total probability for the normal distribution is 1.00, or 100%): P (Z ≥ 1.5) = 1 – P (Z < 1.5) = 1 – 0.9332 = 0.0668 What is P (–0.5 ≤ Z ≤ 1.0) ? Answer: 0.5328 To find the probability that Z is between two values, use the Z-table to find the probabilities corresponding to each z-value, and then find the difference between the probabilities. Here, you want the probability that Z is between –0.5 and 1.0. First, use the Z-table to find the value where the row for –0.5 intersects with the column for 0.00, which is 0.3085. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413. Because the Z-table gives you only "less than" probabilities, find the difference between the probability less than 1.0 and the probability less than –0.5: P (–0.5 ≤ Z ≤ 1.0) = P (Z ≤ 1.0) – P (Z ≤ –0.50) = 0.8413 – 0.3085 = 0.5328 What is P (–1.0 ≤ Z ≤ 1.0) ? Answer: 0.6826 To find the probability that Z is between two values, use the Z-table to find the probabilities corresponding to each z-value, and then find the difference between the probabilities. Here, you want the probability that Z is between –1.0 and 1.0. First, use the Z-table to find the value where the row for –1.0 intersects with 0.00, which is 0.1587. Then, find the value where the row for 1.0 intersects with the column for 0.00, which is 0.8413. Because the Z-table gives you only "less than" probabilities, find the difference between probability less than 1.0 and the probability less than –1.0: P (–1.0 ≤ Z ≤ 1.0) = P (Z ≤ 1.0) – P (Z ≤ –1.0) = 0.8413 – 0.1587 = 0.6826

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