Statistics Workbook For Dummies with Online Practice book cover

Statistics Workbook For Dummies with Online Practice

By: Deborah J. Rumsey Published: 04-23-2019

Practice your way to a higher statistics score 

The adage that "practice makes perfect" is never truer than with math problems. Statistics Workbook For Dummies with Online Practice provides succinct content reviews for every topic, with plenty of examples and practice problems for each concept, in the book and online.  

Every lesson begins with a concept review, followed by a few example problems and plenty of practice problems. There's a step-by-step solution for every problem, with tips and tricks to help with comprehension and retention. New for this edition, free online practice quizzes for each chapter provide extra opportunities to test your knowledge and understanding. 

  • Get FREE access to chapter quizzes in an online test bank
  • Work along with each chapter or use the test bank for final exam review
  • Discover which statistical measures are most meaningful

Scoring high in your Statistics class has never been easier! 

Articles From Statistics Workbook For Dummies with Online Practice

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6 results
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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Statistical T-Distribution — The “T-Table”

Article / Updated 07-08-2021

As a statistical tool, a t-table lists critical values for two-tailed tests. You then use these values to determine confidence values. The following t-table shows degrees of freedom for selected percentiles from the 90th to the 99th: Degrees of Freedom 90th Percentile (a = .10) 95th Percentile (a = .05) 97.5th Percentile (a = .025) 98th Percentile (a = .02) 99th Percentile (a = .01) 1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.333 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.989 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 60 1.296 1.671 2.000 2.390 2.660 Z-values 1.282 1.645 1.960 2.326 2.576

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10 Steps to a Better Math Grade with Statistics

Article / Updated 08-23-2019

Statistics and math are very different subjects, but you use a certain amount of mathematical tools to do statistical calculations. Sometimes you can understand the statistical idea but get bogged down in the formulas and calculations and end up getting the wrong answer. Avoid making the common math mistakes that can cost you points on homework and exams. Read on to increase your confidence with the math tools you need for statistics. Know your math symbols The most basic math symbols are +, –, ∙ (multiplication), and / (division); but have you ever seen the following sign? It means plus or minus and indicates a lower bound and an upper bound for your answer. Other commonly used math symbols involve the Greek letter “capital” sigma, which stands for summation. In math formulas, you often leave out the multiplication sign; for example, 2x means 2 × x. If you come across a math symbol that you don’t understand, ask for help. You can never get comfortable with that symbol until you know exactly what you use it for and why. You may be surprised that after you lift the mystique, math symbols aren’t really as hard as they seem to be. They simply provide you with a shorthand way of expressing something that you need to do. Uproot roots and powers Remember that squaring a number means multiplying it by itself two times, not multiplying by two. And taking the square root means finding the number whose square gives you your result; it doesn’t mean dividing the number by 2. Using math notation, x2 means square the value (so for x = 3, you have 32 = 9); and means take the square root (for x = 9, this means the square root of 9 is 3). You can’t take the square root of a negative number, because you can’t square anything to get a negative number back. So, anything under a square root sign has to be a nonnegative quantity (that is, it has to be greater than or equal to 0). These ideas may seem straightforward, but like everything else, they can get complex very fast. If you need to find the square root of an entire expression, put everything under the square root sign in parentheses so your calculator knows to take the square root of the entire expression, not just part of it. Statistics often deal with percentages — numbers that in decimal form are between 0 and 1. You need to know that numbers between 0 and 1 often act differently than large numbers do. For example, numbers larger than 1 get smaller when you take the square root, but numbers between 0 and 1 get larger when you take the square root. For example, the square root of 4 is 2 (which is smaller than 4), but the square root of 1/4 is 1/2 (which is larger). And when you take powers, the opposite happens. Numbers larger than 1 that you square get larger; for example, 3 squared is 9 (which is larger than 3). Numbers between 0 and 1 that you square get smaller; for example, 1/3 squared is 1/9 (which is smaller). Treat fractions with extra care Every fraction contains a top (numerator) and a bottom (denominator). For example, in the fraction 3/7, 3 is the numerator and 7 is the denominator. But what does a fraction really mean? It means division. The fraction 3/7 means take the number 3 and divide it by 7. A common mistake is to read fractions upside down in terms of what you divide by what. The fraction 1/10 means 1 divided by 10, not 10 divided by 1. If you can hold on to an example like this that you know is correct, it can stop you from making this mistake again later when the formulas get more complicated. Obey the order of operations To follow the order of math operations, remember “PEMDAS”: Parentheses, Exponents (powers of a number), Multiplication and Division (interchangeable), and Addition and Subtraction. Failing to follow the order of operations can result in a big mistake. To remember the letters in PEMDAS for the order of operations, try this: “Please Excuse My Dear Aunt Sally.” Suppose, for example, that you need to calculate the following: . First, calculate what’s in parentheses. You can either type it just as it looks into your calculator or do separately and then plug it in as –6 + 5 + 0.5 – 8 + 10. You should get 3/2 or 1.5. Next, divide by 5 to get or which equals 0.3. Avoid rounding errors Rounding errors can seem small, but they can really add up — literally. Many statistical formulas contain several different types of operations that you can do either all at once, using parentheses properly, or separately, as many students elect to do. Doing the operations separately and writing them down with each step is fine, as long as you don’t round off numbers too much at each stage. For example, suppose that you have to calculate You want to write down each step separately rather than calculate the equation all at once. Suppose that you round off to one digit after the decimal point on each calculation. First, you take the square root of 200 (which rounds to 14.1), and then you take 5.2 divided by 14.1, which is 0.369; you round this to 0.4. Next, you take 1.96 times 0.4 to get 0.784, which you round to 0.8. The actual answer, if you do all the calculations at once with no rounding, is 0.72068, which safely rounds to 0.72. What a huge difference! What would this difference cost you on an exam? At worst, your professor would reject your answer outright, because it strays too far from the correct one. At best, he would take off some points, because your answer isn’t precise enough. Instead of rounding to one digit after the decimal point, suppose that you round to two digits after the decimal point each time. This still gives you the incorrect answer of 0.73. You’ve come closer to the correct answer, but you’re still technically off, and points may be lost. Statistics is a quantitative field, and teachers expect precise answers. What should you do if you want to do calculation steps separately? Keep at least two significant digits after the decimal point during each step, and at the very end, round off to two digits after the decimal point. Don’t round off too much too soon, especially in formulas where many calculations are involved. Your best bet is to use parentheses and use all the decimal places in your calculator. Otherwise, keep at least two significant digits after the decimal point until the very end. Get comfortable with statistical formulas Don’t let basic math and statistical formulas get in your way. Think of them as mathematical shorthand. Suppose that you want to find the average of some numbers. You sum the numbers and divide by n (the size of your data set). If you have only a few numbers, writing out all the instructions is easy, but what if you have 1,000 numbers? Mathematicians have come up with formulas as a way of saying quickly what they want you to do, and the formulas work no matter the size of your data set. The key is getting familiar with formulas and practicing them. Stay calm when formulas get tough Suppose that you encounter a formula that’s a little complicated? How do you remain calm and cool? By starting with small formulas, learning the ropes, and then applying the same rules to the bigger formulas. That’s why you need to understand how the “easy” formulas work and be able use them as formulas; you shouldn’t just figure them out in your head, because you don’t need the formula in that case. The easy formulas build your skills for when things get tougher. Feel fine about functions Many times in math and statistics, different variables are related to each other. For example, to get the area of a square, you take the length of one of the sides and multiply it by itself. In mathematical notation, the formula looks like this: A = s2. This formula really represents a function. It says that the area of the square depends on the length of its sides. It also means that all you have to know is the length of one of the sides to get the area of the square. In math jargon, you say that the area of a square is a function of the length of its sides. Function just means “depends on.” Suppose that you have a line with the equation y = 2x + 3. The equation conveys that x and y are related, and you know how they’re related. If you take any value of x, multiply it by two and add three, you get the corresponding value for y. Suppose that you want to find y when x is –2. To find y for a given x, plug in that number for x and simplify it. In this case, you have y = (2)(–2) + 3. This simplifies to y = –4 + 3 = –1. You can also take this same function and plug in any value for y to get its corresponding value for x. For example, suppose that you have y = 2x + 3, and you’re given y = 4 and asked to solve for x. Plugging in 4 for y, you get 4 = 2x + 3. The only difference is, you normally see the unknown on one side of the equation and the number part on the other. In this case, you see it the other way around. Don’t worry about how it looks; remember what you need to do. You need to get x alone on one side, so use your algebra skills to make that happen. In this case, subtract 3 from each side to get 4 – 3 = 2x, or 1 = 2x. Now divide each side by 2 to get 0.5 = x. You have your answer. You can use a formula in many different ways. If you have all the other pieces of information, you can always solve for the remaining part, no matter where it sits in the equation. Just keep your cool and use your algebra skills to get it done. Certain commonly used functions have names. For example, an equation that has one x and one y is called a linear function, because when you graph it, you get a straight line. Statistics uses lines often, and you need to know the two major parts of a line: the slope and the y-intercept. If the equation of the line is in the form y = mx + b, m is the slope (the change in y over change in x), and b is the y-intercept (the place where the line crosses the y-axis). Suppose that you have a line with the equation y = –2x – 10. In this case, the y-intercept is –10, and the slope is–2. The slope is the number in front of the x in the equation y = mx + b. If you rewrite the previous equation as y = –10 – 2x, the slope is still–2, because –2 is the number that goes with the x. And –10 is still the y-intercept. Know when your answer is wrong You should always look at your answer to see whether it makes sense, in terms of what kind of number you expect to get. Can the number you’re calculating be negative? Can it be a large number or a fraction? Does this number make sense? All these questions can help you catch mistakes on exams and homework before your instructor does. In any fraction, if the numerator (top) is larger than the denominator (bottom), the result is greater than 1. If the numerator (top) is smaller than the denominator (bottom), the result is less than 1. And if the numerator (top) and denominator (bottom) are exactly equal, the result is exactly 1. Show your work You see the instructions “Show your work!” on your exams, and your instructor harps and harps on it, but still, you don’t quite believe that showing your work can be that important. Take it from a seasoned professor, it is. Here’s why: Showing your work helps the person grading your paper see exactly what you tried to do, even if the answer is wrong. This works to your advantage if your work was on the right track. The only way to get partial credit for your work is to show that you had the right idea, and you must do this in writing. Not showing your work makes it hard on the person grading your paper and can cost you points in an indirect way. Grading is a tremendous amount of work. Here's how the “grading effect” on your teacher ultimately affects you. Your teacher has a big pile of papers to grade and only so much time (and energy) to grade them all. A paper with a big messed up area of scribbling, erasing, crossing out, and smudging rears its ugly head. It has no clear tracks as to what’s happening or what the student was thinking. Numbers are pushed around every which way with no clear-cut steps or pattern to follow. How much time can (will) teachers spend trying to figure out this problem? Teachers have to move on at some point; we can only do so much to try to figure out what students were thinking during an exam. Here’s another typical situation. A teacher looks at two papers, both with the right answer. One person wrote out all the steps, labeled everything, and circled the answer, but the other person simply wrote down the answer. Do you give both people full credit? Some teachers do, but many don’t. Why? Because the instructor isn’t sure whether you did the work yourself. Teachers don’t typically advocate doing math “in your head.” We want you to show your work, because someday, even for you, the formulas will get so complicated that you can’t rely on your mind alone to solve them. Plus, you do need to show evidence that the work is your own. What if you write down the answer, and the answer is wrong, but only a tiny little mistake led to the error? With no tracks to show what you were thinking, the teacher can’t give you partial credit, and the littlest of mistakes can cost you big time. Showing your work establishes good habits that last a lifetime. Each time you work a problem, whether you’re working in class, on homework, to study for an exam, or on an exam, if you follow the same procedure each time, good things will happen. Here’s a great way to work a math-related statistics problem: Write out the formula you plan to use, in its entirety (letters included). Clearly write down what number you plug in for each variable in the formulas; for example, x = 2 and y = 6. Work out the calculations in a step-by-step manner, showing each step clearly. Circle your final answer clearly. The biggest argument students give for not showing their work is that it takes too much time. Yes, showing your work takes a little more time in the short run. But showing your work actually saves time in the long run, because it helps you organize your ideas clearly the first time, cuts down on the errors you make the first time around, and lessens your need to have to go back and double check everything at the end. If you do have time to double-check your answers, you have an easier time seeing what you did and finding a potential mistake. Showing your work is a win-win situation. Try showing your work a little more clearly, and see how it impacts your grades.

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Statistics and Histograms

Article / Updated 08-23-2019

A histogram is a bar graph made for quantitative data. Because the data are numerical, you divide it into groups without leaving any gaps in between (so the bars are connected). The Y-axis shows either frequencies (counts) or relative frequencies (percents) of the data that fall into each group. How to create a histogram To make a histogram, you first divide your data into a reasonable number of groups of equal length. Tally up the number of values in the data set that fall into each group (in other words, make a frequency table). If a data point falls on the boundary, make a decision as to which group to put it into, making sure you stay consistent (always put it in the higher of the two, or always put it in the lower of the two). Make a bar graph, using the groups and their frequencies — a frequency histogram. If you divide the frequencies by the total sample size, you get the percentage that falls into each group. A table that shows the groups and their percents is a relative frequency table. The corresponding histogram is a relative frequency histogram. You can use Minitab or a different software package to make histograms, or you can make your histograms by hand. Either way, your choice of interval widths (called bins by computer packages) may be different from the ones seen in the figures, which is fine, as long as yours look similar. And they will, as long as you don’t use an unusually low or high number of bars and your bars are of equal width. You may also choose different start/end points for each interval, and that’s fine as well. Just be sure to label everything clearly so your instructor can see what you’re trying to do. And be consistent about values that end up right on a border; always put them in the lower grouping, or always put them in the upper grouping. If you do have a choice, however, make your histograms by using a computer package like Minitab. It makes your task much easier. See the following for an example of making the two types of histograms. Test scores for a class of 30 students are shown in the following table. Scores Frequency 70–79 8 80–89 16 90–99 6 Frequency histograms and relative frequency histograms look the same; they’re just done using different scales on the Y-axis. The frequency histogram for the scores data is shown in the following figure. You find the relative frequencies by taking each frequency and dividing by 30 (the total sample size). The relative frequencies for these three groups are 8 / 30 = 0.27 or 27%; 16 / 30 = 0.53 or 53%; and 6 / 30 = 0.20 or 20%, respectively. A histogram based on relative frequencies looks the same as the histogram (of the same data). The only difference is the label on the Y-axis. Making sense of histograms A histogram gives you general information about three main features of your quantitative (numerical) data: the shape, center, and spread. The shape of a histogram is shown by its general pattern. Many patterns are possible, and some are common, including the following: Bell-shaped: Looks like a bell — a big lump in the middle and tails that go down on each side at about the same rate. (Figure a) Right skewed: A big part of the data is set off to the left, with a few larger observations trailing off to the right. (Figure b) Left skewed: A big part of the data is set off to the right, with a few smaller observations trailing off to the left. (Figure c) Uniform: All the bars have a similar height. (Figure d) Bimodal: Two peaks, or (Figure e) U-shaped: Bimodal with the two peaks at the low and high ends, with less data in the middle. (See Figure 4-1 (Figure f) Symmetric: Looks the same on each side when you split it down the middle; bell-shaped, uniform, and U-shaped histograms are all examples of symmetric data. (Figures a, d, and f) You can view the center of a histogram in two ways. One is the point on the x-axis where the graph balances, taking the actual values of the data into account. This point is called the average, and you can find it by locating the balancing point (imagine the data are on a teeter-totter). The other way to view center is locating the line in the histogram where 50 percent of the data lies on either side. The line is called the median, and it represents the physical middle of the data set. Imagine cutting the histogram in half so that half of the area lies on either side of the line. Spread refers to the distance between the data, either relative to each other or relative to some central point. One crude way to measure spread is to find the range, or the distance between the largest value and the smallest value. Another way is to look for the average distance from the middle, otherwise known as the standard deviation. The standard deviation is hard to come up with by just looking at a histogram, but you can get a rough idea if you take the range divided by 6. If the heights of the bars close to the middle seem very tall, that means most of the values are close to the mean, indicating a small standard deviation. If the bars appear short, you may have a larger standard deviation. You can do actual summary statistics to calculate the quantitative data, but a histogram can give you a general direction for finding these milestones. And like pie charts and bar graphs, not all histograms are fair, complete, and accurate. You have to know what to look for to evaluate them. How to straighten out skewed data with histograms You need to make special considerations for skewed data sets, in terms of which statistics are the most appropriate to use and when. You should also be aware of how using the wrong statistics can provide misleading answers. You can relate the mean and median to learn about the shape of your data. Having the mean and median close to being equal will create a shape that is roughly symmetric The mean is affected by outliers in the data, but the median is not. If the mean and median are close to each other, the data aren’t skewed and likely don’t contain outliers on one side or the other. That means that the data look about the same on each side of the middle, which is the definition of symmetric data (see a, d, or f in the preceding figure). The fact that the mean and median being close tells you the data are roughly symmetric can be used in a different type of test question. Suppose that someone asks you whether the data are symmetric, and you don’t have a histogram, but you do have the mean and median. Compare the two values of the mean and median, and if they are close, the data are symmetric. If they aren’t, the data are not symmetric. How to spot a misleading histogram Readers can be misled by a histogram in ways that aren’t possible with a bar graph. Remember that a histogram deals with numerical data, not categorical data, which means you have to determine how you want the numerical data broken down into groups to display on the horizontal axis. And how you determine those groupings can make the graph look very different. Watch for histograms that use scale to mislead readers. As with bar graphs, you can exaggerate differences by using a smaller scale on the vertical axis of a histogram, and you can downplay differences by using a larger scale.

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What is Categorical Data and How is It Summarized?

Article / Updated 08-23-2019

What is categorical data? Basically, it is data in which individuals are placed into groups or categories — for example gender, region, or type of movie. Summarizing categorical data involves boiling down all the information into just a few numbers that tell its basic story. Because categorical data involves pieces of data that belong in categories, you have to look at how many individuals fall into each group and summarize the numbers appropriately. Here, you learn about making, interpreting, and evaluating frequency and relative frequency tables for categorical data. Counting on the frequency One way to summarize categorical data is to simply count, or tally up, the number of individuals that fall into each category. The number of individuals in any given category is called the frequency (or count) for that category. If you list all the possible categories along with the frequency for each, you create a frequency table. The total of all the frequencies should equal the size of the sample (because you place each individual in one category). See the following for an example of summarizing data by using a frequency table. Suppose that you take a sample of 10 people and ask them all whether they own a cellphone. Each person falls into one of two categories: yes or no. The data is shown in the following table. Person # Cellphone Person # Cellphone 1 Y 6 Y 2 N 7 Y 3 Y 8 Y 4 N 9 N 5 Y 10 Y Data summaries boil down the data quickly and clearly. A data summary allows you to see patterns in the data, which aren’t clear if you look only at the original data. Own a Cellphone? Frequency Y 7 N 3 Total 10 Relating categorical data with percentages Another way to summarize categorical data is to show the percentage of individuals who fall into each category, thereby creating a relative frequency. The relative frequency of a given category is the frequency (number of individuals in that category) divided by the total sample size, multiplied by 100 to get the percentage. For example, if you survey 50 people and 10 are in favor of a certain issue, the relative frequency of the “in-favor” category is 10 / 50 = 0.20 × 100, which gives you 20 percent. If you list all the possible categories along with their relative frequencies, you create a relative frequency table. The total of all the relative frequencies should equal 100 percent (subject to possible round-off error). See the following for an example of summarizing data by using a relative frequency table. Using the cellphone data from the following table, make a relative frequency table and interpret the results. Person # Cellphone Person # Cellphone 1 Y 6 Y 2 N 7 Y 3 Y 8 Y 4 N 9 N 5 Y 10 Y The following table shows a relative frequency table for the cellphone data. Seventy percent of the people sampled reported owning cellphones, and 30 percent admitted to being technologically behind the times. Own a Cellphone? Relative Frequency Y 70% N 30% You get the 70 percent by taking 7 / 10 × 100, and you calculate the 30 percent by taking 3 / 10 × 100. Watch for total sample sizes when given a relative frequency table. Don’t be misled by percentages alone, thinking they’re always based on large sample sizes, because many are not. Interpreting counts and percents with caution Not all summaries of categorical data are fair and accurate. Knowing what to look for can help you keep your eyes open for misleading and incomplete information. Instructors often ask you to “interpret the results.” In this case, your instructor wants you to use the statistics available to talk about how they relate to the given situation. In other words, what do the results mean to the person who collects the data? See the following for an example of critiquing a data summary. You watch a commercial where the manufacturer of a new cold medicine (“Nocold”) compares it to the leading brand. The results are shown in the following table. How Nocold Compares Percentage Much better 47% At least as good 18% The table about “Nocold” does “Nogood.” This table is an incomplete relative frequency table. The remaining category is “not as good” for the Nocold brand, and the advertiser doesn’t show it. But you can do the math and see that 100% – (47% + 18%) = 35% of the people say that the leading brand is better. If you put the two groups together, 65% of the patients say that Nocold is at least as good as the leading brand, and almost half of the patients say Nocold is much better. What’s missing? The remaining percentage (to keep all possible results in perspective). But more importantly, the total sample size is missing. You don’t know whether the surveyors sampled 10 people, 100 people, or 1,000 people. This means that the precision of the results is unknown. (Precision means how consistent the results will be from sample to sample; it’s related to sample size.) With relative frequency tables, don’t forget to check whether all categories sum to 1 or 100 percent (subject to round-off error), and remember to look for some indicator as to total sample size. If you're interested in knowing how to represent categorical data in a graph, see "How to Summarize and Graph Categorical Data."

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Understanding the Statistical Properties of the Normal Distribution

Article / Updated 08-23-2019

Statisticians call a distribution with a bell-shaped curve a normal distribution. You may have heard of a bell curve. A bell curve describes data from a variable that has an infinite (or very large) number of possible values distributed among the population in a bell shape. This basically means a big group of individuals gravitate near the middle, with fewer and fewer individuals trailing off as you move away from the middle in either direction. You can see a normal distribution’s shape in this figure. Every normal distribution has certain properties. You can use these properties to determine the relative standing of any particular result on the distribution. When you understand the properties of the normal distribution, you'll find it easier to interpret statistical data. A continuous random variable X has a normal distribution if its values fall into a smooth (continuous) curve with a bell-shaped pattern. Each normal distribution has its own mean, denoted by the Greek letter μ and its own standard deviation, denoted by the Greek letter σ. But no matter what their means and standard deviations are, all normal distributions have the same basic bell shape. The properties of any normal distribution (bell curve) are as follows: The shape is symmetric. The distribution has a mound in the middle, with tails going down to the left and right. The mean is directly in the middle of the distribution. (The mean of the population is designated by the Greek letter μ.) The mean and the median are the same value because of the symmetry. The standard deviation is the distance from the center to the saddle point (the place where the curve changes from an “upside-down-bowl” shape to a “right-side-up-bowl” shape. (The standard deviation of the population is designated by the Greek letter σ.) About 68 percent of the values lie within one standard deviation of the mean, about 95 percent lie within two standard deviations, and most of the values (99.7 percent or more) lie within three standard deviations by the empirical rule. Each normal distribution has a different mean and standard deviation that make it look a little different from the rest, yet they all have the same bell shape. Take a look at the following figure. Three normal distributions, with means and standard deviations of a) 90 and 30; b) 120 and 30; and c) 90 and 10, respectively. To compare and contrast the distributions shown in the figure, you first see they are all symmetric with the signature bell shape. Examples (a) and (b) have the same standard deviation, but their means are different; the mean in Example (b) is located 30 units to the right of the mean in Example (a) because its mean is 120 compared to 90. Examples (a) and (c) have the same mean (90), but Example (a) has more variability than Example (c) due to its higher standard deviation (30 compared to 10). Because of the increased variability, most of the values in Example (a) lie between 0 and 180 (approximately), while most of the values in Example (c) lie only between 60 and 120. Finally, Examples (b) and (c) have different means and different standard deviations entirely; Example (b) has a higher mean which shifts the graph to the right, and Example (c) has a smaller standard deviation; its data values are the most concentrated around the mean. Note that the mean and standard deviation are important in order to properly interpret numbers located on a particular normal distribution. For example, you can compare where the value 120 falls on each of the normal distributions in the above figure. In Example (a), the value 120 is one standard deviation above the mean (because the standard deviation is 30, you get 90 + 1[30] = 120). So on this first distribution, the value 120 is the upper value for the range where the middle 68% of the data are located, according to the Empirical Rule. In Example (b), the value 120 lies directly on the mean, where the values are most concentrated. In Example (c), the value 120 is way out on the rightmost fringe, 3 standard deviations above the mean (because the standard deviation this time is 10, you get 90 + 3[10] = 120). In Example (c), values beyond 120 are very unlikely to occur because they are beyond the range where the middle 99.7% of the values should be, according to the Empirical Rule.

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