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How Sample Size Affects the Margin of Error

In statistics, the two most important ideas regarding sample size and margin of error are, first, sample size and margin of error have an inverse relationship; and second, after a point, increasing the [more…]

Choosing a Confidence Level for a Population Sample

In statistics, every confidence interval (and every margin of error, for that matter) has a percentage associated with it, called a confidence level. This percentage represents how confident you are that [more…]

How to Determine the Minimum Size Needed for a Statistical Sample

The margin of error of a confidence interval (CI) is affected by size of the statistical sample; as the size increases, margin of error decreases. Looking at this the other way around, if you want a smaller [more…]

Creating a Confidence Interval for the Difference of Two Means with Known Standard Deviations

If you know the standard deviations for two population samples, then you can find a confidence interval (CI) for the difference between their means, or averages. The goal of many statistical surveys and [more…]

How to Create a Confidence Interval for the Difference of Two Means with Unknown Standard Deviations and/or Small Sample Sizes

You can find a confidence interval (CI) for the difference between the means, or averages, of two population samples, even if the population standard deviations are unknown and/or the sample sizes are [more…]

How to Estimate the Difference between Two Proportions

To estimate the difference between two population proportions with a confidence interval, you can use the Central Limit Theorem when the sample sizes are large enough [more…]

How to Find the Cutoff Point for Rejecting a Null Hypothesis

In statistics, if you want to draw conclusions about a null hypothesis H0 (reject or fail to reject) based on a p-value, you need to set a predetermined cutoff point where only those [more…]

How to Test a Hypothesis for One Population Mean

You can use a hypothesis test to examine or challenge a statistical claim about a population mean if the variable is numerical (for example, age, income, time, and so on) and only one population or group [more…]

How to Use the t-Test to Handle Small Samples and Unknown Standard Deviations

When using a test statistic for one population mean, there are two cases where you must use the t-distribution instead of the Z-distribution. The first case is where the sample size is small [more…]

How to Test a Null Hypothesis Based on One Population Proportion

You can use a hypothesis test to test a statistical claim about a population proportion when the variable is categorical (for example, gender or support/oppose) and only one population or group is being [more…]

How to Compare Two Independent Population Averages

You can compare numerical data for two statistical populations or groups (such as cholesterol levels in men versus women, or income levels for high school versus college grads) to test a claim about the [more…]

How to Test for an Average Difference Using the Paired t-Test

You can test for an average difference using the paired t-test when the variable is numerical (for example, income, cholesterol level, or miles per gallon) and the individuals in the statistical sample [more…]

How to Calculate a Correlation

Can one statistic measure both the strength and direction of a linear relationship between two variables? Sure! Statisticians use the correlation coefficient [more…]

How to Interpret a Correlation Coefficient r

In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of [more…]

Using Linear Regression to Predict an Outcome

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 [more…]

How to Calculate a Regression Line

In statistics, you can calculate a regression line for two variables if their scatterplot shows a linear pattern and the correlation between the variables is very strong [more…]

How to Interpret a Regression Line

In statistics, once you have calculated the slope and y-intercept to form the best-fitting regression line in a scatterplot, you can then interpret their values. [more…]

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

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 [more…]

How to Compare Two Population Proportions

For statistical purposes, you can compare two populations or groups when the variable is categorical (for example, smoker/nonsmoker, Democrat/Republican, support/oppose an opinion, and so on) and you’re [more…]

Working with Statistical Two-Way Tables

To explore the links between two categorical variables, you first need to organize the data that’s been collected, and a table is a great way to do that. A [more…]

Why the Statistical Mean and Median of a Histogram Often Have Different Centers

A histogram gives you a rough idea of where the "center" of the data lies. The word center is in quotes because many different statistics are used to designate the center. The two most common measures [more…]

How to Find the Interquartile Range for a Statistical Sample

To obtain a measure of variation based on the five-number summary of a statistical sample, you can find what's called the interquartile range, or IQR.

The purpose of the five-number summary is to give descriptive [more…]

How a Pie Chart Reflects Categorical Data in a Statistical Data Set

A pie chart takes categorical data from a statistical sample and breaks them down by group, showing the percentage of individuals that fall into each group. Because a pie chart takes on the shape of a [more…]

How to Interpret a Statistical Bar Graph

A bar graph (or bar chart) is perhaps the most common statistical data display used by the media. A bar graph breaks categorical data down by group, and represents these amounts by using bars of different [more…]

How Graphs Can Distort Statistics

A statistical graph can give you a false picture of the statistics on which it is based. For example, it can be misleading through its choice of scale on the frequency/relative frequency axis [more…]


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