Statistics By Jim

Making statistics intuitive

interpreting results

How the Chi-Squared Test of Independence Works

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Chi-squared tests of independence determine whether a relationship exists between two categorical variables. Do the values of one categorical variable depend on the value of the other categorical variable? If the two variables are independent, knowing the value of one variable provides no information about the value of the other variable.

I’ve previously written about Pearson’s chi-square test of independence using a fun Star Trek example. Are the uniform colors related to the chances of dying? You can test the notion that the infamous red shirts have a higher likelihood of dying. In that post, I focus on the purpose of the test, applied it to this example, and interpreted the results.

In this post, I’ll take a bit of a different approach. I’ll show you the nuts and bolts of how to calculate the expected values, chi-square value, and degrees of freedom. Then you’ll learn how to use the chi-squared distribution in conjunction with the degrees of freedom to calculate the p-value. [Read more…] about How the Chi-Squared Test of Independence Works

How to Test Variances in Excel

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Use a variances test to determine whether the variability of two groups differs. In this post, we’ll work through a two-sample variances test that Excel provides. Even if Excel isn’t your primary statistical software, this post provides an excellent introduction to variance tests. Excel refers to this analysis as F-Test Two-Sample for Variances. [Read more…] about How to Test Variances in Excel

How to do Two-Way ANOVA in Excel

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Use two-way ANOVA to assess differences between the group means that are defined by two categorical factors. In this post, we’ll work through two-way ANOVA using Excel. Even if Excel isn’t your main statistical package, this post is an excellent introduction to two-way ANOVA. Excel refers to this analysis as two factor ANOVA. [Read more…] about How to do Two-Way ANOVA in Excel

How to do t-Tests in Excel

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Excel can perform various statistical analyses, including t-tests. It is an excellent option because nearly everyone can access Excel. This post is a great introduction to performing and interpreting t-tests even if Excel isn’t your primary statistical software package.

In this post, I provide step-by-step instructions for using Excel to perform t-tests. Importantly, I also show you how to select the correct form of t-test, choose the right options, and interpret the results. I also include links to additional resources I’ve written, which present clear explanations of relevant t-test concepts that you won’t find in Excel’s documentation. And, I use an example dataset for us to work through and interpret together! [Read more…] about How to do t-Tests in Excel

Revisiting the Monty Hall Problem with Hypothesis Testing

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The Monty Hall Problem is where Monty presents you with three doors, one of which contains a prize. He asks you to pick one door, which remains closed. Monty opens one of the other doors that does not have the prize. This process leaves two unopened doors—your original choice and one other. He allows you to switch from your initial choice to the other unopened door. Do you accept the offer?

If you accept his offer to switch doors, you’re twice as likely to win—66% versus 33%—than if you stay with your original choice.

Mind-blowing, right?

The solution to the Monty Hall Problem is tricky and counter-intuitive. It did trip up many experts back in the 1980s. However, the correct answer to the Monty Hall Problem is now well established using a variety of methods. It has been proven mathematically, with computer simulations, and empirical experiments, including on television by both the Mythbusters (CONFIRMED!) and James Mays’ Man Lab. You won’t find any statisticians who disagree with the solution.

In this post, I’ll explore aspects of this problem that have arisen in discussions with some stubborn resisters to the notion that you can increase your chances of winning by switching!

The Monty Hall problem provides a fun way to explore issues that relate to hypothesis testing. I’ve got a lot of fun lined up for this post, including the following!

  • Using a computer simulation to play the game 10,000 times.
  • Assessing sampling distributions to compare the 66% percent hypothesis to another contender.
  • Performing a power and sample size analysis to determine the number of times you need to play the Monty Hall game to get an answer.
  • Conducting an experiment by playing the game repeatedly myself, record the results, and use a proportions hypothesis test to draw conclusions! [Read more…] about Revisiting the Monty Hall Problem with Hypothesis Testing

Random Assignment in Experiments

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Random assignment uses chance to assign subjects to the control and treatment groups in an experiment. This process helps ensure that the groups are equivalent at the beginning of the study, which makes it safer to assume the treatments caused any differences between groups that the experimenters observe at the end of the study. [Read more…] about Random Assignment in Experiments

Using Confidence Intervals to Compare Means

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To determine whether the difference between two means is statistically significant, analysts often compare the confidence intervals for those groups. If those intervals overlap, they conclude that the difference between groups is not statistically significant. If there is no overlap, the difference is significant.

While this visual method of assessing the overlap is easy to perform, regrettably it comes at the cost of reducing your ability to detect differences. Fortunately, there is a simple solution to this problem that allows you to perform a simple visual assessment and yet not diminish the power of your analysis.

In this post, I’ll start by showing you the problem in action and explain why it happens. Then, we’ll proceed to an easy alternative method that avoids this problem. [Read more…] about Using Confidence Intervals to Compare Means

Can High P-values Be Meaningful?

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Can high p-values be helpful? What do high p-values mean?

Typically, when you perform a hypothesis test, you want to obtain low p-values that are statistically significant. Low p-values are sexy. They represent exciting findings and can help you get articles published.

However, you might be surprised to learn that higher p-values, the ones that are not statistically significant, are also valuable. In this post, I’ll show you the potential value of a p-value that is greater than 0.05, or whatever significance level you’re using. [Read more…] about Can High P-values Be Meaningful?

Using Post Hoc Tests with ANOVA

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Post hoc tests are an integral part of ANOVA. When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all of the group means are equal. However, ANOVA results do not identify which particular differences between pairs of means are significant. Use post hoc tests to explore differences between multiple group means while controlling the experiment-wise error rate.

In this post, I’ll show you what post hoc analyses are, the critical benefits they provide, and help you choose the correct one for your study. Additionally, I’ll show why failure to control the experiment-wise error rate will cause you to have severe doubts about your results. [Read more…] about Using Post Hoc Tests with ANOVA

One-Tailed and Two-Tailed Hypothesis Tests Explained

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Choosing whether to perform a one-tailed or a two-tailed hypothesis test is one of the methodology decisions you might need to make for your statistical analysis. This choice can have critical implications for the types of effects it can detect, the statistical power of the test, and potential errors.

In this post, you’ll learn about the differences between one-tailed and two-tailed hypothesis tests and their advantages and disadvantages. I include examples of both types of statistical tests. In my next post, I cover the decision between one and two-tailed tests in more detail.
[Read more…] about One-Tailed and Two-Tailed Hypothesis Tests Explained

Introduction to Bootstrapping in Statistics with an Example

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Bootstrapping is a statistical procedure that resamples a single dataset to create many simulated samples. This process allows you to calculate standard errors, construct confidence intervals, and perform hypothesis testing for numerous types of sample statistics. Bootstrap methods are alternative approaches to traditional hypothesis testing and are notable for being easier to understand and valid for more conditions.

In this blog post, I explain bootstrapping basics, compare bootstrapping to conventional statistical methods, and explain when it can be the better method. Additionally, I’ll work through an example using real data to create bootstrapped confidence intervals. [Read more…] about Introduction to Bootstrapping in Statistics with an Example

Practical vs. Statistical Significance

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Important ink stamp that relates to the concept of practical significance.You’ve just performed a hypothesis test and your results are statistically significant. Hurray! These results are important, right? Not so fast. Statistical significance does not necessarily mean that the results are practically significant in a real-world sense of importance.

In this blog post, I’ll talk about the differences between practical significance and statistical significance, and how to determine if your results are meaningful in the real world.
[Read more…] about Practical vs. Statistical Significance

Understanding Probability Distributions

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A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying probability distribution.

Suppose you draw a random sample and measure the heights of the subjects. As you measure heights, you can create a distribution of heights. This type of distribution is useful when you need to know which outcomes are most likely, the spread of potential values, and the likelihood of different results.

In this blog post, you’ll learn about probability distributions for both discrete and continuous variables. I’ll show you how they work and examples of how to use them. [Read more…] about Understanding Probability Distributions

Interpreting Correlation Coefficients

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A correlation between variables indicates that as one variable changes in value, the other variable tends to change in a specific direction.  Understanding that relationship is useful because we can use the value of one variable to predict the value of the other variable. For example, height and weight are correlated—as height increases, weight also tends to increase. Consequently, if we observe an individual who is unusually tall, we can predict that his weight is also above the average. [Read more…] about Interpreting Correlation Coefficients

Estimating a Good Sample Size for Your Study Using Power Analysis

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Determining a good sample size for a study is always an important issue. After all, using the wrong sample size can doom your study from the start. Fortunately, power analysis can find the answer for you. Power analysis combines statistical analysis, subject-area knowledge, and your requirements to help you derive the optimal sample size for your study.

Statistical power in a hypothesis test is the probability that the test will detect an effect that actually exists. As you’ll see in this post, both under-powered and over-powered studies are problematic. Let’s learn how to find a good sample size for your study! [Read more…] about Estimating a Good Sample Size for Your Study Using Power Analysis

Comparing Hypothesis Tests for Continuous, Binary, and Count Data

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In a previous blog post, I introduced the basic concepts of hypothesis testing and explained the need for performing these tests. In this post, I’ll build on that and compare various types of hypothesis tests that you can use with different types of data, explore some of the options, and explain how to interpret the results. Along the way, I’ll point out important planning considerations, related analyses, and pitfalls to avoid. [Read more…] about Comparing Hypothesis Tests for Continuous, Binary, and Count Data

Understanding Interaction Effects in Statistics

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Interaction effects occur when the effect of one variable depends on the value of another variable. Interaction effects are common in regression analysis, ANOVA, and designed experiments. In this blog post, I explain interaction effects, how to interpret them in statistical designs, and the problems you will face if you don’t include them in your model. [Read more…] about Understanding Interaction Effects in Statistics

Using Log-Log Plots to Determine Whether Size Matters

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Log-log plots display data in two dimensions where both axes use logarithmic scales. When one variable changes as a constant power of another, a log-log graph shows the relationship as a straight line. In this post, I’ll show you why these graphs are valuable and how to interpret them. [Read more…] about Using Log-Log Plots to Determine Whether Size Matters

When Do You Need to Standardize the Variables in a Regression Model?

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Standardization is the process of putting different variables on the same scale. In regression analysis, there are some scenarios where it is crucial to standardize your independent variables or risk obtaining misleading results.

In this blog post, I show when and why you need to standardize your variables in regression analysis. Don’t worry, this process is simple and helps ensure that you can trust your results. In fact, standardizing your variables can reveal essential findings that you would otherwise miss! [Read more…] about When Do You Need to Standardize the Variables in a Regression Model?