Statistics By Jim

Making statistics intuitive

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Answering the Birthday Problem in Statistics

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The Birthday Problem in statistics asks, how many people do you need in a group to have a 50% chance that at least two people will share a birthday? Go ahead and think about that for a moment. The answer surprises many people. We’ll get to that shortly.

In this post, I’ll not only answer the birthday paradox, but I’ll also show you how to calculate the probabilities for any size group, run a computer simulation of it, and explain why the answer to the Birthday Problem is so surprising. [Read more…] about Answering the Birthday Problem in Statistics

Coronavirus: Exponential Growth and Hospital Beds

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UPDATED March 24, 2020: As the number of confirmed coronavirus cases continues to grow exponentially, the capacity of the hospital system to treat these cases is becoming a concern. The goal of “flattening the curve” is that testing, isolation, and social distancing will slow the increase of new cases. Hopefully, these efforts reduce the numbers of new patients who require hospitalization to a rate that hospitals can handle.

In this post, I’ll identify the top 10 states in the United States that have the greatest likelihood of experiencing hospital capacity problems if coronavirus cases continue to grow exponentially. To recognize these states, I’ll assess per capita rates for both coronavirus infections and hospital beds. I’m looking for states that have a relatively large number of coronavirus cases given the size of their population and have a relatively low number of hospital beds. [Read more…] about Coronavirus: Exponential Growth and Hospital Beds

Coronavirus Curves and Different Outcomes

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Microscopic photograph of the coronavirus
Coronavirus particles as seen by negative stain electron microscopy. Notice the characteristic club-like projections on the membrane.

UPDATED May 9, 2020. The coronavirus, or COVID19, has swept around the world. However, not all countries have had the same experiences. Outcomes have varied by the number of cases, the rate of increase, and how countries have responded.

In this post, I present coronavirus growth curves for 15 countries and their per capita values, graph their new cases per day, daily coronavirus deaths, and describe how each country approached controlling the virus. You can see the differences in outcomes and when the effects of coronavirus mitigation efforts started taking effect. I also include the per capita values for these countries in a table near the end.

At this time, there is plenty of good news with evidence that many of the 15 countries have slowed the growth rate of new cases. However, several other countries have reason to worry. And, we have one new cautionary tale about a country that had the virus contained but is now seeing a spike in new cases. [Read more…] about Coronavirus Curves and Different Outcomes

Failing to Reject the Null Hypothesis

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Failing to reject the null hypothesis is an odd way to state that the results of your hypothesis test are not statistically significant. Why the peculiar phrasing? “Fail to reject” sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? There’s an excellent reason for the odd wording!

In this post, learn what it means when you fail to reject the null hypothesis and why that’s the correct wording. While accepting the null hypothesis sounds more straightforward, it is not statistically correct! [Read more…] about Failing to Reject the Null Hypothesis

Understanding Significance Levels in Statistics

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Significance levels in statistics are a crucial component of hypothesis testing. However, unlike other values in your statistical output, the significance level is not something that statistical software calculates. Instead, you choose the significance level. Have you ever wondered why?

In this post, I’ll explain the significance level conceptually, why you choose its value, and how to choose a good value. Statisticians also refer to the significance level as alpha (α). [Read more…] about Understanding Significance Levels in Statistics

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 based on 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

Guidelines for Removing and Handling Outliers in Data

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Outliers are unusual values in your dataset, and they can distort statistical analyses and violate their assumptions. Unfortunately, all analysts will confront outliers and be forced to make decisions about what to do with them. Given the problems they can cause, you might think that it’s best to remove them from your data. But, that’s not always the case. Removing outliers is legitimate only for specific reasons. [Read more…] about Guidelines for Removing and Handling Outliers in Data

5 Ways to Find Outliers in Your Data

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Outliers are data points that are far from other data points. In other words, they’re unusual values in a dataset. Outliers are problematic for many statistical analyses because they can cause tests to either miss significant findings or distort real results.

Unfortunately, there are no strict statistical rules for definitively identifying outliers. Finding outliers depends on subject-area knowledge and an understanding of the data collection process. While there is no solid mathematical definition, there are guidelines and statistical tests you can use to find outlier candidates. [Read more…] about 5 Ways to Find Outliers in Your Data

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

Low Power Tests Exaggerate Effect Sizes

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If your study has low statistical power, it will exaggerate the effect size. What?!

Statistical power is the ability of a hypothesis test to detect an effect that exists in the population. Clearly, a high-powered study is a good thing just for being able to identify these effects. Low power reduces your chances of discovering real findings. However, many analysts don’t realize that low power also inflates the effect size.

In this post, I show how this unexpected relationship between power and exaggerated effect sizes exists. I’ll also tie it to other issues, such as the bias of effects published in journals and other matters about statistical power. I think this post will be eye-opening and thought provoking! As always, I’ll use many graphs rather than equations. [Read more…] about Low Power Tests Exaggerate Effect Sizes

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

Causation versus Correlation in Statistics

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Causation indicates that an event affects an outcome. Do fatty diets cause heart problems? If you study for a test, does it cause you to get a higher score?

In statistics, causation is a bit tricky. As you’ve no doubt heard, correlation doesn’t necessarily imply causation. An association or correlation between variables simply indicates that the values vary together. It does not necessarily suggest that changes in one variable cause changes in the other variable. Proving causality can be difficult.

If correlation does not prove causation, what statistical test do you use to assess causality? That’s a trick question because no statistical analysis can make that determination. In this post, learn about why you want to determine causation and how to do that. [Read more…] about Causation versus Correlation in Statistics

Observational Studies Explained

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Observational studies use samples to draw conclusions about a population when the researchers do not control the treatment, or independent variable, that relates to the primary research question.

In my previous post, I show how random assignment reduces systematic differences between experimental groups at the beginning of the study, which increases your confidence that the treatments caused any differences between groups you observe at the end of the study.

Unfortunately, using random assignment is not always possible. For these cases, you can conduct an observational study. In this post, learn about observational studies, why these studies must account for confounding variables, and how to do so. I’ll close this post by reviewing a published observational study about vitamin supplement usage. [Read more…] about Observational Studies Explained

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

5 Steps for Conducting Scientific Studies with Statistical Analyses

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The scientific method is a proven procedure for expanding knowledge through experimentation and analysis. It is a process that uses careful planning, rigorous methodology, and thorough assessment. Statistical analysis plays an essential role in this process.

In an experiment that includes statistical analysis, the analysis is at the end of a long series of events. To obtain valid results, it’s crucial that you carefully plan and conduct a scientific study for all steps up to and including the analysis. In this blog post, I map out five steps for scientific studies that include statistical analyses. [Read more…] about 5 Steps for Conducting Scientific Studies with Statistical Analyses