UPDATED! April 3, 2020. The coronavirus mortality rate varies significantly by country. In this post, I look at the mortality rates for ten countries and assess factors that affect these numbers. After discussing the trends, I provide a rough estimate for where the actual fatality rate might lie. [Read more…] about Coronavirus Mortality Rates by Country
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Coronavirus: Exponential Growth and Hospital Beds
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 โflattening the curveโ 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, Iโll 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, Iโll assess per capita rates for both coronavirus infections and hospital beds. Iโm 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
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
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? โFail to rejectโ sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? Thereโs an excellent reason for the odd wording!
In this post, learn what it means when you fail to reject the null hypothesis and why thatโs 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
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, Iโll 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 (ฮฑ). [Read more…] about Understanding Significance Levels in Statistics
How the Chi-Squared Test of Independence Works
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.
Iโve previously written about Pearsonโs 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, Iโll take a bit of a different approach. Iโll show you the nuts and bolts of how to calculate the expected values, chi-square value, and degrees of freedom. Then youโll 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
Use a variances test to determine whether the variability of two groups differs. In this post, weโll work through a two-sample variances test that Excel provides. Even if Excel isnโt 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
Use two-way ANOVA to assess differences between the group means that are defined by two categorical factors. In this post, weโll work through two-way ANOVA using Excel. Even if Excel isnโt 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
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 itโs best to remove them from your data. But, thatโs 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
Outliers are data points that are far from other data points. In other words, theyโre 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 One-Way ANOVA in Excel
Use one-way ANOVA to test whether the means of at least three groups are different. Excel refers to this test as Single Factor ANOVA. This post is an excellent introduction to performing and interpreting a one-way ANOVA test even if Excel isnโt your primary statistical software package. [Read more…] about How to do One-Way ANOVA in Excel
How to do t-Tests in Excel
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 Iโve written, which present clear explanations of relevant t-test concepts that you wonโt find in Excelโs 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
New eBook Release! Introduction to Statistics: An Intuitive Guide
Iโm thrilled to release my new book! Introduction to Statistics: An Intuitive Guide for Analyzing Data and Unlocking Discoveries. [Read more…] about New eBook Release! Introduction to Statistics: An Intuitive Guide
Low Power Tests Exaggerate Effect Sizes
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 donโt realize that low power also inflates the effect size. Learn more about Statistical Power.
In this post, I show how this unexpected relationship between power and exaggerated effect sizes exists. Iโll 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
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 doorsโyour 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, youโre twice as likely to winโ66% versus 33%โthan 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 Maysโ Man Lab. You won’t find any statisticians who disagree with the solution.
In this post, Iโll 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 in Statistics: Hill’s Criteria
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 youโve no doubt heard, correlation doesnโt 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? Thatโs 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 in Statistics: Hill’s Criteria
What is an Observational Study: Definition & Examples
What is an Observational Study?
An observational study uses sample data to find correlations in situations where the researchers do not control the treatment, or independent variable, that relates to the primary research question. The definition of an observational study hinges on the notion that the researchers only observe subjects and do not assign them to the control and treatment groups. That’s the key difference between an observational study vs experiment. These studies are also known as quasi-experiments and correlational studies.
True experiments assign subject to the experimental groups where the researchers can manipulate the conditions. Unfortunately, random assignment is not always possible. For these cases, you can conduct an observational study.
In this post, learn about the types of observational studies, why they are susceptible to confounding variables, and how they compare to experiments. Iโll close this post by reviewing a published observational study about vitamin supplement usage. [Read more…] about What is an Observational Study: Definition & Examples
Random Assignment in Experiments
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
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, itโs 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
Percentiles: Interpretations and Calculations
Percentiles indicate the percentage of scores that fall below a particular value. They are a type of quantile that splits the data into 100 equal-sized groups and tells you where a score stands relative to other scores. For example, a person with an IQ of 120 is at the 91st percentile, which indicates that their IQ is higher than 91 percent of other scores.
Percentiles are a great tool to use when you need to know the relative standing of a value. Where does a value fall within a distribution of values? While the concept behind percentiles is straight forward, there are different mathematical methods for calculating them. In this post, learn about percentiles, special percentiles and their surprisingly flexible uses, and the various procedures for calculating them. [Read more…] about Percentiles: Interpretations and Calculations