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

# conceptual

## 5 Ways to Find Outliers in Your Data

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

## 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 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

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

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

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

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, 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

## Percentiles: Interpretations and Calculations

Percentiles indicate the percentage of scores that fall below a particular value. They tell you where a score stands relative to other scores. For example, a person with an IQ of 120 is at the 91^{st }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

## Using Confidence Intervals to Compare Means

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?

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

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

## When Can I Use One-Tailed Hypothesis Tests?

One-tailed hypothesis tests offer the promise of more statistical power compared to an equivalent two-tailed design. While there is some debate about when you can use a one-tailed test, the general consensus among statisticians is that you should use two-tailed tests unless you have concrete reasons for using a one-tailed test.

In this post, I discuss when you should and should not use one-tailed tests. I’ll cover the different schools of thought and offer my own opinion. [Read more…] about When Can I Use One-Tailed Hypothesis Tests?

## One-Tailed and Two-Tailed Hypothesis Tests Explained

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

## Central Limit Theorem Explained

The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable’s distribution in the population.

Unpacking the meaning from that complex definition can be difficult. That’s the topic for this post! I’ll walk you through the various aspects of the central limit theorem (CLT) definition, and show you why it is so important in the field of statistics. [Read more…] about Central Limit Theorem Explained

## Introduction to Bootstrapping in Statistics with an Example

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

## Confounding Variables Can Bias Your Results

Omitted variable bias occurs when a regression model leaves out relevant independent variables, which are known as confounding variables. This condition forces the model to attribute the effects of omitted variables to variables that are in the model, which biases the coefficient estimates. [Read more…] about Confounding Variables Can Bias Your Results

## Sample Statistics Are Always Wrong (to Some Extent)!

Here’s some shocking information for you—sample statistics are *always* wrong! When you use samples to estimate the properties of populations, you never obtain the correct values exactly. Don’t worry. I’ll help you navigate this issue using a simple statistical tool! [Read more…] about Sample Statistics Are Always Wrong (to Some Extent)!

## Populations, Parameters, and Samples in Inferential Statistics

Inferential statistics lets you draw conclusions about populations by using small samples. Consequently, inferential statistics provide enormous benefits because typically you can’t measure an entire population.

However, to gain these benefits, you must understand the relationship between populations, subpopulations, population parameters, samples, and sample statistics.

In this blog post, I discuss these concepts, and how to obtain representative samples using random sampling.

**Related post**: Difference between Descriptive and Inferential Statistics

[Read more…] about Populations, Parameters, and Samples in Inferential Statistics

## Types of Errors in Hypothesis Testing

Hypothesis tests use sample data to make inferences about the properties of a population. You gain tremendous benefits by working with random samples because it is usually impossible to measure the entire population.

However, there are tradeoffs when you use samples. The samples we use are typically a miniscule percentage of the entire population. Consequently, they occasionally misrepresent the population severely enough to cause hypothesis tests to make errors.

In this blog post, you will learn about the two types of errors in hypothesis testing, their causes, and how to manage them. [Read more…] about Types of Errors in Hypothesis Testing