What are some common misconceptions about t-tests and z-tests?

1 T-Test Basics

A t-test is a statistical test used to compare the means of two groups or to compare a sample mean to a known value when the population standard deviation is unknown. A widespread misconception is that t-tests can only be used for small sample sizes. While it's true that they are particularly useful when dealing with smaller samples, t-tests are robust and can be applied to larger samples as well. The key is that the sample data should be approximately normally distributed, and if this assumption holds, a t-test can yield reliable results regardless of sample size.

3 Sample Size Myth

The myth that t-tests are for small samples and z-tests are for large samples is pervasive but misleading. The choice between a t-test and a z-test does not hinge solely on sample size. It also depends on whether the population standard deviation is known and if the data meets certain distribution conditions. You might use a t-test for a large sample if the population standard deviation is unknown, or a z-test for a smaller sample if it is known and the sample comes from a normal distribution.

4 Normality Assumption

Another misconception is that both t-tests and z-tests require the data to be normally distributed. While the normality assumption is more critical for a t-test, especially with small sample sizes, both tests are quite robust to deviations from normality with larger samples. The Central Limit Theorem helps here, as it states that the distribution of sample means will tend to be normal if the sample size is large enough, even if the data itself is not normally distributed.

5 Equal Variances

The assumption of equal variances (homoscedasticity) is often misunderstood. Some believe that t-tests cannot be used unless the variances of the two groups are equal. However, there's a version of the t-test, called Welch's t-test, which does not assume equal variances and can be used when this assumption is violated. It's important to test for equal variances before deciding on the type of t-test to use, as ignoring this can lead to inaccurate results.

6 Misuse of Tests

Lastly, a common error is misusing t-tests and z-tests altogether by applying them in inappropriate situations. For example, using a t-test for dependent samples when the samples are independent, or vice versa, leads to incorrect conclusions. Understanding the types of t-tests—paired, independent two-sample, and one-sample—and when to use them is crucial. Similarly, employing a z-test without knowing the population standard deviation or when the sample does not represent the population well can invalidate your findings.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?