# Applied Statistics Lesson of the Day – The Independent 2-Sample t-Test with Unequal Variances (Welch’s t-Test)

March 13, 2014 Leave a comment

A common problem in statistics is determining whether or not the means of 2 populations are equal. The **independent 2-sample t-test** is a popular parametric method to answer this question. (In an earlier Statistics Lesson of the Day, I discussed how data collected from a completely randomized design with 1 binary factor can be analyzed by an independent 2-sample t-test. I also discussed its possible use in the discovery of argon.) I have learned 2 versions of the independent 2-sample t-test, and they differ on the variances of the 2 samples. The 2 possibilities are

- equal variances
- unequal variances

Most statistics textbooks that I have read elaborate at length about the independent 2-sample t-test with equal variances (also called **Student’s t-test**). However, **the assumption of equal variances needs to be checked using the chi-squared test before proceeding with the Student’s t-test**, yet this check does not seem to be universally done in practice. Furthermore, conducting one test based on the results of another can **inflate the probability of committing a Type 1 error** (Ruxton, 2006).

Some books give due attention to the **independent 2-sample t-test with unequal variances** (also called **Welch’s t-test**), but some barely mention its value, and others do not even mention it at all. I find this to be puzzling, because the assumption of equal variances is often violated in practice, and Welch’s t-test provides an easy solution to this problem. There is a seemingly intimidating but straightforward calculation to approximate the number of **degrees of freedom** for Welch’s t-test, and this calculation is automatically incorporated in most software, including R and SAS. Finally, Welch’s t-test **removes the need to check for equal variances**, and it is **almost as powerful as Student’s t-test** when the variances are equal (Ruxton, 2006).

For all of these reasons, **I recommend Welch’s t-test** when using the parametric approach to compare the means of 2 populations.

### Reference

Graeme D. Ruxton. “The unequal variance *t*-test is an underused alternative to Student’s *t*-test and the Mann–Whitney *U* test“. Behavioral Ecology (July/August 2006) 17 (4): 688-690 first published online May 17, 2006

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