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

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 comparing the means of 2 populations.


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

Discovering Argon with the 2-Sample t-Test

I learned about Lord Rayleigh’s discovery of argon in my 2nd-year analytical chemistry class while reading “Quantitative Chemical Analysis” by Daniel Harris.  (William Ramsay was also responsible for this discovery.)  This is one of my favourite stories in chemistry; it illustrates how diligence in measurement can lead to an elegant and surprising discovery.  I find no evidence that Rayleigh and Ramsay used statistics to confirm their findings; their paper was published 13 years before Gosset published about the t-test.  Thus, I will use a 2-sample t-test in R to confirm their result.

Lord Rayleigh                                    William Ramsay

Photos of Lord Rayleigh and William Ramsay

Source: Wikimedia Commons

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