# t-test using a sample less than 30 [duplicate]

When I do the two tailed t-test using a sample less than 30, can I use this result? This is because I do also still see the t-test table which has degrees of freedom far less than 30. How further small can the sample size be?

• Is that 30 per group, or 30 total? If the latter, are the groups 15-15?
– JAD
Jul 27, 2017 at 8:29
• One case I have two groups 11-14, the other case I simply have one group to test with sample size less than 30.
– Eric
Jul 27, 2017 at 8:31
• The smallest possible sample sizes for a two-sample equal variance t-test is 1 and 2 (while many packages implement their tests in a way that precludes one of the samples having only one observation there's no good reason that they should do so -- R will do an equal variance t-test with 1 and 2 though, try: t.test(1,c(3.5,3.2),var.equal=TRUE)). Details are covered here. Jul 27, 2017 at 9:25
• If you're not fixed on it having to specifically be a t-test, even smaller samples are possible. For example, one can adapt the discussion in whuber's answer here (also see here and comments here) to a two sample test of means with $n_1=n_2=1$ Jul 27, 2017 at 9:36
• It's not usually advisable to use such small $n$ but it's not an issue with the test "working" -- it still works as it should. But if your power is very low at anticipated effect sizes, then even if you reject you may have trouble convincing people it's not just a type I error. [Note also that it's not practical to use rank based tests down to sample sizes as low as the $t$ can be used for -- e.g. if you try to do a two-tailed Wilcoxon-Mann-Whitney for n's of 1 & 2 the lowest achievable significance level is $\frac{_2}{^3}$]. Jul 27, 2017 at 9:43

## 1 Answer

T-test are useful if the data is normally distributed and iid (@djima thank you). If the effect size is large you can use the t-test also if the sample size is small. So yes, you can use a t-test with a sample size which is smaller than 30. The effect size can be calculated with Cohen's D.. Under certain circumstances other measures such as the Glass Delta or the Hedges G are more useful.

"The present simulation study showed that there is no fundamental objection to using a regular t-test with extremely small sample sizes. Even a sample size as small as 2 did not pose problems. In most of the simulated cases, the Type I error rate did not exceed the nominal value of 5%. A paired t-test is also feasible with extremely small sample sizes, particularly when the within-pair correlation coefficient is high. (de Winter, 2013, p. 7)

J.C.F. de Winter (2013), "Using the Student’s t-test with extremely small sample sizes," Practical Assessment, Research and Evaluation, 18:10, August, ISSN 1531-7714 http://pareonline.net/getvn.asp?v=18&n=10

• Normally distributed and IID! Correlated samples would inflate t-stats.
– djma
Jul 27, 2017 at 9:26
• Thank you. So if the effect size from the Cohen's D test is 2.254386362 while my sample of two groups are 11-14, the t-test is ok, right?
– Eric
Jul 27, 2017 at 9:28
• @Eric yes you can do Jul 27, 2017 at 9:31
• @Ferdi: then if I have Cohen's D test value of -8.058891649 while each of my two group sample sizes are 9 and 6 respectively, do I conclude that I have enough sample size for t-test?
– Eric
Jul 27, 2017 at 10:04
• Yes. The absolute value of your number is really large. As your sample is very small you can have a look at it "by hand" and it should be evident. Jul 27, 2017 at 10:11