There is no minimum sample size for the t test to be valid. Validity requires that the assumptions for the test statistic hold approximately. Those assumptions are in the one sample case that the data are iid normal (or approximately normal) with mean 0 under the null hypothesis and a variance that is unknown but estimated from the sample. In the two sample case it is that both samples are independent of each other and each sample consists of iid normal variables with the two samples having the same mean and a common unknown variance under the null hypothesis. A pooled estimate of variance is used for the statistic.

In the one sample case the distribution under the null hypothesis is a central t with n-1 degrees of freedom. In the two sample cases with sample sizes n and m not necessarily equal the null distribution of the test statistics is t with n+m-2 degrees of freedom. The increased variability due to low sample size is accounted for in the distribution which has heavier tails when the degrees of freedom is low which corresponds to a low sample size. So critical values can be found for the test statistic to have a given significance level for any sample size (well, at least of size 2 or larger).

The problem with low sample size is with regard to the power of the test. The reviewer may have felt that 15 per group was not a large enough sample size to have high power of detecting a meaningful difference say delta between the two means or a mean greater than delta in absolute value for a one sample problem. Needing 40 would require a specification of a certain power at a particular delta that would be achieved with n equal 40 but not lower than 40.

regards

t-test procedure is not advisable with n < 10 (

Dear Hein,

Thank you so much for your answer, but there are an expensive biological studies apply (for example) t-test for two independent samples instead of Mann Whitney Test because they think the parametric tests are more accurate than non-parametric even though the sample sizes are small.

Best wishes

Huda

There is no minimum sample size for the t test to be valid. Validity requires that the assumptions for the test statistic hold approximately. Those assumptions are in the one sample case that the data are iid normal (or approximately normal) with mean 0 under the null hypothesis and a variance that is unknown but estimated from the sample. In the two sample case it is that both samples are independent of each other and each sample consists of iid normal variables with the two samples having the same mean and a common unknown variance under the null hypothesis. A pooled estimate of variance is used for the statistic.

In the one sample case the distribution under the null hypothesis is a central t with n-1 degrees of freedom. In the two sample cases with sample sizes n and m not necessarily equal the null distribution of the test statistics is t with n+m-2 degrees of freedom. The increased variability due to low sample size is accounted for in the distribution which has heavier tails when the degrees of freedom is low which corresponds to a low sample size. So critical values can be found for the test statistic to have a given significance level for any sample size (well, at least of size 2 or larger).

The problem with low sample size is with regard to the power of the test. The reviewer may have felt that 15 per group was not a large enough sample size to have high power of detecting a meaningful difference say delta between the two means or a mean greater than delta in absolute value for a one sample problem. Needing 40 would require a specification of a certain power at a particular delta that would be achieved with n equal 40 but not lower than 40.

regards

Thank you Hassan,

In fact, the distribution of T is represented the Normal distribution divided by the square root of the Chi-square distribution, so the normality condition (the distribution of data is Normal) is very necessary for applying t-test to control type one error. But I wonder why many importent biological studies ignore this condition.

Huda

My advice doesn't come from the statistical textbook. It is a practical experience, that you should anticipate certain distortion by interpreting results in border cases of too many or too little numbers of cases. See the last paragraph of Hassan Nima's answer, it is right.

What does the procedure t-test mean? It compared two random samples (I think this mode of t-test is the matter here) resp. their mean values with the condition of two classes of cases under a predefined likelihood, wether they belong to the same population. You test it with the assertion they belong to different populations (H1). If you use many many cases than a tiny difference between the mean values of the two samples is sufficient to bring a "significant" result, because this is caused by approximation of the sample to the (theoretically thought) population. With a small sample you have to anticipate big differences to get a "significant" result.

Results you will get at any rate. But I think it is useful to discuss the border of sample sizes, and I can't assert that such problems don't exist.

I had forgotten: The question of applying Chi2-test shouldn't be mixed with the problem of appropriate sample sizes with t-test, because chi-square is a rougher test, working with qualitative variables. it is not useful to apply it if you have got quantitative measured values, if your hypothesis asserts significant differences between events or groups.

Dear Hein,

Thanks for your interest. My question related with quantity variables not qualitative and when the samples drown randomly and identically from Normal distribution (it's population has Normal distribution) but the sizes of samples are very small (say n=2), in this case, can we apply t-test for two independent samples? I have read hundreds of biological research that apply this case.

Regards

If the population under study, follows the normal distribution, then, sample's mean will follows normal distribution also, for any sample size (even if n = 2). T test is robust against sample sizes. So, in this case (with existance of normality condition for the population) I think they can apply t-test with small samples sizes.

Best wishes

Huda

There is no minimum prescribed sample size for t test. However, the sample is to be such that the associated degrees of freedom becomes should not become very small ( i.e. should not be close to zero). more the degrees of freedom, more the power of the test. Thus, sample size is to be selected accordingly.