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As I've learnt, a T - test is used to compare 2 populations’ means, whereas an F-test (ANOVA) is used to compare 2/> populations’ variances.

At the end is this doing the same thing?

My background is from biology and no strong math/stat background. I wonder because whenever I used ANOVA (comparing >2 groups) followed by postHOC Tukey and not observing sig. differences, supervisor asking to use multiple t-test every time. Is this acceptable way of doing statistics.

I see there are many publications in biology where they do not follow statistics taught in the Textbooks.

The appropriateness of the statistical test depends on the research hypothesis. If, as you suggest in your question, the research hypothesis is that there is a difference in means between at least two groups when there are strictly more than two groups to be compared, then the $F$-test arising from ANOVA is an appropriate test under additional assumptions, because the null hypothesis would be $$H_0 : mu_1 = mu_2 = ldots = mu_k$$ where $k > 2$ represents the number of groups, and $mu_i$ is the true mean of group $i$. A level $alpha$ test would control Type I error for the alternative (research) hypothesis. But the result of such a test would not formally tell you which groups differ from each other in a pairwise sense; thus the need for the Tukey post hoc test, or you could use pairwise $t$ tests with multiplicity correction.

As an illustration of the importance of the research hypothesis, if you have a control group against which different treatments are compared, you could use Dunnett's test instead of ANOVA, as the only comparisons of interest are the treatments against control, not treatments against each other.

The central issue underlying the subsequent identification of statistically significant pairwise differences after an omnibus test is that of multiple comparisons--e.g., even with as few as $4$ groups, you would have $binom{4}{2} = 6$ pairwise comparisons and the Type I error would be inflated without multiplicity correction such as the Bonferroni adjustment.

To simply do pairwise tests before ANOVA would, in my opinion, be ill-advised from the perspective of statistical rigor, although as I have implied, it is not the more serious methodological flaw. It may be useful for exploratory purposes, but adjustment for multiple comparisons is absolutely necessary in order to make inferential claims that could withstand scrutiny.

One final note: a "$t$ test" does not require an assumption of equal variances; the Welch $t$ test (using the Satterthwaite estimate of the degrees of freedom) is one way to address the issue of unequal group variances, and the test statistic is compared to a Student $t$ distribution, so I would consider that a $t$ test.

A T-test is a univariate hypothesis test that is applied when standard deviation is not known and the sample size is small. The T-statistic follows Student t-distribution under null hypothesis. You use this test for comparing the means of two populations. As @gd1035 mentioned, the t-test assumes equal variances, which you could first check by using an F-test.

The F-test, on the other hand, is statistical test that determines the equality of the variances of the two normal populations. The F-statistic follows the f-distribution under null hypothesis. You use this test for comparing two population variances.