There may not be a theoretical minimum sample size, but the n-1 term in denominator for calculating the variance becomes zero. Division by zero is not usually recommended. The smallest possible sample size is 2.

Parametric tests assume a normal distribution. If this assumed distribution is correct, then a small number of samples is sufficient to characterize the mean and variance.

Non-parametric tests make no assumption about the distribution of the data. It is therefore more difficult to fill in all the observations that you could have taken but didn't.

A sample size of two is generally not considered sufficient for any purpose. A sample size of three will often be unpublishable. The suggested number of replicates depend on the application, journal, reviewers, and your own sense of scientific integrity. A clinical trial with three replicates will not be accepted by anyone no matter how good the statistics might look.

The more replication, the greater the confidence that your results are not just a random event.

Parametric tests are more powerful than non-parametric tests so they might be preferable for small sample sizes.  T-tests are specifically designed for small sample sizes and there is no theoretical minimum sample size for which a t-test is valid.

(Edit - following Timothy's answer below of course you do need enough data points to be able to estimate a variance, so 2 is the theoretical minimum - or perhaps 3 in total as you need to compare the between and within groups variance estimate)

They also associated with estimates of effect sizes, which non-parametric tests are not so they can be more useful, but only in cases where the assumptions underlying the tests are not violated.

There may not be a theoretical minimum sample size, but the n-1 term in denominator for calculating the variance becomes zero. Division by zero is not usually recommended. The smallest possible sample size is 2.

Parametric tests assume a normal distribution. If this assumed distribution is correct, then a small number of samples is sufficient to characterize the mean and variance.

Non-parametric tests make no assumption about the distribution of the data. It is therefore more difficult to fill in all the observations that you could have taken but didn't.

A sample size of two is generally not considered sufficient for any purpose. A sample size of three will often be unpublishable. The suggested number of replicates depend on the application, journal, reviewers, and your own sense of scientific integrity. A clinical trial with three replicates will not be accepted by anyone no matter how good the statistics might look.

The more replication, the greater the confidence that your results are not just a random event.

Thanks for all,

For more illustration / Suppose we want test the differences between three levels of concentrations. We use three samples each sample of size (n = 2 , the total number of observations will be 6) ,then applying  the One way analysis of variance (ANOVA) in a comparison between the three concentrations. Can we apply ANOVA directly by assuming that, data satisfies the Normality condition?

So you have a treatment. For illustration say nitrogen fertilization. This treatment is applied at three levels: say 50, 100, and 150 kg/ha. At each level you have two observations. These are two replicates. Key words: replicates, treatments, levels.

If this is the case, you don't have anything that anyone else will be interested in. The problem is that biological variability is such that your results are probably random chance - no matter how good they look to you right now.

That said, if this is preliminary data and you are trying to learn how the analysis works before spending great effort to gather the real data, then yes you can run the analysis. You don't have sufficient data to get a meaningful test for normality, so you will just have to assume normality. With such a small data set, you might make the calculations by hand as well as using a statistical analysis program. This way you can check to make sure that the program is giving you answers that are what you want. One common problem is to switch the independent and dependent variables when there is only one of each. You then get a "surprising result" that disagrees with the published literature.

Just try the usual Mann & Whitney test on the two samples (-50, -49, -48) and (1000, 1001, 1002) which are evidently different, and you'll have your answer.

Non-parametric tests just can't lead p

Therefore, according to the example of Timothy, there is no problem for applying  ANOVA .

Many thanks for all.

Just to be clear, there is no problem running the numbers through the data analysis program. That is not the same as saying that it is an activity worth doing. This is a great class exercise, or preliminary data. It would take extraordinary circumstances to make it worth anything beyond that.