The Student Newman Keuls test is a posthoc test that is used to check for pairwise equality of means for several populations. It is typically used after we reject the null hypothesis in the ANOVA procedure.

It is similar to Tukeys posthoc test in the sense that both of these tests are used to check for pairwise equality of means.

The difference between the two tests is that the Student Newman Keuls test is more powerful meaning that it is much more likely to identify when the alternative hypothesis is true compared to Tukey’s posthoc test. Of course, this increases the probability that the Student Newman Keuls test incorrectly rejects the null hypothesis even though H_{0} is true.

**Assumptions behind the Student Newman Keuls test**:

- The parent populations follow the normal distribution.
- The variances of all the parent populations are equal.
- The sample values are chosen independently.

**Procedure to conduct the Student Newman Keuls test**:

**1.** Identify the two populations with the greatest and smallest sample means. Let the sample means be denoted as X̄_{A} and X̄_{B}.

**2.** Formulate the null and alternative hypothesis as follows,

H_{0}: The two population means are equal **vs** H_{1}: The two population means are unequal

**3.** Calculate the value of the test statistic using the formula,

where, MSE is the Mean standard error and ‘n’ is the sample size

If the sample sizes are unequal use the formula,

where, n_{A} and n_{B} are the sample sizes

**4.** Find the critical q table value for the required level of significance.

**5.** If the test statistic is less than the table value, accept the null hypothesis and conclude that the two means are equal. Since the sample means with the greatest difference are equal we conclude that all the means are equal and we are done.

**6.** If the test statistic is greater than the table value, we reject the null hypothesis and conclude that the two means are equal. Remove the population with the largest mean and repeat the test to check for the inequality of means of the remaining populations.