# How to correctly state a multiple-comparison hypothesis pair

I need to compare multiple treatments over a predefined set of benchmark instances.
However, I'm facing some difficulties on how to correctly state my hypothesis pair.

I want to verify if there are differences in the average results of $$m$$ treatments.
Assume that the set $$M = \{\mu_1, \mu_2, \ldots, \mu_m\}$$ contains the average results given by treatment $$i \in M$$.

Thus, I set the following hypothesis pair

$$\begin{equation} \label{eq:hypothesis1} \left\{\begin{matrix} H_0: & \mu_i = \mu_{j} \\ H_1: & \mu_i \neq \mu_{j} \end{matrix}\right.\qquad \forall (\mu_i, \mu_j) \in M, \end{equation}$$

where the null hypothesis ($$H_0$$) states that the results of all treatments do not significantly differ. On the other hand, the alternative hypothesis ($$H_1$$) states that the results of at least a treatment statistically differ from the others.

I'm really not confident that the manner I stated my hypothesis pair is OK, but I can't figure out how to improve my notation and explanation.
• You can simply say that $H_a$ is "not all equal." Alternatively, you can define $\bar \mu = \frac 1m\sum_i \mu_i,\, \theta = \sum_i(\mu_i - \bar \mu)^2.$ Then $H_0: \theta = 0$ and $H_a: \theta > 0.$ – BruceET Sep 11 at 0:17