Known Variance (σ², shared by all groups):

Comparing μ Across Groups

The table below shows the probability (as a percentage) that each group’s mean μ is the highest or lowest among all groups (approximated via Monte Carlo simulation).

Group	Prob(Highest μ)	Prob(Lowest μ)

The Normal Distribution Across Multiple Groups

\[ f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x - \mu)^2}{2\sigma^2} }, \quad \mu \sim \mathcal{N}(\mu_0, \sigma_0^2). \]

When you have multiple groups of normally distributed data with a known variance, each group may have a different underlying mean \(\mu_i\). By treating each group’s unknown \(\mu_i\) as a random variable with a Normal posterior, you can visualize the PDFs together and compare their likelihoods of being the highest or lowest among the groups.

When the variance is unknown, it is estimated separately for each group. With a non-informative prior, each group’s posterior for \(\mu_i\) is a Student-t distribution with \(n_i - 1\) degrees of freedom, centred on the group’s sample mean \(\bar{x}_i\) with scale \(s_i/\sqrt{n_i}\).