This is part of probstat.

Variance estimations

Estimator 1. The first estimator for $\sigma ^{2}$ is

${\frac {1}{m}}\sum _{i=1}^{m}\sum _{j=1}^{n}{\frac {(X_{ij}-{\bar {X}}_{i\centerdot })^{2}}{n-1}}.$

This estimator is always correct. (Even when $H_{0}$ is not true.)

Estimator 2. The second estimator is calculated from the sample means. It will be correct only when $H_{0}$ is true.

$n\sum _{i=1}^{m}{\frac {({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}}{m-1}}$

Hypothesis testing

Then compute

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Error

Too many requests (f061ab2)

SS_{w}=\sum _{i=1}^{m}\sum _{j=1}^{n}(X_{ij}-{\bar {X}}_{i\centerdot })^{2}

and

$SS_{b}=n\sum _{i=1}^{m}({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}$

Recall that

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Error

Too many requests (f061ab2)

{\frac {SS_{w}}{\sigma ^{2}}}\sim \chi _{m(n-1)}^{2}

,

and

.

Also, since $S_{i}^{2}$ is independent of $Wikimedia Error * { margin: 0; padding: 0; } body { background: #fff; font: 15px/1.6 sans-serif; color: #333; } .content { margin: 7% auto 0; padding: 2em 1em 1em; max-width: 640px; display: flex; flex-direction: row; flex-wrap: wrap; } .footer { clear: both; margin-top: 14%; border-top: 1px solid #e5e5e5; background: #f9f9f9; padding: 2em 0; font-size: 0.8em; text-align: center; } img { margin: 0 2em 2em 0; } a img { border: 0; } h1 { margin-top: 1em; font-size: 1.2em; } .content-text { flex: 1; } p { margin: 0.7em 0 1em 0; } a { color: #0645ad; text-decoration: none; } a:hover { text-decoration: underline; } code { font-family: sans-serif; } summary { font-weight: bold; cursor: pointer; } details[open] { background: #970302; color: #dfdedd; } .text-muted { color: #777; } @media (prefers-color-scheme: dark) { a { color: #9e9eff; } body { background: transparent; color: #ddd; } .footer { border-top: 1px solid #444; background: #060606; } #logo { filter: invert(1) hue-rotate(180deg); } .text-muted { color: #888; } }$