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Let $\{x_{i,j} : i=1..7,j=1,..n\}$ be a set of samples from $n$ weeks (where $i$ denotes the day of the week). Is there any interesting information to be gleaned from the relationship (ratio, difference, etc.) between $ \mathbb{E}_{i} [std(x_i)]$ and $std(x)$?

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This is closely related to ANOVA, which essentially compares a pooled within-group (within-week, here) variance estimate with the between-group (between-week) variance estimate. If $\text{var}(x_i)$ is much smaller than $\text{var}(x)$ then there is evidence that the mean differs from group to group (week to week).

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