Variance and Standard Deviation for Grouped Data

The variance and standard deviation formulas extend to grouped data exactly as the mean deviation formulas did in the previous section — every squared deviation is weighted by the frequency with which it occurs.

Discrete frequency distribution. For distinct values $x_1,\ldots,x_k$ with frequencies $f_1,\ldots,f_k$ and $N=\sum f_i$,

Continuous frequency distribution. Exactly as for grouped mean deviation, each class is represented by its midpoint $x_i$, and the same discrete-style formulas are applied using these midpoints:

The computational shortcut, extended to grouped data. The identity derived in the previous section extends directly (the derivation is identical, with every $x_i$ replaced by frequency-weighted sums):

— again, "the (frequency-weighted) mean of the squares minus the square of the (frequency-weighted) mean." This is frequently the fastest route to the variance of grouped data, since it needs only two running column totals, $\sum f_i x_i$ and $\sum f_i x_i^2$, computed once each.

Steps to compute (either discrete or continuous, direct method).

1. Find $N=\sum f_i$ and the mean $\bar x = \dfrac1N\sum f_i x_i$ (using class midpoints for continuous data).
2. Find $x_i-\bar x$ (or, for the shortcut identity, simply $x_i^2$) for every distinct value/midpoint.
3. Form the column $f_i(x_i-\bar x)^2$ (or $f_i x_i^2$ for the shortcut).
4. Sum this column, divide by $N$ — and, for the shortcut identity, subtract $\bar x^2$ at the very end.
5. Take the square root of the result to get the standard deviation. …