Variance and standard deviation for grouped data

The formula for variance and standard deviation for grouped data is very similar to the one for ungrouped data. Below, we show the formula for ungrouped data and grouped data.

Variance for grouped data

Variance Population Sample
Ungrouped data $$ Variance = \frac{Σ(x - µ)^{2}} {N} $$ $$ Variance = \frac{Σ(x - \bar x)^{2}} {n - 1} $$
Grouped data $$ Variance = \frac{Σf(m - µ)^{2}} {N} $$ $$ Variance = \frac{Σf(m - \bar x)^{2}} {n - 1} $$

For grouped data, m is the midpoint of a class and f is the frequency of a class.

Did you notice the similarity? Recall that a class is a group of values such as 1-3 containing 1, 2, and 3.

For grouped data, we use the midpoint of a class instead of x or the exact value. Then, just like the mean, we multiply the numerator by f or the frequency before taking the sum.

To get the standard deviation, just take the square root of the variance. By the same token, to get the variance, just raise the standard deviation to the power of 2.

Standard deviation for grouped data

Standard deviation Population Sample
Ungrouped data $$ σ = \sqrt {\frac{Σ(x - µ)^{2}} {N} } $$ $$ s = \sqrt {\frac{Σ(x - \bar x)^{2}} {n - 1} } $$
Grouped data $$ σ = \sqrt {\frac{Σf(m - µ)^{2}} {N} } $$ $$ s = \sqrt {\frac{Σf(m - \bar x)^{2}} {n - 1} } $$

Let s represent the sample standard deviation, then  s² is the sample variance.

$$ s = \sqrt {s^{2}}$$

Let σ represent the population standard deviation, then σ² is the population variance.

$$ σ = \sqrt {σ^{2}}$$

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