Standard Deviation of a Discrete Frequency Distribution

Standard Deviation of a Discrete Frequency Distribution

When data is presented as a discrete frequency distribution, each value $x_i$ occurs with a frequency $f_i$. The total number of observations is $N = \sum_{i=1}^{n} f_i$, and the mean is $\bar{x} = \frac{1}{N} \sum_{i=1}^{n} f_i x_i$.

The standard deviation for such a distribution is defined as:

This is the natural extension of the standard deviation formula for ungrouped data — each squared deviation $(x_i - \bar{x})^2$ is weighted by its frequency $f_i$, and the sum is divided by the total frequency $N$ instead of the number of distinct values.

The variance $\sigma^2$ is simply the square of this expression:

Worked Example

Example 9. Find the variance and standard deviation for the following data:

Solution. The calculations are organised in a table. First compute $f_i x_i$ for each row, then find the mean $\bar{x}$. Next compute the deviations $x_i - \bar{x}$, square them, multiply by $f_i$, and sum.

From the table:

The mean is:

The variance is:

The standard deviation is:

$\sigma = \sqrt{45.8} \approx 6.77$ …