# Population Variance Formula and Calculation by Hand

This tutorial shows the formula for population variance and the steps for calculating population variance by hand.

## Formula

Population variance is the measure of the variability of a population. The following is the formula for population variance.

$\sigma^2=\frac{\sum_{i=1}^N (x_i-\mu)^2}{N}$

where,

• $$N$$ is the number of data points in the population.
• $$x_i$$ is the ith data point.
• $$\mu$$ is the mean of the population.

## Population vs. Sample Data

The following is the population of a set of data. It has 11 observations, whereas the sample has 6. Typically, sample data is a subset of the population.

## Calculating Population Variance

### Step 1

The following is to calculate the population mean of $$\mu$$, which is 79.64.

$$\mu = (80+90+81+…+77+89)/11=79.64$$

### Step 2

The following calculates the population variance.

$$\sigma^2=\frac{\sum_{i=1}^N (x_i-\mu)^2}{N} = \frac{(80-79.64)^2+(90-79.640^2+…+(89-79.64)^2}{11}=46.23$$