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 i
^{th}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.

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

**Further Reading**

It should be noted that the formulas for **population variance** and **sample variance** are different. The following is a tutorial is about sample variance.