# Sample Variance Formula and Calculation by Hand

Sample variance is the measure of the variability in a given sample. A sample is a set of observations that are a subset of a population.

## Sample Variance Formula

The following is the formula for sample variance.

$s^2=\frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n-1}$

where,

• $$n$$ is the number of observations in the sample.
• $$x_i$$ is the ith observation.
• $$\bar{x}$$ is the mean of the obervations.

## Data Example

The following is a sample of 6 students with math scores.

## Calculating sample variance by hand

### Step 1: the mean

The following calculates the mean of the sample. The numerator is the sum of all observations, and the denominator is the number of the sample.

$$\bar{x} = \frac{80+90+81+82+78+71}{6} = 80.33$$

### Step 2: sample variance

The following is to calculate the sample variance. Note that, the denominator is 6-1, rather than 6.

$$s^2=\frac{(80-80.33)^2+(90-80.33)^2+(81-80.33)^2+(82-80.33)^2+(78-80.33)^2+(71-80.33)^2}{6-1} = 37.87$$