# Variance Calculator

Variance calculator and sample variance calculator with a step-by-step solution and APA format.

Enter comma , space or Enter after each data value.
The tool doesn't count empty cells or non-numeric cells.

## Variance Calculator

Use the variance calculator to compute both sample variance and population variance, complete with a step-by-step solution, and then present the results in APA format.

### What is Variance?

Variance is a parameter that measures the variability of data. It represents the average of squared differences between each value and the mean.

#### Population variance formula

To calculate the population variance, you need the entire dataset.

$\mathrm{Var\left(x\right)}={\sigma }^{2}=\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n}$

### What Is Sample Variance? (S2)

When you do not have data for the entire population, you calculate the sample variance from the sampled data. Unlike population variance, when calculating the sample variance, you divide by (n - 1); in this case, the resulting statistic is unbiased.

#### Sample variance formula

Following the sample variance formula

${S}^{2}=\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n-1}$

### What statistic should I use, variance or sample variance?

Use the variance (σ²) if the data contains the entire population, otherwise use the sample variance (S²).

### What is the sample variance

Usually, you don't have access to the entire population's data because it can be costly to gather all the data or may damage the sample. In such cases, you calculate the sample variance based on a random sample of the data.

When calculating the sample variance, you use a sample average (x̄) instead of the population average (µ). The sample average is a bit closer to the center of the sample than the population average. As a result, if you were to divide by n, on average, the sample average would be greater than the population variance. Dividing by (n-1) will correct the biased estimation of the variance, and partially correct the biased estimation of the variance (Bessel's correction). The sample variance is still biased, but this correction makes it the best simple formula.

#### Simulation Example

Normal(µ=10, σ=4) distribution with a sample size of n=20, with 1,000,000 repetitions.
The averages of the statistics:

StatisticDividing by (n-1)Dividing by n
Standard Deviation3.9583.849
Variance16.00815.208

For an unbiased statistic, we expect to get a standard deviation of 4 and a variance of 16.
You may notice that dividing by (n-1) yields better results than dividing by n. The result for the variance is not biased; it is very close to 16, while the result for the standard deviation is biased.

### Glossary

n - Sample size, the number of values.
Mean - Average.
S - Sample standard deviation, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Standard Deviation.
S2 - Sample variance, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Variance.
σ - Population standard deviation, if the list of values you entered is the entire population, this is the exact Standard deviation.
σ2 - Population variance, if the list of values you entered is the entire population, this is the exact Variance.