Variance and Standard Deviation Calculator

The Variance and Standard Deviation Calculator is a free online tool used to calculate the variance and standard deviation of a set of numbers.

In statistics, variance is the average of the sum of squares of the differences between the actual value and the expected value (the mean). Usually denoted by σ². It is used to describe the degree of dispersion of a random variable. It occupies a very important position in statistics.

The expected value has a population mean or a sample mean, and different expected values have different calculation formulas.

The formula for population variance is

Among them, σ² is the population variance, μ is the population mean, N is the total number of data.

The formula for sample variance is

Where S² is the sample variance, x is the sample mean, n is the total number of data.

There are several steps to calculating the variance:

First, find the sum of a set of numbers.

Second, calculate the arithmetic mean of the data set.

Third, calculate the squared difference of each data value from the mean.

Fourth, add all the squared differences together.

Fifth, calculate the population variance or sample variance, divide by N or n – 1.

For example, a sample data [3, 5, 6, 8, 9, 5], find the variance.

Calculate the sum of the data

3 + 5 + 6 + 8 + 9 + 5 = 36

Calculate the average.

366 = 6

calculate the squared difference

(3 – 6)² = 9

(5 – 6)² = 1

(6 – 6)² = 0

(8 – 6)² = 4

(9 – 6)² = 9

(5 – 6)² = 1

Add all the squared differences together.

9 + 1 + 0 + 4 + 9 + 1 = 24

Calculation results

The population variance is

246 = 4

The sample variance is

246 – 1 = 4.8

The square root of the variance is called the standard deviation, represented by σ. The standard deviation can reflect the dispersion of a data set. Two sets of data with the same mean do not necessarily have the same standard deviation.

Similarly, the standard deviation is also divided into population standard deviation and sample standard deviation.

The population standard deviation formula is

The sample standard deviation formula is

The standard deviation is the square root of the variance, so in addition to all the steps of computing the variance, there is an extra step, which is computing the square root of the variance.

In the above example, the data set is [3, 5, 6, 8, 9, 5]. We have calculated their variance. The population variance is 4 and the sample variance is 4.8.

So, the population standard deviation is

√4 = 2

The sample standard deviation is

√4.8 = 2.19

Using the calculator is simple, follow these 2 steps:

In the above example, just enter 3, 5, 6, 8, 9, 5 in the input box and then click calculate button. As shown in the figure, the calculation results are consistent with the above manual calculation.