# Variance Calculator

The variance calculator is used to find the sample and population variance, standard deviation, and mean of a set of data.

Result:

 Variance: Standard Deviation: Mean:

## How to Use the Variance Calculator?

1. Firstly, enter the set of values or the data with commas or spaces separated. For example 3 5 7 8 10 or 3, 5, 7, 8, 10.
2. Select the data type whether it's sample or population.
3. Lastly, press the 'Calculate' button.
4. As a result, the tool shows the variance result, standard deviation, and mean.
5. Press the 'Reset' button to clear all the input and output fields.

### What is Variance?

The variance is a statistical measure representing the dispersion or spread in a set of data points. It measures how far each element in a data set is from the mean (average) and thus from every other element in a data set. The high variance indicates the more spread out the data points around the mean. It's widely used in statistics, finance, quality control, and many more fields.

### Variance Formula

The formula for calculating variance is:

##### Sample Variance:
S2 =
 ∑ ( xi - x̄ )2 n - 1

Where,
s2 = Sample variance,
xi = Value of ith element,
x̄ = Mean value of all elements,
n = Number of elements.

##### Population Variance:
σ 2 =
 ∑ ( xi - μ )2 N

Where,
σ2 = Population variance,
xi = Value of ith element,
μ = Mean value of all elements,
N = Number of elements.

### How to Calculate the Variance?

To understand the calculation, let's take an example.

##### Example:

Calculate the sample variance for the data set of { 46 69 32 60 52 41 }.

##### Solution:

1. Firstly, calculate the mean (average) of the dataset.

Mean (x̄) = (46 + 69 + 32 + 60 + 52 + 41) / 6 = 50.

2. Now subtract the mean from each data point to get the deviation from the mean.

ScoreDeviation from the Mean
4646 - 50 = -4
6969 - 50 = 19
3232 - 50 = -18
6060 - 50 = 10
5252 - 50 = 2
4141 - 50 = -9

3. Square the result.

Squared Deviation
(-4)2 = 16
(19)2 = 361
(-18)2 = 324
(10)2 = 100
(2)2 = 4
(-9)2 = 81

4. Sum all the squared differences and divide it by the number of data points minus one. That is (n - 1) = (6 - 1) = 5.

Sample Variance (s2) = (16 + 361 + 324 + 100 + 4 + 81) / 5 = 886/5 = 177.22.

Similarly, you can find the population variance using the same method.