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Variance Calculator



How does Variance Calculator work?

  • Select the population or sample variance
  • Input the comma-separated values of the data set.
  • Hit The Calculate Button.
  • Use the Reset button to calculate new values.




Variance calculator finds the variance of sample and population data values with steps. The sample variance calculator also provides the standard deviation, mean, and sum of squares.


What is a variance?


In statistics, the variance is the measure of dispersion i.e., measure the spread of data values from the expected value. It is the expectation of the squared deviation of a random variable from its sample or population mean.

It is denoted by s2 and σ2 for sample and population data respectively.


Variance formula


The formula for the sample variance is:

Sample variance formula

The formula for the population variance is:

population variance formula


How to calculate the variance?


Below are a few examples of variance solved by our mean and variance calculator.


Example 1: For sample variance


Find the variance of the given sample data.

2, 4, 7, 12, 15

Solution

Step 1: Calculate the sample mean of the given data.

Sample mean = x̅ = Σx/n

Sample mean = x̅ = [2 + 4 + 7 + 12 + 15]/5

Mean = x̅ = 40/5

Mean = x̅ = 8

Step 2: Now measure the dispersion and squares of deviation.

Data values (x)xi - x̅(xi - x̅)2
22 – 8 = -6(-6)2 = 36
44 – 8 = -4(-4)2 = 16
77 – 8 = -1(-1)2 = 1
1212 – 8 = 4(4)2 = 16
1515 – 8 = 7(7)2 = 49

Step 3: Find the summation of the squared deviations.

Σ(xi - x̅)2 = 36 + 16 + 1 + 16 + 49

Σ(xi - x̅)2 =  118

Step 4: Now divide the sum of squares by n-1.

Σ(xi - x̅)2/n-1 = 118/5-1

Σ(xi - x̅)2 / n-1 = 118/4

Σ(xi - x̅)2 / n-1 = 29.5


Example 2: For population variance


Find the variance of the given population data.

1, 14, 19, 25, 26, 35

Solution

Step 1: Calculate the population mean of the given data.

Population mean = µ = Σx/n

= [1 + 14 + 19 + 25 + 26 + 35]/6

= 120/6

= 20

Step 2: Now measure the dispersion and squares of deviation.

Data values (x)

xi - µ

(xi - µ)2

1

1 – 20 = -19

(-19)2 = 361

14

14 – 20 = -6

(-6)2 = 36

19

19 – 20 = 1

(1)2 = 1

25

25 – 20 = 5

(5)2 = 25

26

26 – 20 = 6

(6)2 = 36

35

35 – 20 = 15

(15)2 = 225

Step 3: Find the summation of the squared deviations.

Σ(xi - µ)2 = 361 + 36 + 1 + 25 + 36 + 225

Σ(xi - µ)2 =  684

Step 4: Now divide the sum of squares by n.

Σ(xi - µ)2/n = 684/6

Σ(xi - µ)2/n = 136.8


References


Wikimedia Foundation. (2022, June 20). What is a variance? Wikipedia.