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Variance in Statistics: Definition, Formula, and Examples

Variance in Statistics Definition Formula, and Examples
Variance in Statistics Definition Formula, and Examples

Variance is a topic of statistics, used to evaluate the spread of numbers in a set of data relating to the mean of the data. To measure the squeezed or stretched distribution, we use variance.

The dispersion of the observation of the data from the mean is dependent on the value of the variance. More value of the variance causes more scatter from its mean. While the low value of the variance concludes less scattered from the mean. 

In this article, we will learn about the definition and formulas of the variance with a lot of examples.

What is a variance?

According to Wikipedia, a variance is the expectation of the squared deviation of a random variable for its population mean or sample. Variance is taken for two kinds of data, population data, and sample data. 

If the variance is used in population data, then it is known as population variance. While if the variance is used for sample data, then it is known as sample variance. Variance can also be stated as the expected value of the squared differences of the number of observations and from the mean.

The main work of the variance is to measure the distance for each observation from the mean. The distance decides whether the number is closer to the mean or far. It is usually denoted by sigma square for the population variance written as σ2. For sample variance, it is denoted by s2. We can also write variance such as var(x). 

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Formulas of the variance

The equations used for population variance and sample variance are given below. 

  • Variance of the population data = σ2 = Σ (x – µ)2/N

In the above formula, x is the observations of the population data, µ is the population mean, N is the total number of observations, and σ2 is the population variance.

  1. Variance of the sample data = s2 = Σ (x – )2/n – 1

In the above equation, x is the observations of the sample data, µ is the sample mean, n is the total number of observations, and s2 is the sample variance.

To get the results according to the above formulas, use a variance calculator. This tool will provide you variance as well standard deviation of the given data.

How to find variance?

By using formulas, we can easily find the variance for the population data or sample data. Let us take some examples.

Example 1: For population variance

Find the population variance of the given population data, 11, 13, 14, 8, 12, 16, and 10?

Solution 

Step 1: Write the given population data.

11, 13, 14, 8, 12, 16, 10

Step 2: Now find the sample mean “µ” of the given data.

Sum of population data = 11 +13 + 14 + 8 + 12 + 16 + 10

Sum of population data = 84

Total number of observations = N = 7

Sample mean = µ = Sum of sample data / Total number of observations

Sample mean = µ = 84/7 = 12

Step 3: Now find the sum of squares of the population data.

Population data (x) x – µ (x – µ)2
11 11 – 12 = -1 (-1)2 = 1
13 13 – 12 = 1 (1)2 = 1
14 14 – 12 = 2 (2)2 = 4
8 8 – 12 = -4 (-4)2 = 16
12 12 – 12 = 0 (0)2 = 0
16 16 – 12 = 4 (4)2 = 16
10 10 – 12 = -2 (-2)2 = 4
Σ x = 84 Σ (x – µ)2 = 42
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Step 4: Now take the general population variance formula for population data.

Variance of the population data = σ2 = Σ (x – µ)2/N

Step 5: Put the sum of square and number of observations in the above formula.

σ2 = Σ (x – µ)2/N

σ2 = 42/7

σ2 = 6

Example 2: For sample variance

Find the sample variance of the given sample data, 3, 6, 12, 16, 21, 24, and 29?

Solution 

Step 1: Write the given sample data.

3, 6, 12, 16, 21, 24, 29

Step 2: Now evaluate the mean of the given data.

Sum of sample data = 3 + 6 + 12 + 16 + 21 + 24 + 29

Sum of sample data = 111

Total number of observations = n = 7

Sample mean = Sum of sample data / Total number of observations

Sample mean = 111/7 = 15.8571

Step 3: Now find the square of difference of sample observation and sample mean.

Sample data (x) x – (x – )2
3 3 – 15.8571 = -12.8571 (-12.8571)2 = 165.3050
6 6 – 15.8571= -9.8571 (-9.8571)2 = 97.1624
12 12 – 15.8571 = -3.8571 (-3.8571)2 = 14.8772
16 16 – 15.8571 = 0.1429 (0.1429)2 = 0.0204
21 21 – 15.8571= 5.1429 (5.1429)2 = 26.4494
24 24 – 15.8571 = 8.1429 (8.1429)2 = 66.3068
29 29 – 15.8571 = 13.1429 (13.1429)2 = 172.7358
Σ x = 111 Σ (x – )2 = 542.8571

 

Step 4: Now take the general standard deviation formula for sample data.

Variance of the sample data = s2 = Σ (x – )2/n – 1

Step 5: Put all the calculated values in the above formula.

s2 = Σ (x – )2/n – 1

s2 = 542.8571/7 – 1 

s2 = 542.8571/6

s2 = 90.4762

Example 3

Find the sample variance of the given sample data, 13, 16, 22, 26, 31, and 39?

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Solution 

Step 1: Write the given sample data.

13, 16, 22, 26, 31, 39

Step 2: Now evaluate the mean of the given data.

Sum of sample data = 13 + 16 + 22 + 26 + 31 + 39

Sum of sample data = 147

Total number of observations = n = 6

Sample mean = Sum of sample data / Total number of observations

Sample mean = 147/7 = 24.5

Step 3: Now find the square of difference of sample observation and sample mean.

Sample data (x) x – (x – )2
13 13 – 24.5= -11.5 (-11.5)2 = 132.25
16 16 – 24.5= -8.5 (-8.5)2 = 72.25
22 22 – 24.5= -2.5 (-2.5)2 = 6.25
26 26 – 24.5= 1.5 (1.5)2 = 2.25
31 31 – 24.5= 6.5 (6.5)2 = 42.25
39 39 – 24.5= 14.5 (14.5)2 = 210.25
Σ x = 147 Σ (x – )2 = 465.5

 

Step 4: Now take the general standard deviation formula for sample data.

Variance of the sample data = s2 = Σ (x – )2/n – 1

Step 5: Put all the calculated values in the above formula.

s2 = Σ (x – )2/n – 1

s2 = 465.5/6 – 1 

s2 = 465.5/5

s2 = 93.1

Summary

In this article, we have learned about variance and its formulas. By practising the above examples, you can easily solve the variance. You can easily find the sample variance and population variance by using the variance formulas.

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