How variance is calculated for sample and population data? Explained with solution

In descriptive statistics, there are various types that are used to calculate the nature and spread of the data values. The variance is one of them. It is frequently used to measure the spread of sample and population data values. 

The standard deviation is another type of statistics that is used to measure the distribution of data values from the mean. It is more accurate than the variance as it is measured in single units while the variance is measured in squared units.

In this post, we will learn the basics of variance along with its types, formulas, and examples with solutions.

What is the variance?

The variance is the subtype of descriptive statistics that is used to measure the variation of data values from the expected value. It depends on the nature of the data sets such as population or sample data values used in this subtype of descriptive statistics.

The population set is the set that is taken from the whole set of observations taken from the whole observations under consideration.

While the sample is taken from the whole set under consideration. It is taken to ease up the calculations and avoid the larger set of observations to find the estimated result of the whole population.

Types of variance

There are further two types of variance one is the sample variance while the other is the population variance.

1. Population variance

The data set taken from the population and to measure the spread of data values from the average value of the whole set of data is known as population variance. It is calculated by taking the difference of the data values from the population mean.

And after that take the square of the deviation to make them positive and add all the squared deviations to evaluate the sum of squares and in the end, find the quotient of the sum of the squares and the total number of population data

2. Sample variance

The sample data set taken from the population and to measure the spread of data values from the mean value of the sample set of data is known as sample variance. It is calculated by taking the difference of the data values from the sample mean.

And after that take the square of the deviation to make them positive and add all the squared deviations to evaluate the sum of squares. In the end, find the quotient of the sum of the squares and the total number of sample data.

Formulas of variance

Here are the general formulas to calculate the variance of sample and population data.

Population variance formula

The formula for finding the population variance is:

σ² = [∑ (y_i – μ)²/N]

Sample variance formula

The formula for finding the sample variance is:

s² = [∑ (y_i – ȳ)²/N]

A variance calculator can be used to find the variance of sample and population data with just a single click according to the above formulas.

How to calculate the variance of sample and population data?

The variance can be calculated by using the formulas of sample and population. Let’s take a few examples of calculating the variance of sample and population data manually.

Example 1: for calculating the variance from sample data

Evaluate the given sample data values to calculate the sample variance.

5, 7, 10, 11, 14, 16, 17, 23, 25, 27, 28, 33

Solution

Step 1: First of all, add all the sample observations and divide them by the total number of observations.

Sum = 5 + 7 + 10 + 11 + 14 + 16 + 17 + 23 + 25 + 27 + 28 + 33

Sum = 216

Total number of observation = N = 12

Sample Mean = ȳ = 216/12 = 108/6 = 54/3

Sample Mean = ȳ = 18

Step 2: Now subtract the given observations from the sample mean (ȳ) and take the square of the outputs of each subtraction to make them positive.

Data values	yi – ȳ	(yi – ȳ)2
5	5 – 18 = -13	(-13)2 = 169
7	7 – 18 = -11	(-11)2 = 121
10	10 – 18 = -8	(-8)2 = 64
11	11 – 18 = -7	(-7)2 = 49
14	14 – 18 = -4	(-4)2 = 16
16	16 – 18 = -2	(-2)2 = 4
17	17 – 18 = -1	(-1)2 = 1
23	23 – 18 = 5	(5)2 = 25
25	25 – 18 = 7	(7)2 = 49
27	27 – 18 = 9	(9)2 = 81
28	28 – 18 = 10	(10)2 = 100
33	33 – 18 = 15	(15)2 = 225

Step 3: Now calculate the sum of squares by adding the above-squared deviations.  

∑ (y_i – ȳ)² = 169 + 121 + 64 + 49 + 16 + 4 + 1 + 25 + 49 + 81 + 100 + 225

∑ (y_i – ȳ)² = 904

Step 4: Now take the quotient of the sum of squared differences and the total number of observations decreased by one. This will give you the result of the variance.

∑ (y_i – ȳ)² / N – 1 = 904 / 12 – 1

∑ (y_i – ȳ)² / N – 1 = 904 / 11

∑ (y_i – ȳ)² / N – 1 = 82.18

Example 2: for calculating the variance from population data

Evaluate the given sample data values to calculate the population variance. 

5, 7, 9, 12, 15, 17, 19, 22, 25, 27, 29

Solution

Step 1: First of all, add all the population observations and divide them by the total number of observations.

Sum = 5 + 7 + 9 + 12 + 15 + 17 + 19 + 22 + 25 + 27 + 29

Sum = 187

Total number of observation = N = 11

Population Mean = µ = 187/11

Population Mean = µ = 17

Step 2: Now subtract the given observations from the population mean (µ) and take the square of the outputs of each subtraction to make them positive.

Data values	yi – µ	(yi – µ)2
5	5 – 17 = -12	(-12)2 = 144
7	7 – 17 = -10	(-10)2 = 100
9	9 – 17 = -8	(-8)2 = 64
12	12 – 17 = -5	(-5)2 = 25
15	15 – 17 = -2	(-2)2 = 4
17	17 – 17 = 0	(0)2 = 0
19	19 – 17 = 2	(2)2 = 4
22	22 – 17 = 5	(5)2 = 25
25	25 – 17 = 8	(8)2 = 64
27	27 – 17 = 10	(10)2 = 100
29	29 – 17 = 12	(12)2 = 144

Step 3: Now calculate the sum of squares by adding the above-squared deviations. 

∑ (y_i – µ)² = 144 + 100 + 64 + 25 + 4 + 0 + 4 + 25 + 64 + 100 + 144

∑ (y_i – µ)² = 674

Step 4: Now take the quotient of the sum of squared differences and the total number of observations. This will give you the result of the variance.

∑ (y_i – µ)² / N = 674 / 11

∑ (y_i – µ)² / N = 61.27

Wrap up

Now you can get all the basics for calculating the variance by using the sample and population data values. In this lesson, we have discussed all the basic concepts that are necessary to solve the sample and population variance problems.

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