What is variance in simple words?

Variance is the average of the squared differences from the mean. It shows how much data values spread out.

What does a high variance mean?

A high variance indicates that data points are far from the mean, i.e., more variability.

What is statistical variance used for?

Statistical variance is used to analyze the spread of data, compare distributions, and perform data modeling in research and machine learning.

Is variance always positive?

Yes, because you square the differences, the variance is always non-negative.

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Variance

Variance is a statistical tool used to measure the spread or dispersion of a set of data points around the mean. It shows how much the values deviate from the average, helping us understand data variability. The larger the variance, the more spread out the data is. From basic statistics to complex data science models, analyzing variance is essential in identifying consistency, trends, and outliers.

1.0Variance Meaning in Statistics

In statistics, variance refers to the average of the squared deviations from the mean of a data set. It quantifies how much the numbers differ from the average value.

Mathematically, the statistical variance of a data set tells us whether the data points are tightly grouped or widely scattered.

Formula for Calculating the Variance

For a Population (σ²): $σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2}$

For a Sample (s²): $s^{2} = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2}$

Where:

$x_{i}$ = each value
$μ$ = population mean
$\overset{x}{ˉ}$ = sample mean
N = population size
n = sample size

2.0How to Find the Variance – Step-by-Step

Find the Mean of the data.
Subtract the Mean from each data point (deviation).
Square each deviation.
Sum all squared deviations.
Divide the sum by:

n (for population)
n - 1 (for sample)

3.0Example of Variance (Sample)

Q: Find the variance of the following data: 5, 7, 3, 7, 10

Step 1: Find the Mean ( $\overset{x}{ˉ}$ )

$\overset{x}{ˉ} = \frac{5 + 7 + 3 + 7 + 10}{5} \frac{32}{5} = 6.4$

Step 2: Subtract Mean and Square

$(5 - 6.4)^{2} = 1.96 (7 - 6.4)^{2}$

$= 0.36 (3 - 6.4)^{2}$

$= 11.56 (7 - 6.4)^{2}$

$= 0.36 (10 - 6.4)^{2}$

$= 12.96$

Step 3: Add the Squares

Sum= 1.96+0.36+11.56+0.36+12.96 = 27.2

Step 4: Divide by n - 1

$s^{2} = \frac{27.2}{4} = 6.8$

Answer: Variance = 6.8

4.0Solved Examples of Variance

Example 1: If the variance of a random variable X is 4, then find the variance of the variable Y = 3X + 5.

Solution:
We know that:

$Va r (a X + b) = a^{2} . Va r (X)$

Here, a = 3, b = 5, and Var(X) = 4

$Va r (Y) = 3^{2} .4 = 9 \times 4 = 36$

Example 2: Find the sample variance of the data: 8, 10, 6, 4, 12

Solution:
Step 1: Find the Mean

$\overset{x}{ˉ} = \frac{8 + 10 + 6 + 4 + 12}{5} = \frac{40}{5} = 8$

Step 2: Calculate squared deviations

$(8 - 8)^{2} = 0$

$(10 - 8)^{2} = 4$

$(6 - 8)^{2} = 4$

$(4 - 8)^{2} = 16$

$12 - 8)^{2} = 16$

Sum = 0 + 4 + 4 + 16 + 16 = 40

Step 3: Sample Variance = $s^{2} = \frac{40}{5 - 1} = \frac{40}{4} = 10$

Example 3: If X is a random variable and kk is a constant, then what is Var(X+k)?

A. Var(X)

B. Var(X) + k

C. Var(X) - k

D. $Va r (X) + k^{2}$

Answer: A

Explanation: Adding a constant does not change the variance.

$Va r (X + k) = Va r (X)$

Correct Option: A

Example 4: Two groups of students have the following data:

Group A: $n_{1}$ = 5, mean = 4, variance = 2
Group B: $n_{2}$ = 7, mean = 6, variance = 3

Find the combined variance.

Solution:
Use the formula:

$Combined Variance = \frac{n _{1} σ _{1}^{2} + n _{2} σ _{2}^{2} + \frac{n _{1} n _{2}}{n _{1} + n _{2}} ( x ˉ _{1} - x ˉ _{2} ) ^{2}}{n _{1} + n _{2}} .$

Plug in values:

$= \frac{5 \cdot 2 + 7 \cdot 3 + \frac{5 \cdot 7}{12} ( 4 - 6 ) ^{2}}{12} .$

$= \frac{10 + 21 + \frac{35}{12} \cdot 4}{12}$

$= \frac{31 + \frac{140}{12}}{12} .$

$= \frac{31 + 11.67}{12}$

$= \frac{42.67}{12} \approx 3.56$

Example 5: A frequency distribution is given as:

x	2	4	6	8	10
f	1	2	3	2	2

Find the variance.

Solution:

Step 1: Find the mean $\overset{x}{ˉ}$

$\overset{x}{ˉ} = \frac{\sum f x}{\sum f} = \frac{( 2 ) ( 1 ) + ( 4 ) ( 2 ) + ( 6 ) ( 3 ) + ( 8 ) ( 2 ) + ( 10 ) ( 2 )}{1 + 2 + 3 + 2 + 2}$

$\overset{x}{ˉ} = \frac{2 + 8 + 18 + 16 + 20}{10}$

$\overset{x}{ˉ} = \frac{64}{10} = 6.4$

Step 2: Find $\sum f (x - \overset{x}{ˉ})^{2}$

$2 (2 - 6.4)^{2} = 19.36 \cdot 1 = 19.36$

$(4 - 6.4)^{2} = 5.76 \cdot 2 = 11.52$

$(6 - 6.4)^{2} = 0.16 \cdot 3 = 0.48$

$(8 - 6.4)^{2} = 2.56 \cdot 2 = 5.12$

$(10 - 6.4)^{2} = 12.96 \cdot 2 = 25.92$

Total = 19.36 + 11.52 + 0.48 + 5.12 + 25.92 = 62.4

Step 3: Variance = $\frac{62.4}{10} = 6.24$

5.0Why is Variance Important?

It helps in analyzing data consistency.
Used in risk assessment in finance and insurance.
Essential in machine learning models for feature analysis.
Forms the basis for standard deviation and hypothesis testing.

6.0Difference Between Variance and Standard Deviation

Feature	Variance	Standard Deviation
Definition	Average of squared deviations	Square root of variance
Units	Squared units	Same units as original data
Symbol	$σ^{2}, s^{2}$	$σ, s$
Use Case	Mathematical computation	Easy interpretation

7.0Practice: Find the Variance of the Following Data

Find the variance of: 4, 6, 8, 10
Find the variance of: 12, 15, 18, 21, 24
Find the sample variance of: 9, 7, 5, 3

8.0Sample Question on Variance

Q1: How do I find variance in statistics?

Use the formula: $s^{2} = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2}$

First, find the mean, then compute squared deviations and average them.

Also Read:

Correlation and Regression	Mean Deviation
Median of Grouped Data	Central Tendency

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Variance

Variance is a statistical tool used to measure the spread or dispersion of a set of data points around the mean. It shows how much the values deviate from the average, helping us understand data variability. The larger the variance, the more spread out the data is. From basic statistics to complex data science models, analyzing variance is essential in identifying consistency, trends, and outliers.

1.0Variance Meaning in Statistics

In statistics, variance refers to the average of the squared deviations from the mean of a data set. It quantifies how much the numbers differ from the average value.

Mathematically, the statistical variance of a data set tells us whether the data points are tightly grouped or widely scattered.

Formula for Calculating the Variance

For a Population (σ²): $σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2}$

For a Sample (s²): $s^{2} = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2}$

Where:

$x_{i}$ = each value
$μ$ = population mean
$\overset{x}{ˉ}$ = sample mean
N = population size
n = sample size

2.0How to Find the Variance – Step-by-Step

Find the Mean of the data.
Subtract the Mean from each data point (deviation).
Square each deviation.
Sum all squared deviations.
Divide the sum by:

n (for population)
n - 1 (for sample)

3.0Example of Variance (Sample)

Q: Find the variance of the following data: 5, 7, 3, 7, 10

Step 1: Find the Mean ( $\overset{x}{ˉ}$ )

$\overset{x}{ˉ} = \frac{5 + 7 + 3 + 7 + 10}{5} \frac{32}{5} = 6.4$

Step 2: Subtract Mean and Square

$(5 - 6.4)^{2} = 1.96 (7 - 6.4)^{2}$

$= 0.36 (3 - 6.4)^{2}$

$= 11.56 (7 - 6.4)^{2}$

$= 0.36 (10 - 6.4)^{2}$

$= 12.96$

Step 3: Add the Squares

Sum= 1.96+0.36+11.56+0.36+12.96 = 27.2

Step 4: Divide by n - 1

$s^{2} = \frac{27.2}{4} = 6.8$

Answer: Variance = 6.8

4.0Solved Examples of Variance

Example 1: If the variance of a random variable X is 4, then find the variance of the variable Y = 3X + 5.

Solution:
We know that:

$Va r (a X + b) = a^{2} . Va r (X)$

Here, a = 3, b = 5, and Var(X) = 4

$Va r (Y) = 3^{2} .4 = 9 \times 4 = 36$

Example 2: Find the sample variance of the data: 8, 10, 6, 4, 12

Solution:
Step 1: Find the Mean

$\overset{x}{ˉ} = \frac{8 + 10 + 6 + 4 + 12}{5} = \frac{40}{5} = 8$

Step 2: Calculate squared deviations

$(8 - 8)^{2} = 0$

$(10 - 8)^{2} = 4$

$(6 - 8)^{2} = 4$

$(4 - 8)^{2} = 16$

$12 - 8)^{2} = 16$

Sum = 0 + 4 + 4 + 16 + 16 = 40

Step 3: Sample Variance = $s^{2} = \frac{40}{5 - 1} = \frac{40}{4} = 10$

Example 3: If X is a random variable and kk is a constant, then what is Var(X+k)?

A. Var(X)

B. Var(X) + k

C. Var(X) - k

D. $Va r (X) + k^{2}$

Answer: A

Explanation: Adding a constant does not change the variance.

$Va r (X + k) = Va r (X)$

Correct Option: A

Example 4: Two groups of students have the following data:

Group A: $n_{1}$ = 5, mean = 4, variance = 2
Group B: $n_{2}$ = 7, mean = 6, variance = 3

Find the combined variance.

Solution:
Use the formula:

$Combined Variance = \frac{n _{1} σ _{1}^{2} + n _{2} σ _{2}^{2} + \frac{n _{1} n _{2}}{n _{1} + n _{2}} ( x ˉ _{1} - x ˉ _{2} ) ^{2}}{n _{1} + n _{2}} .$

Plug in values:

$= \frac{5 \cdot 2 + 7 \cdot 3 + \frac{5 \cdot 7}{12} ( 4 - 6 ) ^{2}}{12} .$

$= \frac{10 + 21 + \frac{35}{12} \cdot 4}{12}$

$= \frac{31 + \frac{140}{12}}{12} .$

$= \frac{31 + 11.67}{12}$

$= \frac{42.67}{12} \approx 3.56$

Example 5: A frequency distribution is given as:

x	2	4	6	8	10
f	1	2	3	2	2

Find the variance.

Solution:

Step 1: Find the mean $\overset{x}{ˉ}$

$\overset{x}{ˉ} = \frac{\sum f x}{\sum f} = \frac{( 2 ) ( 1 ) + ( 4 ) ( 2 ) + ( 6 ) ( 3 ) + ( 8 ) ( 2 ) + ( 10 ) ( 2 )}{1 + 2 + 3 + 2 + 2}$

$\overset{x}{ˉ} = \frac{2 + 8 + 18 + 16 + 20}{10}$

$\overset{x}{ˉ} = \frac{64}{10} = 6.4$

Step 2: Find $\sum f (x - \overset{x}{ˉ})^{2}$

$2 (2 - 6.4)^{2} = 19.36 \cdot 1 = 19.36$

$(4 - 6.4)^{2} = 5.76 \cdot 2 = 11.52$

$(6 - 6.4)^{2} = 0.16 \cdot 3 = 0.48$

$(8 - 6.4)^{2} = 2.56 \cdot 2 = 5.12$

$(10 - 6.4)^{2} = 12.96 \cdot 2 = 25.92$

Total = 19.36 + 11.52 + 0.48 + 5.12 + 25.92 = 62.4

Step 3: Variance = $\frac{62.4}{10} = 6.24$

5.0Why is Variance Important?

It helps in analyzing data consistency.
Used in risk assessment in finance and insurance.
Essential in machine learning models for feature analysis.
Forms the basis for standard deviation and hypothesis testing.

6.0Difference Between Variance and Standard Deviation

Feature	Variance	Standard Deviation
Definition	Average of squared deviations	Square root of variance
Units	Squared units	Same units as original data
Symbol	$σ^{2}, s^{2}$	$σ, s$
Use Case	Mathematical computation	Easy interpretation

7.0Practice: Find the Variance of the Following Data

Find the variance of: 4, 6, 8, 10
Find the variance of: 12, 15, 18, 21, 24
Find the sample variance of: 9, 7, 5, 3

8.0Sample Question on Variance

Q1: How do I find variance in statistics?

Use the formula: $s^{2} = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2}$

First, find the mean, then compute squared deviations and average them.

Also Read:

Correlation and Regression	Mean Deviation
Median of Grouped Data	Central Tendency

Frequently Asked Questions

What is variance in simple words?

What does a high variance mean?

What is statistical variance used for?

Is variance always positive?

Frequently Asked Questions

What is variance in simple words?

What does a high variance mean?

What is statistical variance used for?

Is variance always positive?

Join ALLEN!

Variance

1.0Variance Meaning in Statistics

Formula for Calculating the Variance

2.0How to Find the Variance – Step-by-Step

3.0Example of Variance (Sample)

4.0Solved Examples of Variance

5.0Why is Variance Important?

6.0Difference Between Variance and Standard Deviation

7.0Practice: Find the Variance of the Following Data

8.0Sample Question on Variance

Table of Contents

Join ALLEN!

Variance

1.0Variance Meaning in Statistics

Formula for Calculating the Variance

2.0How to Find the Variance – Step-by-Step

3.0Example of Variance (Sample)

4.0Solved Examples of Variance

5.0Why is Variance Important?

6.0Difference Between Variance and Standard Deviation

7.0Practice: Find the Variance of the Following Data

8.0Sample Question on Variance

Table of Contents