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): ${\sigma }^2=\frac{1}{N}{\sum }_{i=1}^N(x_i-\mu {)}^2$

For a Sample (s²): $s^2=\frac{1}{n-1}\sum (x_i-\bar{x}{)}^2$

Where:

• $x_i$ = each value
• $\mu$= population mean
• $\bar{x}$ = sample mean
• N = population size
• n = sample size

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

1. Find the Mean of the data.
2. Subtract the Mean from each data point (deviation).
3. Square each deviation.
4. Sum all squared deviations.
5. 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 ( $\bar{x}$ )

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

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:

$Var(aX+b)=a^2.Var(X)$

Here, a = 3, b = 5, and Var(X) = 4 

$Var(Y)=3^2.4=9\times 4=36$

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

Solution:
Step 1: Find the Mean

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

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. $Var(X)+k^2$

Answer: A

Explanation: Adding a constant does not change the variance.

$Var(X+k)=Var(X)$

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:

$\text{Combined Variance}=\frac{n_1{\sigma }_1^2+n_2{\sigma }_2^2+\frac{n_1{n}_2}{n_1+n_2}{\left({\bar{x}}_1-{\bar{x}}_2\right)}^2}{n_1+n_2}.$

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: 

Find the variance.

Solution:

Step 1: Find the mean $\bar{x}$

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

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

$\bar{x}=\frac{64}{10}=6.4$ 

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

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

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

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

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

$(10-6.4{)}^2=12.96\cdot 2=25.92$ 

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

7.0Practice: Find the Variance of the Following Data

1. Find the variance of: 4, 6, 8, 10
2. Find the variance of: 12, 15, 18, 21, 24
3. 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: $s^2=\frac{1}{n-1}\sum (x_i-\bar{x}{)}^2$

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

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