Variance: Overview, Questions, Preparation

Variance is defined as the expected value of a random variable’s square deviation from its mean value when expressed in probability theory and statistics. In simpler words, it measures how far apart a given set of numbers is from its average value. 

As a concept, variance finds its application in statistics when performing statistical inference, descriptive statistics, hypotheses testing, or even Monte Carlo sampling alongside science. Statistical analysis is performed on data sets. 

Formula - Variance 

Considering that we are aware of the fact that variance is the square of standard deviation, we can say that - 

Variance = (Standard deviation)²= σ²

From this, the 2 corresponding formulas  - 

Here; x = value of observations
μ = Population mean of values 
n = Number of Observations 
x = Sample mean 
N = total number of observations in a population 

Properties of Variance 

The variance - Var(x) of a random variable known as ‘X’ consists of the following properties - 

1. Var (x+a) = Var(x); a is constant 
2. V(aX) = a² V(X) ; here a is constant 
3. Var (aX+b) = a².Var(X) ; here both a and b are constant 
4. In case of independent variables; 

Var (X₁+X₂+X₃…..) = Var (X₁) + Var (X₂) 

Calculation for Variance 

To calculate variance, you need to follow the below-mentioned steps - 

1. Find the mean by calculating the average of a given number set.
2. For each number in the data set, subtract the mean and then square the result. 
3. Find out the average of squared differences. 

As a topic, Variance comes under the Statistics and Probability section in Class 11th and carries a total weightage of 15 marks. The students will learn about the process of calculating variance for grouped and ungrouped data apart from analysing frequency distributions with equal mean but different variances. 

1: Find the variance of the following numbers - 

4, 15, 10, 7, 6, 12, 18, 5, 8, 9

Solution:

Mean = 4+15+10+7+6+12+18+5+8+9/ 10 = 94/10 = 9.4 

       σ² = ∑(X−μ)²/N = 184.87/10 = 18.487

2: Find population variance for the following - 14, 9, 21, 15, 8

Solution:

Mean = 14+9+21+15+8/5 = 67/5 = 13.4

  σ² = ∑(X−μ)²/N = 96.6/5 = 19.32

3: Find variance when heights of 5 dogs are given as follows - 

600mm, 100mm, 700mm, 350mm, 400mm 

Solution:

Mean = 600+100+700+ 350+400/5 = 2150/5 = 430

= (230)2 + (-330)2+ (330)2+ (80)2+ (30)2     = 52900 - 108900 + 108900+ 6400+900

Variance : 56,000+ 108900+ 6400+900/5 = 172200/5 =34,440

Q; What is variance used for in statistical analysis?

Q: What is the symbol used to represent variance? 

Q: What is the standard deviation? 

Q: What is the formula for calculating variance? 

Q: How are the standard deviation and variance related?