Variance and standard deviation: Class 11 practice
Variance σ² = Σ(x − x̄)²/n = (Σx²/n) − x̄². Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Mean
- Squares
The idea behind variance and standard deviation
Variance σ² = Σ(x − x̄)²/n = (Σx²/n) − x̄². Standard deviation σ is its square root. It measures how spread out the data is.
Step-by-step method
- Find the mean x̄.
- Square each deviation (x − x̄) and add.
- Divide by n for variance; take √ for SD.
Worked example 1 (Basic)
Find the mean and variance of 4, 9, 6, 6.
- x̄ = 6.25
- Σ(x − x̄)² = 12.75
- σ² = 3.19
- Answer: x̄ = 6.25, σ² = 3.19
Worked example 2 (Standard)
Find the variance of 16, 14, 13, 20, 17, 10, 8.
- x̄ = 14
- Σ(x − x̄)² = 102
- σ² = 14.57
- Answer: 14.57
Worked example 3 (Challenge)
Find the standard deviation of 9, 20, 17, 5.
- x̄ = 12.75
- Σ(x − x̄)² = 144.75
- σ² = 36.19
- σ = 6.02
- Answer: 6.02
Common mistakes and misconceptions
- Dividing by n − 1 (sample) when the syllabus uses n
- Forgetting to square deviations
Practice questions
- (Basic) Find the mean and variance of 14, 8, 6, 16.
- (Basic) Find the mean and variance of 8, 4, 13, 16, 20, 10, 4.
- (Standard) Find the variance of 20, 3, 6, 17, 20.
- (Standard) Find the variance of 7, 14, 7, 19.
- (Challenge) Find the standard deviation of 16, 8, 18, 7, 15, 18.
- (Challenge) Find the standard deviation of 16, 2, 14, 4, 20, 13.
Answer key with full working
- Q1: x̄ = 11 → Σ(x − x̄)² = 68 → σ² = 17 → Answer: x̄ = 11, σ² = 17
- Q2: x̄ = 10.71 → Σ(x − x̄)² = 217.43 → σ² = 31.06 → Answer: x̄ = 10.71, σ² = 31.06
- Q3: x̄ = 13.2 → Σ(x − x̄)² = 262.8 → σ² = 52.56 → Answer: 52.56
- Q4: x̄ = 11.75 → Σ(x − x̄)² = 102.75 → σ² = 25.69 → Answer: 25.69
- Q5: x̄ = 13.67 → Σ(x − x̄)² = 121.33 → σ² = 20.22 → σ = 4.5 → Answer: 4.5
- Q6: x̄ = 11.5 → Σ(x − x̄)² = 247.5 → σ² = 41.25 → σ = 6.42 → Answer: 6.42
Syllabus note
This skill is listed under "Statistics" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is variance and standard deviation in?
"Statistics", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.