Variance & Standard Deviation – Solved Problems
Variance and Standard Deviation are measures of dispersion. They tell us how much the values in a data set deviate from the mean.
Key Formulas:
Mean: μ = Σx / n
Variance: σ² = Σ(x − μ)² / n
Standard Deviation: σ = √σ²
Mean: μ = Σx / n
Variance: σ² = Σ(x − μ)² / n
Standard Deviation: σ = √σ²
AdSense Placeholder – Top Banner
Problem 1: Find variance and standard deviation
Find the variance and standard deviation of the data: 2, 4, 6, 8
Solution:
Mean μ = (2 + 4 + 6 + 8) / 4 = 5
Σ(x − μ)² = 20
Variance = 20 / 4 = 5
Standard Deviation = √5 ≈ 2.24
Mean μ = (2 + 4 + 6 + 8) / 4 = 5
| x | x − μ | (x − μ)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 6 | 1 | 1 |
| 8 | 3 | 9 |
Variance = 20 / 4 = 5
Standard Deviation = √5 ≈ 2.24
Problem 2: Data with equal spacing
Calculate the standard deviation of the first five natural numbers.
Mean μ = (1 + 2 + 3 + 4 + 5) / 5 = 3
Σ(x − μ)² = 4 + 1 + 0 + 1 + 4 = 10
Variance = 10 / 5 = 2
Standard Deviation = √2 ≈ 1.41
Σ(x − μ)² = 4 + 1 + 0 + 1 + 4 = 10
Variance = 10 / 5 = 2
Standard Deviation = √2 ≈ 1.41
AdSense Placeholder – In-Content Ad
Problem 3: Compare two data sets
Which data set has greater variability?
A: 5, 5, 5, 5
B: 2, 4, 6, 8
A: 5, 5, 5, 5
B: 2, 4, 6, 8
Set A: Mean = 5 → All values same → Variance = 0
Set B: Variance = 5 (from Problem 1)
Set B has greater variability
Set B: Variance = 5 (from Problem 1)
Set B has greater variability
Problem 4: Find variance when standard deviation is known
The standard deviation of a data set is 6. Find the variance.
Variance = (Standard Deviation)² = 6² = 36
Problem 5: Find standard deviation from variance
The variance of a data set is 25. Find the standard deviation.
Standard Deviation = √25 = 5
AdSense Placeholder – Mid Page
Problem 6: Word problem
The marks scored by a student in five tests are: 40, 50, 60, 70, 80.
Find the standard deviation.
Mean = (40+50+60+70+80)/5 = 60
Σ(x − μ)² = 400 + 100 + 0 + 100 + 400 = 1000
Variance = 1000 / 5 = 200
Standard Deviation = √200 ≈ 14.14
Σ(x − μ)² = 400 + 100 + 0 + 100 + 400 = 1000
Variance = 1000 / 5 = 200
Standard Deviation = √200 ≈ 14.14
Problem 7: Zero variance case
Find the variance of the data: 7, 7, 7, 7
Mean = 7
All deviations = 0
Variance = 0
Standard Deviation = 0
All deviations = 0
Variance = 0
Standard Deviation = 0
Problem 8: Effect of scaling data
If the standard deviation of a data set is 4, what will be the standard deviation if each value is multiplied by 3?
When data is multiplied by a constant k,
New Standard Deviation = |k| × Old Standard Deviation
= 3 × 4 = 12
New Standard Deviation = |k| × Old Standard Deviation
= 3 × 4 = 12
AdSense Placeholder – Bottom Banner
Key Observations
- Variance measures spread, not central value
- Standard deviation is always non-negative
- Zero variance means all values are identical
- Scaling data scales standard deviation proportionally