Question 1

The standard deviation of the dataset {5, 5, 5, 5, 5} is:

Accepted Answer

Step 1: All values are identical (5). Step 2: Mean = 5. Step 3: Each deviation from mean = 5 − 5 = 0. Step 4: Sum of squared deviations = 0. Step 5: Variance = 0 ÷ 5 = 0. Step 6: Standard Deviation = √0 = 0.

Question 2

The variance of {1, 2, 3} is 2/3. What is the variance of {2, 4, 6} (each value doubled)?

Accepted Answer

Step 1: Recall the property: if each data point is multiplied by k, variance is multiplied by k². Step 2: Here k = 2, so new variance = (2/3) × 2² = (2/3) × 4 = 8/3.

Question 3

The variance of the dataset {2, 4, 6, 8, 10} is:

Accepted Answer

Step 1: Find the mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6. Step 2: Find squared deviations from mean: (2−6)² = 16, (4−6)² = 4, (6−6)² = 0, (8−6)² = 4, (10−6)² = 16. Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 4

If the standard deviation of a dataset increases from 4 to 8, by what factor has the variance increased?

Accepted Answer

Step 1: Original variance = 4² = 16. Step 2: New variance = 8² = 64. Step 3: Factor of increase = 64 ÷ 16 = 4.

Question 5

The mean of a dataset is 10 and the sum of squared deviations from the mean is 80. If there are 8 observations, the standard deviation is:

Accepted Answer

Step 1: Variance = (Sum of squared deviations) ÷ n = 80 ÷ 8 = 10. Step 2: Standard Deviation = √Variance = √10 ≈ 3.16.

Question 6

If the standard deviation of a dataset is 5, what is the variance?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 5² = 25.

Question 7

The standard deviation of five numbers is 4. If each number is decreased by 2, what is the new standard deviation?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from each observation does NOT change the standard deviation (it only shifts the mean). Step 2: New standard deviation = 4 (unchanged).

Question 8

The standard deviation of a set of 10 observations is 6. The sum of squared deviations from the mean is ___.

Accepted Answer

Step 1: Standard deviation σ = √(Variance). Step 2: Variance = σ² = 6² = 36. Step 3: Variance = (Sum of squared deviations) ÷ n. Step 4: 36 = (Sum of squared deviations) ÷ 10. Step 5: Sum of squared deviations = 36 × 10 = 360.

Question 9

Two datasets have the same mean of 50. Dataset A has variance 25, and Dataset B has variance 100. Which statement is correct?

Accepted Answer

Step 1: Variance measures the spread of data around the mean. Step 2: Dataset A (variance 25) has standard deviation √25 = 5. Step 3: Dataset B (variance 100) has standard deviation √100 = 10. Step 4: Dataset B has greater spread/dispersion around the mean.

Question 10

The mean of a dataset {8, 12, 16, 20, 24} is 16. What is the variance of this dataset?

Accepted Answer

Step 1: Mean = (8 + 12 + 16 + 20 + 24) ÷ 5 = 80 ÷ 5 = 16 ✓. Step 2: Deviations from mean: -8, -4, 0, 4, 8. Step 3: Squared deviations: 64, 16, 0, 16, 64. Step 4: Variance = (64 + 16 + 0 + 16 + 64) ÷ 5 = 160 ÷ 5 = 32.

Question 11

The variance of a dataset is 36. If each observation is multiplied by 5, what will be the new variance?

Accepted Answer

Step 1: Recall that when each observation is multiplied by a constant k, the variance is multiplied by k². Step 2: New variance = 36 × 5² = 36 × 25 = 900.

Question 12

A dataset has standard deviation 8. After subtracting 5 from each observation and then dividing by 2, the new standard deviation is:

Accepted Answer

Step 1: Subtracting a constant from all observations does not change standard deviation (it only shifts the mean). Step 2: Dividing all observations by 2 scales the standard deviation by |2|. Step 3: New standard deviation = 8/2 = 4.

Question 13

A dataset has 5 observations: 10, 15, 20, 25, 30. If each observation is multiplied by 3, what is the ratio of the new variance to the original variance?

Accepted Answer

Step 1: Original mean = (10+15+20+25+30)/5 = 20. Step 2: Original variance = [(10-20)² + (15-20)² + (20-20)² + (25-20)² + (30-20)²]/5 = [100+25+0+25+100]/5 = 50. Step 3: When each observation is multiplied by k, variance is multiplied by k². Here k=3, so new variance = 50 × 9 = 450. Step 4: Ratio = 450/50 = 9.

Question 14

A frequency distribution has mean 40 and variance 36. The sum of all frequencies is 100. If the sum of squared deviations from mean is recalculated after removing 4 observations of value 40, what is the new variance (approximately)?

Accepted Answer

Step 1: Original variance = 36, so sum of squared deviations = 36 × 100 = 3600. Step 2: Removing 4 observations of value 40 (which equals the mean) removes 0 squared deviation. Step 3: New sum of squared deviations = 3600 - 0 = 3600. Step 4: New total frequency = 100 - 4 = 96. Step 5: New variance = 3600/96 = 37.5.

Question 15

Three groups have sizes 20, 30, and 50 with means 10, 20, and 30 respectively. The overall mean is 23. What is the combined variance if each group has internal variance of 4?

Accepted Answer

Step 1: Verify overall mean = (20×10 + 30×20 + 50×30)/(20+30+50) = (200+600+1500)/100 = 2300/100 = 23. ✓ Step 2: Combined variance = [Σ(nᵢ × σᵢ²) + Σ(nᵢ × (μᵢ - μ)²)]/N. Step 3: Within-group variance component = (20×4 + 30×4 + 50×4)/100 = 400/100 = 4. Step 4: Between-group variance component = [20(10-23)² + 30(20-23)² + 50(30-23)²]/100 = [20×169 + 30×9 + 50×49]/100 = [3380+270+2450]/100 = 6100/100 = 61. Step 5: Combined variance = 4 + 61 = 65.

SSC MTS Standard Deviation & Variance

SSC MTS Standard Deviation & Variance — Past Exam Questions

Standard Deviation & Variance— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Standard Deviation & Variance