Question 1

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 2

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 3

The standard deviation of the dataset {3, 3, 3, 3, 3} is:

Accepted Answer

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

Question 4

The variance of {1, 2, 3, 4, 5} is 2. If each value is multiplied by 3, the new variance becomes:

Accepted Answer

Step 1: Recall the property: if each data point is multiplied by a constant k, the new variance = k² × (original variance). Step 2: Here, k = 3 and original variance = 2. Step 3: New variance = 3² × 2 = 9 × 2 = 18.

Question 5

The standard deviation of a dataset is 4. If a constant 10 is added to each value, the new standard deviation is:

Accepted Answer

Step 1: Recall the property: adding a constant to all data points does NOT change the standard deviation (or variance). Step 2: This is because adding a constant shifts all values equally, preserving the spread. Step 3: New standard deviation = 4.

Question 6

For a dataset with mean 20 and standard deviation 5, what is the coefficient of variation (CV)?

Accepted Answer

Step 1: Coefficient of Variation (CV) = (Standard Deviation ÷ Mean) × 100%. Step 2: CV = (5 ÷ 20) × 100% = 0.25 × 100% = 25%.

Question 7

The standard deviation of five numbers is 4. If the sum of the numbers is 50, what is the sum of their squares?

Accepted Answer

Step 1: Standard deviation σ = 4, so variance σ² = 16. Step 2: Mean = 50/5 = 10. Step 3: Variance formula: σ² = (Σx²)/n - (mean)². Step 4: 16 = (Σx²)/5 - 100. Step 5: (Σx²)/5 = 116. Step 6: Σx² = 580. Therefore, the answer is 580.

Question 8

Two datasets have the same mean of 20. Dataset A has a standard deviation of 3, and Dataset B has a standard deviation of 6. Which statement is correct?

Accepted Answer

Step 1: Both datasets have mean = 20. Step 2: Dataset A: σ_A = 3, so variance = 9. Step 3: Dataset B: σ_B = 6, so variance = 36. Step 4: Since variance of B (36) > variance of A (9), Dataset B has greater spread/dispersion around the mean. Step 5: Dataset B is more dispersed/variable. Therefore, the answer is that Dataset B has greater variability.

Question 9

The variance of the dataset {2, 4, 6, 8, 10} is calculated. If a constant 3 is added to each observation, what is the change in variance?

Accepted Answer

Step 1: When a constant is added to each observation, the mean shifts by that constant, but the deviations from the mean remain unchanged. Step 2: Variance depends only on deviations from the mean, not on the absolute values. Step 3: Adding a constant does not change variance. Step 4: Change in variance = 0. Therefore, the answer is 0.

Question 10

A dataset has 10 observations with a mean of 15 and a sum of squared deviations from the mean of 400. What is the standard deviation?

Accepted Answer

Step 1: Sum of squared deviations = 400. Step 2: Variance = (Sum of squared deviations) / n = 400 / 10 = 40. Step 3: Standard deviation = √variance = √40 = √(4 × 10) = 2√10 ≈ 6.32. However, checking for clean answers: √40 = 2√10. Step 4: If the problem expects a simplified form, SD = 2√10. If numerical: SD ≈ 6.32. For SSC exams, the answer is typically 2√10 or approximately 6.32. Therefore, the answer is 2√10.

Question 11

The standard deviation of a dataset is 5. If each observation is decreased by 2 and then multiplied by 3, what is the new standard deviation?

Accepted Answer

Step 1: Original standard deviation σ = 5. Step 2: Subtracting a constant does not change standard deviation (only shifts the mean). Step 3: After subtracting 2: new σ = 5. Step 4: Multiplying by 3 scales the standard deviation by the absolute value of the multiplier. Step 5: New σ = 3 × 5 = 15. Therefore, the answer is 15.

Question 12

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

Accepted Answer

Step 1: Original variance = 36. Step 2: When each observation is multiplied by a constant k, the new variance becomes k² × (original variance). Step 3: New variance = 5² × 36 = 25 × 36 = 900. Therefore, the answer is 900.

Question 13

Two datasets A and B have the same mean of 50. Dataset A has 8 observations with variance 16, and Dataset B has 12 observations with variance 36. What is the combined variance when both datasets are merged?

Accepted Answer

Step 1: Use the formula for combined variance: σ²_combined = [n₁σ₁² + n₂σ₂² + n₁(x̄₁ - x̄_combined)² + n₂(x̄₂ - x̄_combined)²] / (n₁ + n₂). Step 2: Since both means are equal (50), x̄_combined = 50, so the last two terms vanish. Step 3: σ²_combined = [8(16) + 12(36)] / 20 = [128 + 432] / 20 = 560 / 20 = 28.

Question 14

A dataset has mean 40 and standard deviation 6. If the dataset is standardized (converted to z-scores), what will be the variance of the standardized dataset?

Accepted Answer

Step 1: Standardization transforms each observation x to z = (x - mean)/SD. Step 2: The standardized dataset has mean 0 and standard deviation 1 by definition. Step 3: Variance = (Standard Deviation)² = 1² = 1.

Question 15

A dataset has standard deviation 12. If each observation is divided by 4 and then multiplied by 3, what is the new standard deviation?

Accepted Answer

Step 1: Original standard deviation σ = 12. Step 2: Transformation: X' = (X/4) × 3 = (3/4)X. Step 3: When data is multiplied by a constant k, the new standard deviation = |k| × original SD. Step 4: Here k = 3/4, so new SD = (3/4) × 12 = 9.

Question 16

The variance of a dataset is 144. If each observation is multiplied by 5 and then 3 is subtracted from each result, what will be the new variance?

Accepted Answer

Step 1: Recall that variance is affected only by multiplication, not by addition/subtraction. Step 2: When each observation is multiplied by k, the new variance becomes k² times the original variance. Step 3: Here k = 5, so new variance = 5² × 144 = 25 × 144 = 3600. The subtraction of 3 does not affect variance.

SSC GD Constable Standard Deviation & Variance

SSC GD Constable Standard Deviation & Variance — Past Exam Questions

Standard Deviation & Variance— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Standard Deviation & Variance