Question 1

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

Accepted Answer

Step 1: Recall the relationship between standard deviation and variance: Variance = (Standard Deviation)². Step 2: Variance = 5² = 25.

Question 2

The variance of {1, 2, 3} is 2/3. What is the standard deviation (rounded to 2 decimal places)?

Accepted Answer

Step 1: Given variance = 2/3 ≈ 0.667. Step 2: Standard Deviation = √Variance = √(2/3) = √0.667 ≈ 0.816 ≈ 0.82 (rounded to 2 decimal places).

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

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

Accepted Answer

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

Question 5

Two datasets have the same mean of 10. Dataset A has a standard deviation of 2, and Dataset B has a standard deviation of 4. Which statement is true?

Accepted Answer

Step 1: Both datasets have mean = 10. Step 2: Dataset A has SD = 2, so Variance_A = 2² = 4. Step 3: Dataset B has SD = 4, so Variance_B = 4² = 16. Step 4: Since Variance_B > Variance_A, Dataset B has greater spread/dispersion around the mean. Step 5: Dataset B is more dispersed than Dataset A.

Question 6

If every value in a dataset is increased by 5, how does the standard deviation change?

Accepted Answer

Step 1: Adding a constant to all values shifts the entire dataset but does not change the spread. Step 2: Standard deviation measures spread around the mean. Step 3: When all values increase by 5, the mean also increases by 5, so all deviations from the mean remain unchanged. Step 4: Therefore, the standard deviation remains the same.

Question 7

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 8

The variance of the dataset {2, 4, 6, 8, 10} is 8. If a new observation 12 is added to the dataset, which of the following is closest to the new variance? (Assume population variance formula)

Accepted Answer

Step 1: Original mean = (2+4+6+8+10)/5 = 30/5 = 6. Step 2: New dataset = {2, 4, 6, 8, 10, 12}, new mean = (30+12)/6 = 42/6 = 7. Step 3: Sum of squared deviations from new mean = (2-7)² + (4-7)² + (6-7)² + (8-7)² + (10-7)² + (12-7)² = 25 + 9 + 1 + 1 + 9 + 25 = 70. Step 4: New variance = 70/6 ≈ 11.67.

Question 9

Two datasets have the same mean of 60. Dataset A has variance 16, and Dataset B has variance 64. If both datasets have 25 observations, what is the ratio of the standard deviation of A to the standard deviation of B?

Accepted Answer

Step 1: SD of A = √16 = 4. Step 2: SD of B = √64 = 8. Step 3: Ratio = 4 : 8 = 1 : 2.

Question 10

A distribution has mean 100 and standard deviation 15. If the data is standardized (converted to z-scores), what will be the mean and standard deviation of the standardized data?

Accepted Answer

Step 1: Standardization formula: Z = (X - mean) / SD. Step 2: When all data is standardized, the new mean = 0 and new SD = 1 (by definition of z-scores).

Question 11

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 (only shifts the mean). Step 2: New standard deviation = 4 (unchanged).

Question 12

A dataset has mean 50 and standard deviation 10. The sum of squared deviations from the mean is 4000. How many observations are in the dataset?

Accepted Answer

Step 1: Variance = (Sum of squared deviations) / n. Step 2: SD² = Variance, so 10² = 100 = 4000 / n. Step 3: n = 4000 / 100 = 40.

Question 13

A dataset has 5 observations with mean 20 and variance 16. If each observation is multiplied by 3 and then 5 is added to each result, what is the new variance?

Accepted Answer

Step 1: When a constant is added to each observation, variance remains unchanged. Step 2: When each observation is multiplied by a constant k, the new variance becomes k² times the original variance. Step 3: Here, multiply by 3 then add 5. Only the multiplication affects variance. Step 4: New variance = 3² × 16 = 9 × 16 = 144.

Question 14

Two datasets A and B have the same mean of 50. Dataset A has 8 observations with variance 36, and Dataset B has 12 observations with variance 64. What is the variance of the combined dataset (A ∪ B)?

Accepted Answer

Step 1: For combined variance with equal means, use: σ²_combined = [n₁σ₁² + n₂σ₂²] / (n₁ + n₂). Step 2: Since both datasets have the same mean, there is no between-group variance component. Step 3: σ²_combined = (8 × 36 + 12 × 64) / (8 + 12) = (288 + 768) / 20 = 1056 / 20 = 52.8.

Question 15

Three datasets have variances 9, 16, and 25 respectively, with equal sample sizes of 20 each. A new combined dataset is formed by merging all three. If all three datasets have the same mean of 30, what is the variance of the combined dataset?

Accepted Answer

Step 1: When combining datasets with equal means and equal sample sizes, the combined variance is the simple average of individual variances. Step 2: Combined variance = (9 + 16 + 25) / 3 = 50 / 3 ≈ 16.67.

Question 16

A sample of 100 observations has mean 50 and variance 100. If 10 new observations, each with value 50, are added to the sample, what is the new variance?

Accepted Answer

Step 1: Original: n₁=100, μ₁=50, σ₁²=100. Step 2: New observations: n₂=10, all values = 50 (mean=50, variance=0). Step 3: Combined mean = (100×50 + 10×50)/(100+10) = 5500/110 = 50 (unchanged). Step 4: Since combined mean equals both group means, use: σ²_combined = [n₁σ₁² + n₂σ₂²]/(n₁+n₂) = [100×100 + 10×0]/110 = 10000/110 ≈ 90.91.

Question 17

A dataset of 50 numbers has standard deviation 12. If each number is decreased by 20% and then increased by 8, what is the new standard deviation?

Accepted Answer

Step 1: Decreasing each number by 20% means multiplying by 0.8. Step 2: Increasing by 8 is adding a constant. Step 3: For transformation X' = 0.8X + 8: SD(X') = |0.8| × SD(X) = 0.8 × 12 = 9.6. Step 4: Adding a constant does not change standard deviation.

SBI Clerk Standard Deviation & Variance

SBI Clerk Standard Deviation & Variance — Past Exam Questions

Standard Deviation & Variance— Rules & Concept

Key Points to Remember

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Standard Deviation & Variance — Practice Questions

60-Second Revision — Standard Deviation & Variance