Question 1

The mean of a dataset {4, 6, 8, 10, 12} is 8. What is the variance of this dataset?

Accepted Answer

Step 1: Mean = 8 (given). Step 2: Calculate deviations from mean: (4−8)² = 16, (6−8)² = 4, (8−8)² = 0, (10−8)² = 4, (12−8)² = 16. Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 2

Two datasets have variances 16 and 9 respectively. What are their standard deviations?

Accepted Answer

Step 1: For first dataset, SD₁ = √16 = 4. Step 2: For second dataset, SD₂ = √9 = 3. Step 3: The standard deviations are 4 and 3 respectively.

Question 3

The variance of a dataset is 49. If each value in the dataset is decreased by 5, what is the variance of the new dataset?

Accepted Answer

Step 1: Original Variance = 49. Step 2: When a constant is added or subtracted from each value, the variance remains unchanged (variance measures spread, not location). Step 3: New Variance = 49.

Question 4

The standard deviation of five numbers is 2. What is the variance of these five numbers?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 2² = 4. Therefore, the variance is 4.

Question 5

The variance of a dataset is 36. What is the standard deviation of this dataset?

Accepted Answer

Step 1: Recall that Standard Deviation (SD) = √Variance. Step 2: SD = √36 = 6. Therefore, the standard deviation is 6.

Question 6

If the standard deviation of a dataset is 5, what is the variance of a new dataset formed by multiplying each value by 3?

Accepted Answer

Step 1: Original SD = 5, so original Variance = 25. Step 2: When each value is multiplied by a constant k, the new variance = k² × original variance. Step 3: New Variance = 3² × 25 = 9 × 25 = 225.

Question 7

A dataset has mean 25 and standard deviation 5. If 10 is subtracted from each observation, what will be the new standard deviation?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from each observation does NOT change the standard deviation (or variance). Step 2: The new standard deviation remains 5.

Question 8

The standard deviation of five numbers is 6. If the sum of the numbers is 50, what is the sum of their squares?

Accepted Answer

Step 1: Use the formula: Variance = (Sum of Squares / n) - (Mean)². Step 2: Standard deviation = 6, so Variance = 36. Step 3: Mean = 50/5 = 10. Step 4: 36 = (Sum of Squares / 5) - 100. Step 5: Sum of Squares / 5 = 136. Step 6: Sum of Squares = 680.

Question 9

Two datasets A and B have the same mean of 20. Dataset A has 8 observations with variance 16, and Dataset B has 12 observations with variance 9. What is the combined variance of both datasets?

Accepted Answer

Step 1: For combined variance when means are equal: Combined Variance = (n₁·Var₁ + n₂·Var₂) / (n₁ + n₂). Step 2: Combined Variance = (8×16 + 12×9) / (8+12) = (128 + 108) / 20 = 236 / 20 = 11.8.

Question 10

The variance of a set of 6 numbers is 25. The sum of the numbers is 60. If one number is removed, the sum becomes 54. What is the variance of the remaining 5 numbers? (Assume the removed number was 6)

Accepted Answer

Step 1: Original: n=6, Sum=60, Variance=25. Step 2: Sum of squares: Var = (ΣX²/n) - (Mean)². Mean = 60/6 = 10. So 25 = (ΣX²/6) - 100 ⟹ ΣX² = 750. Step 3: Removed number = 6. New sum = 54, n = 5. Step 4: New mean = 54/5 = 10.8. Step 5: New ΣX² = 750 - 36 = 714. Step 6: New Variance = (714/5) - (10.8)² = 142.8 - 116.64 = 26.16.

Question 11

The variance of a dataset is 144. If each observation is multiplied by 5, what will be the new variance?

Accepted Answer

Step 1: Recall the property that when each observation is multiplied by a constant k, the variance is multiplied by k². Step 2: Original variance = 144. Step 3: New variance = 144 × 5² = 144 × 25 = 3600.

Question 12

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

Accepted Answer

Step 1: Recall that when data is transformed by Y = aX + b, the new variance = a² × (original variance). Step 2: Here a = 3 (multiplication factor) and b = −5 (additive constant). Step 3: New variance = 3² × 16 = 9 × 16 = 144. The additive constant does not affect variance.

Question 13

Two datasets A and B have the same mean of 50. Dataset A has 8 observations with standard deviation 6, while Dataset B has 12 observations with standard deviation 4. If both datasets are combined, what is the variance of the combined dataset (rounded to nearest integer)?

Accepted Answer

Step 1: For combined variance, use formula: Var(combined) = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)] / (n₁ + n₂), where d₁ and d₂ are deviations of individual means from combined mean. Step 2: Combined mean = (8×50 + 12×50) / 20 = 50. Step 3: d₁ = 50 − 50 = 0, d₂ = 50 − 50 = 0. Step 4: Var(A) = 6² = 36, Var(B) = 4² = 16. Step 5: Var(combined) = [8(36 + 0) + 12(16 + 0)] / 20 = (288 + 192) / 20 = 480 / 20 = 24.

Question 14

In a dataset, the coefficient of variation (CV) is 25%. If the mean is 80, what is the standard deviation? Additionally, if all values are increased by 20%, what will be the new coefficient of variation?

Accepted Answer

Step 1: CV = (σ/mean) × 100. So 25 = (σ/80) × 100, giving σ = 20. Step 2: When all values increase by 20%, new mean = 80 × 1.2 = 96. Step 3: Standard deviation scales proportionally with multiplicative changes: new σ = 20 × 1.2 = 24. Step 4: New CV = (24/96) × 100 = 25%. The coefficient of variation remains unchanged.

SBI PO Standard Deviation & Variance

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Standard Deviation & Variance— Rules & Concept

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60-Second Revision — Standard Deviation & Variance