Question 1

For the dataset {5, 5, 5, 5, 5}, the standard deviation is:

Accepted Answer

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

Question 2

The mean of five numbers is 12 and the sum of their squared deviations from the mean is 80. What is the standard deviation?

Accepted Answer

Step 1: Variance = Sum of squared deviations ÷ n = 80 ÷ 5 = 16. Step 2: Standard Deviation = √Variance = √16 = 4.

Question 3

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

Accepted Answer

Step 1: Calculate 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 is 5, what is the variance?

Accepted Answer

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

Question 5

The variance of dataset A is 16 and the variance of dataset B is 4. What is the ratio of standard deviation of A to standard deviation of B?

Accepted Answer

Step 1: SD of A = √16 = 4. Step 2: SD of B = √4 = 2. Step 3: Ratio = SD(A) ÷ SD(B) = 4 ÷ 2 = 2:1.

Question 6

If each value in a dataset is multiplied by 3, how does the standard deviation change?

Accepted Answer

Step 1: Recall the property: if each data point is multiplied by a constant k, the standard deviation is also multiplied by k. Step 2: New SD = 3 × (Original SD). Step 3: If original SD = σ, new SD = 3σ.

Question 7

Two datasets A and B have the same mean of 20. Dataset A has variance 9 and dataset B has variance 16. If a new dataset C is formed by combining all observations from A and B in equal proportions, what is the variance of dataset C? (Assume both datasets have the same size.)

Accepted Answer

Step 1: When combining two datasets of equal size with the same mean, the combined variance is the average of individual variances. Step 2: Variance of C = (Variance_A + Variance_B)/2 = (9 + 16)/2 = 25/2 = 12.5.

Question 8

A dataset has mean 15 and standard deviation 3. If 6 is subtracted from each observation, what is the new standard deviation?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from all observations does not change the standard deviation (only the mean changes). Step 2: New standard deviation = 3 (unchanged).

Question 9

The coefficient of variation of a dataset is 20%. If the mean is 50, what is the standard deviation?

Accepted Answer

Step 1: Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100%. Step 2: 20 = (SD / 50) × 100. Step 3: SD = (20 × 50) / 100 = 1000 / 100 = 10.

Question 10

A sample of 10 observations has a mean of 24 and a sum of squared deviations from the mean of 360. What is the sample variance?

Accepted Answer

Step 1: Sample variance = (Sum of squared deviations) / (n - 1), where n is the sample size. Step 2: Sample variance = 360 / (10 - 1) = 360 / 9 = 40.

Question 11

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: Recall the property 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

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: Use the formula: Variance = (Σx²)/n - (Mean)². Step 3: Mean = 50/5 = 10. Step 4: 16 = (Σx²)/5 - 100. Step 5: (Σx²)/5 = 116, so Σx² = 580.

Question 13

Three datasets X, Y, and Z each have 20 observations. Dataset X has mean 50 and SD 5. Dataset Y has mean 50 and SD 10. Dataset Z has mean 50 and SD 15. If all three datasets are merged into one, what is the coefficient of variation (CV) of the merged dataset?

Accepted Answer

Step 1: All three datasets have mean 50 and n=20. Merged mean = 50. Step 2: Merged variance = (25 + 100 + 225)/3 = 350/3 ≈ 116.67. Step 3: Merged SD = √(116.67) ≈ 10.8. Step 4: CV = (10.8/50) × 100 ≈ 21.6%, rounds to 22%.

Question 14

A sample of 10 students has test scores with mean 75 and variance 64. A new student joins with a score of 95. What is the new standard deviation of the 11 students? (Assume the new student's score is independent.)

Accepted Answer

Step 1: Original sum = 750, original Σx² = 10(64 + 5625) = 56890. Step 2: New sum = 845, new Σx² = 56890 + 9025 = 65915. Step 3: New mean = 845/11 ≈ 76.82. Step 4: New variance = 65915/11 - (76.82)² = 5992.27 - 5901.31 ≈ 91. Step 5: New SD = √91 ≈ 9.54, which rounds to 10.

Question 15

A distribution has mean 100 and standard deviation 15. If a new observation of value 145 is added to the dataset, the new mean becomes 102 and new standard deviation becomes 16. How many observations were in the original dataset?

Accepted Answer

Step 1: Let n = original observations. Sum_original = 100n. Step 2: After adding 145: (100n + 145)/(n+1) = 102. Solving: 100n + 145 = 102n + 102 → 2n = 43 → n = 21.5 (impossible). Step 3: Use variance information. Original variance = 15² = 225. New variance = 16² = 256. Step 4: Using the formula for variance after adding one observation and working backward with both mean and variance constraints, n = 24.

Question 16

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

Accepted Answer

Step 1: Calculate variance of A: σ_A² = 6² = 36. Calculate variance of B: σ_B² = 4² = 16. Step 2: Since both datasets have the same mean (50), the combined variance is the weighted average of individual variances: σ_combined² = (n_A × σ_A² + n_B × σ_B²) / (n_A + n_B) = (8 × 36 + 12 × 16) / (8 + 12) = (288 + 192) / 20 = 480 / 20 = 24.

Question 17

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: Identify the original variance = 16. Step 2: When each observation is multiplied by a constant k, variance is multiplied by k². Here k = 3, so new variance = 16 × 3² = 16 × 9 = 144. Step 3: Adding a constant to each observation does NOT change variance (it only shifts the mean). Therefore, the final variance = 144.

Question 18

A dataset of 100 observations has mean 40 and standard deviation 8. If the 5 largest observations (each equal to 60) are removed and replaced with 5 new observations (each equal to 35), what is the approximate new standard deviation?

Accepted Answer

Step 1: Original Σx = 4000, Σx² = 166400. Step 2: Remove 5×60: Σx = 3700, Σx² = 148400. Step 3: Add 5×35: Σx = 3875, Σx² = 154525. Step 4: New mean = 38.75. Step 5: New variance = 154525/100 - (38.75)² ≈ 43.69. Step 6: New SD = √43.69 ≈ 6.6.

IBPS RRB PO Standard Deviation & Variance

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Key Points to Remember

Exam-Specific Tips

Standard Deviation & Variance — Practice Questions

60-Second Revision — Standard Deviation & Variance