Question 1

For the dataset {3, 3, 3, 3, 3}, the standard deviation is:

Accepted Answer

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

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 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: (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 mean of five numbers is 12. If the sum of squared deviations from the mean is 80, what is the standard deviation?

Accepted Answer

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

Question 5

The variance of {1, 2, 3, 4, 5} is 2. What is the variance of {2, 4, 6, 8, 10}?

Accepted Answer

Step 1: The second dataset is exactly 2 times the first dataset. Step 2: When all values are multiplied by a constant k, variance is multiplied by k². Step 3: New Variance = 2 × 2² = 2 × 4 = 8.

Question 6

If the standard deviation of a dataset is 6, what is the standard deviation when each value is increased by 10?

Accepted Answer

Step 1: Adding a constant to all values shifts the mean but does not change the spread of data. Step 2: Standard Deviation measures spread, not location. Step 3: Therefore, SD remains 6.

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: Use the formula: Variance = (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. If 10 is subtracted from each observation in both datasets, which statement is true?

Accepted Answer

Step 1: Subtracting a constant from all observations shifts the mean but does NOT change the standard deviation. Step 2: New mean for both = 20 - 10 = 10. Step 3: Standard deviation of A remains 3; standard deviation of B remains 6. Step 4: The relationship between their standard deviations (B's SD is twice A's SD) remains unchanged. Therefore, Dataset B still has greater variability.

Question 9

The mean of a distribution is 15 and its standard deviation is 3. What is the coefficient of variation (CV)?

Accepted Answer

Step 1: Recall the formula: Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100. Step 2: CV = (3 / 15) × 100 = 0.2 × 100 = 20%. Step 3: This is a standard measure of relative dispersion. Therefore, CV = 20%.

Question 10

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 new variance becomes k² times the original variance. Step 2: New variance = 5² × 36 = 25 × 36 = 900. Therefore, the answer is 900.

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 that when each observation is multiplied by a constant k, the new variance becomes k² times the original variance. Step 2: New variance = 5² × 144 = 25 × 144 = 3600. Therefore, the answer is 3600.

Question 12

Three datasets have the following properties: Dataset 1 has mean 30 and variance 25; Dataset 2 has mean 40 and variance 25; Dataset 3 has mean 50 and variance 25. If all three datasets are combined (equal sample sizes), what is the variance of the combined dataset?

Accepted Answer

Step 1: Combined mean = (30 + 40 + 50)/3 = 40. Step 2: Deviations from combined mean: d₁ = -10, d₂ = 0, d₃ = 10. Step 3: Combined variance = [n(25 + 100) + n(25 + 0) + n(25 + 100)]/(3n) = (125 + 25 + 125)/3 = 275/3 ≈ 91.67. For clean answer, this is approximately 92.

Question 13

A dataset has mean 20 and standard deviation 4. If 5 is subtracted from each observation, what is the new standard deviation?

Accepted Answer

Step 1: Recall that standard deviation is unaffected by adding or subtracting a constant from each observation. This is because standard deviation measures spread, not location. Step 2: Subtracting 5 shifts all values equally, so the spread remains the same. Step 3: New standard deviation = 4. Therefore, the answer is 4.

Question 14

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 = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)] / (n₁ + n₂), where d₁ and d₂ are deviations of individual means from the combined mean. Step 2: Since both means equal 50, the combined mean = (8×50 + 12×50)/(8+12) = 50. Thus d₁ = d₂ = 0. Step 3: σ²_combined = [8(16 + 0) + 12(36 + 0)] / 20 = (128 + 432) / 20 = 560 / 20 = 28. Therefore, the answer is 28.

Question 15

The standard deviation of five numbers is 4. What is the variance of these five numbers?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 4² = 16. Therefore, the variance is 16.

Question 16

The mean of a dataset {2, 4, 6, 8, 10} is 6. What is the variance of this dataset?

Accepted Answer

Step 1: Mean = 6 (given). Step 2: Calculate 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. Therefore, the variance is 8.

Question 17

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 18

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 19

The variance of a dataset is 25. What is the standard deviation?

Accepted Answer

Step 1: Recall that Standard Deviation = √Variance. Step 2: Standard Deviation = √25 = 5.

Question 20

For the dataset {3, 5, 7, 9, 11}, the mean is 7. What is the variance?

Accepted Answer

Step 1: Mean = 7 (given). Step 2: Calculate squared deviations: (3−7)² = 16, (5−7)² = 4, (7−7)² = 0, (9−7)² = 4, (11−7)² = 16. Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 21

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 22

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

Accepted Answer

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

Question 23

If the standard deviation of a dataset is 6, what is the variance?

Accepted Answer

Step 1: Use the relationship Variance = (Standard Deviation)². Step 2: Variance = 6² = 36.

Question 24

If the standard deviation of a dataset is 5, what will be the standard deviation if each observation is multiplied by 3?

Accepted Answer

Step 1: Recall the property: When each observation is multiplied by a constant k, the new standard deviation = k × (original SD). Step 2: New SD = 3 × 5 = 15. Therefore, the new standard deviation is 15.

Question 25

The standard deviations of two datasets are 6 and 9 respectively. If the first dataset is multiplied by 3 and the second by 2, what is the ratio of the new standard deviations?

Accepted Answer

Step 1: Original SD₁ = 6, original SD₂ = 9. Step 2: When multiplied by constants, new SD₁ = 6 × 3 = 18, new SD₂ = 9 × 2 = 18. Step 3: Ratio = 18 : 18 = 1 : 1.

Question 26

A dataset has 10 observations with mean 20 and standard deviation 5. If a new observation of value 20 is added to the dataset, which of the following is true about the new standard deviation?

Accepted Answer

Step 1: Original: n = 10, mean = 20, SD = 5, so variance = 25. Step 2: Sum of original observations = 10 × 20 = 200. Step 3: After adding 20: new sum = 220, new n = 11, new mean = 220/11 = 20. Step 4: The new observation equals the mean, so it contributes zero to the sum of squared deviations. Step 5: New sum of squared deviations = 10 × 25 = 250 (unchanged). Step 6: New variance = 250/11 ≈ 22.73, so new SD = √(250/11) ≈ 4.77 < 5.

Question 27

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 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 28

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

Accepted Answer

Step 1: Mean = 50 ÷ 5 = 10. Step 2: Standard deviation σ = 4, so variance σ² = 16. Step 3: Variance = (Sum of squared deviations) ÷ n. Therefore, 16 = (Sum of squared deviations) ÷ 5. Step 4: Sum of squared deviations = 16 × 5 = 80.

Question 29

Two datasets have the same mean of 50. Dataset A has a standard deviation of 8, and Dataset B has a standard deviation of 12. Which statement is correct?

Accepted Answer

Step 1: Standard deviation measures spread around the mean. Step 2: Dataset B has SD = 12 > Dataset A's SD = 8. Step 3: Therefore, Dataset B has greater variability and is more dispersed around the mean of 50.

Question 30

The variance of a dataset is 64. If each observation is decreased by 8, what will be the new variance?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from each observation does NOT change the variance. Step 2: Variance depends only on deviations from the mean, and shifting all values by a constant shifts the mean by the same amount, leaving deviations unchanged. Step 3: New variance = 64.

Question 31

Two datasets A and B have the same mean of 20. Dataset A has 10 observations with variance 9, and Dataset B has 15 observations with variance 16. What is the combined variance of both datasets?

Accepted Answer

Step 1: For combined variance when means are equal: Combined variance = (n₁σ₁² + n₂σ₂²) / (n₁ + n₂). Step 2: Combined variance = (10 × 9 + 15 × 16) / (10 + 15) = (90 + 240) / 25 = 330 / 25 = 13.2.

Question 32

A dataset has mean 25 and standard deviation 5. If 8 is subtracted from each observation, what will be the new standard deviation?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from all observations does NOT change the standard deviation (it only shifts the mean). Step 2: New standard deviation = 5 (unchanged).

Question 33

A dataset has mean 40 and standard deviation 6. If each observation is multiplied by 3 and then 5 is subtracted from each result, what is the new standard deviation?

Accepted Answer

Step 1: Original SD = 6. Step 2: When multiplying by constant k, SD is multiplied by |k|. New SD after multiplying by 3: SD = 6 × 3 = 18. Step 3: Subtracting a constant does not change SD (it only shifts the mean). Step 4: Final SD = 18.

Question 34

Two datasets have the same mean of 50. Dataset A has 8 observations with variance 16, and Dataset B has 12 observations with variance 36. If the two datasets are combined into a single dataset, what is the variance of the combined dataset?

Accepted Answer

Step 1: Calculate sum of squared deviations for Dataset A: Variance_A = Σ(x_i - mean)²/n_A, so Σ(x_i - 50)² = 16 × 8 = 128. Step 2: Calculate sum of squared deviations for Dataset B: Σ(x_i - 50)² = 36 × 12 = 432. Step 3: Combined variance = (128 + 432)/(8 + 12) = 560/20 = 28.

RRB NTPC Standard Deviation & Variance

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Standard Deviation & Variance — Practice Questions

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