Question 1

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 of variance under linear transformation. When each observation is multiplied by a constant k, the variance is multiplied by k². Step 2: New variance = Original variance × k² = 36 × 5² = 36 × 25 = 900. Therefore, the answer is 900.

Question 2

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 of variance under scaling. When each observation is multiplied by a constant k, the variance is multiplied by k². Step 2: Original variance = 36. Step 3: New variance = 36 × 5² = 36 × 25 = 900. Therefore, the answer is 900.

Question 3

For the dataset {1, 3, 5, 7, 9}, the standard deviation is closest to:

Accepted Answer

Step 1: Mean = (1 + 3 + 5 + 7 + 9) ÷ 5 = 25 ÷ 5 = 5. Step 2: Squared deviations: (1-5)² = 16, (3-5)² = 4, (5-5)² = 0, (7-5)² = 4, (9-5)² = 16. Step 3: Sum = 40. Step 4: Variance = 40 ÷ 5 = 8. Step 5: SD = √8 ≈ 2.83.

Question 4

The variance of a dataset is 36. If each observation in the dataset is multiplied by 5 and then 3 is subtracted from each result, 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: When a constant is added or subtracted, variance remains unchanged. Step 3: Original variance = 36. After multiplying by 5: new variance = 36 × 5² = 36 × 25 = 900. Step 4: Subtracting 3 does not change variance. Therefore, the new variance is 900.

Question 5

The variance of a dataset is 36. If each observation in the dataset is multiplied by 5 and then 3 is subtracted from each result, 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: When a constant is added or subtracted from each observation, variance remains unchanged. Step 3: Original variance = 36. After multiplying by 5: new variance = 36 × 5² = 36 × 25 = 900. Step 4: Subtracting 3 does not change variance. Therefore, the new variance is 900.

Question 6

If each observation in a data set is multiplied by 3, and the standard deviation of the original data set is 4, what is the variance of the new data set?

Accepted Answer

Step 1: Recall the property — if each observation is multiplied by a constant k, the new standard deviation = k × original SD. Step 2: New SD = 3 × 4 = 12. Step 3: Variance = (SD)² = 12² = 144. Note: A common mistake is multiplying the original variance (16) by 3, giving 48, which is wrong. Therefore, answer is 144.

Question 7

Two datasets have the same mean of 50 but different standard deviations. Dataset A has SD = 5 and Dataset B has SD = 10. If we increase every value in both datasets by 20, what will be the ratio of the new standard deviations?

Accepted Answer

Step 1: Adding a constant to all values does not change standard deviation (only location, not spread changes). Step 2: Dataset A remains SD = 5, Dataset B remains SD = 10. Step 3: Ratio = 5:10 = 1:2.

Question 8

Two datasets A and B have the same mean of 50. Dataset A consists of 5 observations with variance 16, while Dataset B consists of 10 observations with variance 25. If the two datasets are combined into a single dataset of 15 observations, what is the variance of the combined dataset?

Accepted Answer

Step 1: Calculate the sum of squared deviations for Dataset A: Variance_A = Σ(x - mean)²/n_A, so Σ(x - mean)² = 16 × 5 = 80. Step 2: Calculate the sum of squared deviations for Dataset B: Σ(x - mean)² = 25 × 10 = 250. Step 3: Since both datasets have the same mean (50), when combined, the total sum of squared deviations = 80 + 250 = 330. Step 4: Combined variance = 330 / 15 = 22. Therefore, the variance of the combined dataset is 22.

Question 9

The variance of the dataset {1, 2, 3, 4, 5} is:

Accepted Answer

Step 1: Find the mean = (1 + 2 + 3 + 4 + 5) ÷ 5 = 15 ÷ 5 = 3. Step 2: Calculate squared deviations: (1−3)² = 4, (2−3)² = 1, (3−3)² = 0, (4−3)² = 1, (5−3)² = 4. Step 3: Sum of squared deviations = 4 + 1 + 0 + 1 + 4 = 10. Step 4: Variance = 10 ÷ 5 = 2.

Question 10

The standard deviation of the dataset {2, 4, 6, 8, 10} is 2√2. What is the variance of this dataset?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = (2√2)² = 4 × 2 = 8. Therefore, the variance is 8.

Question 11

The standard deviation of the dataset {5, 5, 5, 5, 5} is:

Accepted Answer

Step 1: All values are identical, so the 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 12

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

Question 13

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 14

If the variance of a dataset is 16, what is the standard deviation?

Accepted Answer

Step 1: Recall the relationship: Standard Deviation = √Variance. Step 2: SD = √16 = 4.

Question 15

Five students scored: 10, 20, 30, 40, 50 marks. The mean is 30. What is the variance of their scores?

Accepted Answer

Step 1: Mean = 30 (given). Step 2: Calculate deviations from mean: (10−30)² = 400, (20−30)² = 100, (30−30)² = 0, (40−30)² = 100, (50−30)² = 400. Step 3: Sum of squared deviations = 400 + 100 + 0 + 100 + 400 = 1000. Step 4: Variance = 1000 ÷ 5 = 200. Therefore, the variance is 200.

Question 16

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 17

The standard deviation of the dataset {3, 6, 9} is approximately:

Accepted Answer

Step 1: Find the mean = (3 + 6 + 9) ÷ 3 = 18 ÷ 3 = 6. Step 2: Calculate squared deviations: (3-6)² = 9, (6-6)² = 0, (9-6)² = 9. Step 3: Sum of squared deviations = 9 + 0 + 9 = 18. Step 4: Variance = 18 ÷ 3 = 6. Step 5: Standard Deviation = √6 ≈ 2.45.

Question 18

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 19

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: Calculate 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 20

A dataset has a standard deviation of 5. If each value in the dataset is multiplied by 3, what will be the new standard deviation?

Accepted Answer

Step 1: Recall the property: when all values are multiplied by a constant k, the standard deviation is also multiplied by k. Step 2: New SD = Original SD × 3 = 5 × 3 = 15. Therefore, the new standard deviation is 15.

Question 21

The standard deviation of a dataset is 4. If each data point is multiplied by 3, the new standard deviation will be:

Accepted Answer

Step 1: Recall the property: when each data point is multiplied by a constant k, the new standard deviation = k × (original standard deviation). Step 2: New SD = 3 × 4 = 12.

Question 22

The variance of the dataset {5, 5, 5, 5, 5} is:

Accepted Answer

Step 1: Find the mean = (5 + 5 + 5 + 5 + 5) ÷ 5 = 25 ÷ 5 = 5. Step 2: Each deviation from mean = 5 − 5 = 0. Step 3: Sum of squared deviations = 0 + 0 + 0 + 0 + 0 = 0. Step 4: Variance = 0 ÷ 5 = 0.

Question 23

The variance of the dataset {3, 6, 9} is:

Accepted Answer

Step 1: Find the mean = (3 + 6 + 9) ÷ 3 = 18 ÷ 3 = 6. Step 2: Find squared deviations: (3−6)² = 9, (6−6)² = 0, (9−6)² = 9. Step 3: Sum of squared deviations = 9 + 0 + 9 = 18. Step 4: Variance = 18 ÷ 3 = 6.

Question 24

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 25

If each value in a dataset is increased by 7, the standard deviation will:

Accepted Answer

Step 1: Adding a constant to all values shifts the mean but does not change the deviations from the mean. Step 2: If original data is {x₁, x₂, ..., xₙ} with mean μ, new data is {x₁+7, x₂+7, ..., xₙ+7} with mean μ+7. Step 3: New deviation = (xᵢ + 7) − (μ + 7) = xᵢ − μ = original deviation. Step 4: Since deviations remain unchanged, variance and standard deviation remain unchanged.

Question 26

The standard deviation of five numbers is 6. If 4 is added to each number, what is the new standard deviation?

Accepted Answer

Step 1: Recall that adding a constant to all observations does not change the standard deviation (it only shifts the mean). Step 2: The new standard deviation remains 6. Therefore, the answer is 6.

Question 27

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 standard deviation is unaffected by adding or subtracting a constant from all observations. Step 2: Only the mean shifts by that constant. Step 3: Original SD = 5. Step 4: New SD = 5 (unchanged).

Question 28

The mean of a dataset is 50 and the standard deviation is 10. If the variance is 100, what is the coefficient of variation (as a percentage)?

Accepted Answer

Step 1: Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100. Step 2: CV = (10 / 50) × 100 = 0.2 × 100 = 20%.

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