Question 1

If each value of a data set is multiplied by 3, the variance of the new data set becomes how many times the original variance?

Accepted Answer

Step 1: Recall the effect of scaling on variance. If each observation is multiplied by a constant k, the new variance becomes k² times the original variance. Step 2: Here k = 3, so the new variance = 3² × original variance = 9 × original variance. Therefore, the variance becomes 9 times the original variance. (Note: Standard deviation would become 3 times, but variance becomes 9 times.)

Question 2

The mean of 10 observations is 20 and the mean of their squares is 450. A new observation of value 30 is added to the dataset. What is the variance of the 11 observations?

Accepted Answer

Step 1: For original 10 observations, Mean = 20, Mean of squares = 450. Variance₁₀ = 450 − 20² = 450 − 400 = 50. Step 2: Sum of original 10 observations = 10 × 20 = 200. Sum of squares = 10 × 450 = 4500. Step 3: After adding 30, new sum = 200 + 30 = 230, new mean = 230/11. New sum of squares = 4500 + 30² = 4500 + 900 = 5400. Step 4: New variance = (5400/11) − (230/11)² = 5400/11 − 52900/121 = (5400 × 11 − 52900)/121 = (59400 − 52900)/121 = 6500/121 ≈ 53.72. Wait — let me use cleaner numbers. New variance = 5400/11 − (230/11)² = (5400 × 121 − 230² × 11)/(11² × 11). Using direct formula: Var = [ΣX²/n − (ΣX/n)²] = 5400/11 − (230/11)² = 490.909 − 437.190 = 53.719. Rounding to clean form = 6500/121. Therefore, variance = 6500/121.

Question 3

The variance of the dataset {1, 3, 5, 7, 9} is:

Accepted Answer

Step 1: Calculate the mean = (1 + 3 + 5 + 7 + 9) ÷ 5 = 25 ÷ 5 = 5. Step 2: Find 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 of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 4

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 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 5

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

Accepted Answer

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

Question 6

The variance of the dataset {5, 5, 5, 5, 5} 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 + 0 + 0 + 0 + 0 = 0. Step 4: Variance = 0 ÷ 5 = 0.

Question 7

The standard deviation of the dataset {1, 3, 5, 7, 9} is:

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 = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8. Step 5: SD = √8 = 2√2 ≈ 2.83.

Question 8

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 9

The variance of the dataset {10, 20, 30} is:

Accepted Answer

Step 1: Mean = (10 + 20 + 30) ÷ 3 = 60 ÷ 3 = 20. Step 2: Squared deviations: (10−20)² = 100, (20−20)² = 0, (30−20)² = 100. Step 3: Sum of squared deviations = 100 + 0 + 100 = 200. Step 4: Variance = 200 ÷ 3 = 66.67 (or 200/3).

Question 10

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

Accepted Answer

Step 1: Calculate the mean = (5 + 5 + 5 + 5 + 5) ÷ 5 = 25 ÷ 5 = 5. Step 2: Find squared deviations: (5−5)² = 0 for all five values. Step 3: Sum of squared deviations = 0. Step 4: Variance = 0 ÷ 5 = 0. Step 5: Standard Deviation = √0 = 0.

Question 11

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 12

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 13

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

Accepted Answer

Step 1: Recall that adding a constant to each observation does not change the standard deviation (it only shifts the mean). Step 2: New standard deviation = 4 (unchanged).

Question 14

The mean of a dataset is 50 and the standard deviation is 10. If the dataset is transformed by the formula Y = 2X − 30, what is the new standard deviation?

Accepted Answer

Step 1: Recall that SD(aX + b) = |a| × SD(X), where a and b are constants. Step 2: Here, a = 2 and b = −30. Step 3: New SD = |2| × 10 = 2 × 10 = 20.

Question 15

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

Accepted Answer

Step 1: Calculate 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 = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 16

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 the spread or dispersion of data around the mean. Step 2: Dataset B (SD = 12) has greater spread than Dataset A (SD = 8). Step 3: Therefore, Dataset B is more dispersed/variable than Dataset A.

Question 17

For a dataset, the mean is 50 and the standard deviation is 8. If the variance is calculated for a new dataset where each value is decreased by 10 and then divided by 2, what is the new variance?

Accepted Answer

Step 1: Original variance = SD² = 8² = 64. Step 2: When subtracting a constant, variance is unchanged. Step 3: When dividing by 2, variance is multiplied by (1/2)² = 1/4. Step 4: New variance = 64 × (1/4) = 16.

Question 18

The coefficient of variation (CV) of a dataset is defined as CV = (SD ÷ Mean) × 100%. If a dataset has mean 40 and standard deviation 8, what is its coefficient of variation?

Accepted Answer

Step 1: Apply the formula CV = (SD ÷ Mean) × 100%. Step 2: CV = (8 ÷ 40) × 100% = 0.2 × 100% = 20%.

Question 19

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. What is the combined variance when both datasets are merged?

Accepted Answer

Step 1: Use the formula for combined variance: σ²_combined = [n₁σ₁² + n₂σ₂² + n₁(x̄₁ − x̄_overall)² + n₂(x̄₂ − x̄_overall)²] / (n₁ + n₂). Step 2: Since both means are 50, x̄_overall = 50, so the deviation terms are zero. Step 3: σ²_combined = [8(16) + 12(36)] / 20 = [128 + 432] / 20 = 560 / 20 = 28.

Question 20

The standard deviation of a set of 5 numbers is 6. If the sum of squares of deviations from the mean is S, what is S?

Accepted Answer

Step 1: Recall that variance σ² = S/n, where S is the sum of squared deviations and n is the number of observations. Step 2: Standard deviation σ = 6, so variance σ² = 36. Step 3: S = σ² × n = 36 × 5 = 180.

Question 21

Three datasets have variances 25, 36, and 49 respectively. If a new dataset is formed by combining all three with equal sample sizes, and the means of all three datasets are identical, what is the standard deviation of the combined dataset?

Accepted Answer

Step 1: When datasets with equal sample sizes and identical means are combined, the combined variance equals the average of individual variances. Step 2: Combined variance = (25 + 36 + 49) / 3 = 110 / 3 ≈ 36.67. Step 3: Combined SD = √36.67 ≈ 6.06 ≈ 6.

Question 22

A frequency distribution has mean 50 and variance 100. The frequencies are 10, 15, and 25 for three class intervals. If the class midpoints are 40, 50, and 60 respectively, verify the variance and find the coefficient of variation.

Accepted Answer

Step 1: Total frequency N = 10 + 15 + 25 = 50. Step 2: Verify mean: (10×40 + 15×50 + 25×60) / 50 = (400 + 750 + 1500) / 50 = 2650 / 50 = 53 (given as 50, so assume it is 53 for this problem). Step 3: Variance = Σf(x − x̄)² / N = [10(40−53)² + 15(50−53)² + 25(60−53)²] / 50 = [10(169) + 15(9) + 25(49)] / 50 = [1690 + 135 + 1225] / 50 = 3050 / 50 = 61. Step 4: Given variance is 100, so SD = 10. Step 5: Coefficient of variation = (SD / mean) × 100 = (10 / 50) × 100 = 20%.

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