Question

Which of the following data sets contains the smallest variance?

A. {12,14,16}
B. {12, 14, 18}
C. {120, 140, 160}
D. {60, 61, 59}
E. {20, 20, 25}

Answer 1

After calculating the variance for each data set, it is found that data set D, which contains the numbers {60, 61, 59}, has the smallest variance of 1.

Step-by-step explanation:

The student has asked which of the following data sets contains the smallest variance: A. {12,14,16} B. {12, 14, 18} C. {120, 140, 160} D. {60, 61, 59} E. {20, 20, 25}. To answer this, we need to calculate the variance for each data set. Variance is a measure of how spread out the numbers in the data set are. It is calculated by taking the average of the squared differences from the mean.

• For A: Mean = 14, Variance = ((2^2 + 0^2 + 2^2) / 3) = 8/3 ≈ 2.67
• For B: Mean = 14.67, Variance = ((2.67^2 + 0.67^2 + 3.33^2) / 3) ≈ 7.56
• For C: Mean = 140, Variance = ((20^2 + 0^2 + 20^2) / 3) = 800/3 ≈ 266.67
• For D: Mean = 60, Variance = ((1^2 + 1^2 + 1^2) / 3) = 3/3 = 1
• For E: Mean = 21.67, Variance = ((1.67^2 + 1.67^2 + 3.33^2) / 3) ≈ 7.56

Data set D has the smallest variance, which is 1.

Answer 2

The data set with the smallest variance is Option A: {12, 14, 16}.

Step-by-step explanation:

Variance is a measure of how spread out a set of values is. It is calculated as the average of the squared differences from the mean. The formula for variance (σ²) is given by:

`\[ \sigma² = (1)/(N) \sum_(i=1)^(N) (x_i - \bar{x})^2 \]`

where ( N ) is the number of data points,( x_i ) is each individual data point, and ( bar{x} ) is the mean of the data set.

For Option A: {12, 14, 16}, the mean (( bar{x} )) is ((12 + 14 + 16) / 3 = 14). The squared differences from the mean are ((12-14)² + (14-14)² + (16-14)² = 4 + 0 + 4 = 8\). The variance is ( sigma² = 8/3 ), which is approximately 2.67.

For the other options, the calculations are as follows:

- Option B: {12, 14, 18} → ( sigma² = 2/3 ) (approximately 0.67)

- Option C: {120, 140, 160} → ( sigma² = 800/3 ) (approximately 266.67)

- Option D: {60, 61, 59} → ( sigma² = 2/3 ) (approximately 0.67)

- Option E: {20, 20, 25} → ( sigma² = 5/3 ) (approximately 1.67)

Comparing the variances, Option A has the smallest variance, making it the data set with the least spread.

Full Question:

Which of the following data sets contains the smallest variance?