Question

The variance of the discrete data 3, 4, 5, 6, 7, 8, 10, 13 is

• A. 7.5
• B. 8
• C. 9.5
• D. 9

Hint:

When calculating variance, you have two main formulas. The definitional formula $\frac{\sum (x_i - \mu)^2}{N}$ is good if the mean is a nice integer. The computational formula $\frac{\sum x_i^2}{N} - \mu^2$ is generally faster if you're using a calculator but can be prone to rounding errors if the mean is not exact. Always double-check your arithmetic, as it's easy to make a small error in summing or squaring.

Answer 1

The Correct Option is D

Step 1: Calculate the Mean (\( \bar{x} \)): Data: 3, 4, 5, 6, 7, 8, 10, 13. Number of terms \( n = 8 \). \[ \bar{x} = \frac{3+4+5+6+7+8+10+13}{8} = \frac{56}{8} = 7 \]
Step 2: Calculate Variance (\( \sigma^2 \)): Formula: \( \sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 \). Calculate squared deviations from mean (7): \( (3-7)^2 = 16 \) \( (4-7)^2 = 9 \) \( (5-7)^2 = 4 \) \( (6-7)^2 = 1 \) \( (7-7)^2 = 0 \) \( (8-7)^2 = 1 \) \( (10-7)^2 = 9 \) \( (13-7)^2 = 36 \) Sum of squared deviations \( = 16 + 9 + 4 + 1 + 0 + 1 + 9 + 36 = 76 \). \[ \sigma^2 = \frac{76}{8} = 9.5 \]