Question

Given the data: 5, 7, 9, 11, 13. The variance of the data, is

• A. 11
• B. 10
• C. 9
• D. 8
• E. 7

Hint:

For an Arithmetic Progression with common difference $d$ and $n$ terms, if $n$ is small, direct calculation is fastest. For larger sets, variance depends only on the number of terms and the common difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Variance ($\sigma^2$) measures the spread of data. It is calculated as the average of the squared deviations from the mean.
Key Formula or Approach:
1. Find Mean ($\mu$).
2. Variance $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$.

Step 2: Detailed Explanation:
Data: 5, 7, 9, 11, 13.
Number of values $n = 5$.
Mean $\mu$:
\[ \mu = \frac{5 + 7 + 9 + 11 + 13}{5} = \frac{45}{5} = 9 \]
Sum of squared deviations:
$(5 - 9)^2 = (-4)^2 = 16$
$(7 - 9)^2 = (-2)^2 = 4$
$(9 - 9)^2 = 0^2 = 0$
$(11 - 9)^2 = 2^2 = 4$
$(13 - 9)^2 = 4^2 = 16$
Sum = $16 + 4 + 0 + 4 + 16 = 40$.
Variance:
\[ \sigma^2 = \frac{40}{5} = 8 \]

Step 3: Final Answer:
The variance is 8.