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 measures how far a set of numbers is spread out from their average value. It is defined as the average of the squared differences from the mean.
Step 2: Key Formula or Approach:
First, calculate the mean (\(\mu\)):
\[ \mu = \frac{\sum x_i}{n} \] Then calculate the variance (\(\sigma^2\)):
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] Step 3: Detailed Explanation:
The dataset is \(5, 7, 9, 11, 13\). There are \(n = 5\) terms.
Calculate the mean:
\[ \mu = \frac{5 + 7 + 9 + 11 + 13}{5} = \frac{45}{5} = 9 \] Now, calculate the squared differences from the mean:
For \(5\): \((5 - 9)^2 = (-4)^2 = 16\)
For \(7\): \((7 - 9)^2 = (-2)^2 = 4\)
For \(9\): \((9 - 9)^2 = 0^2 = 0\)
For \(11\): \((11 - 9)^2 = 2^2 = 4\)
For \(13\): \((13 - 9)^2 = 4^2 = 16\)
Sum the squared differences:
\[ \sum (x_i - \mu)^2 = 16 + 4 + 0 + 4 + 16 = 40 \] Calculate the variance:
\[ \sigma^2 = \frac{40}{5} = 8 \] Step 4: Final Answer:
The variance is 8.