Question

Which of the following is the variance of the data? \[ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20,\ 22,\ 24 \]

• A. \(15\)
• B. \(33\)
• C. \(5.74\)
• D. \(16\)

Hint:

Variance is the mean of squared deviations from the arithmetic mean.

Answer 1

The Correct Option is B

Concept:
Variance measures the average of squared deviations from the mean. \[ \sigma^2=\frac{\sum (x_i-\bar{x})^2}{n} \]

Step 1: Write the data. \[ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20,\ 22,\ 24 \]

Step 2: Find the mean. \[ \bar{x}=\frac{6+8+10+12+14+16+18+20+22+24}{10} \] \[ \bar{x}=\frac{150}{10} \] \[ \bar{x}=15 \]

Step 3: Find deviations from mean. \[ 6-15=-9 \] \[ 8-15=-7 \] \[ 10-15=-5 \] \[ 12-15=-3 \] \[ 14-15=-1 \] \[ 16-15=1 \] \[ 18-15=3 \] \[ 20-15=5 \] \[ 22-15=7 \] \[ 24-15=9 \]

Step 4: Square the deviations. \[ (-9)^2=81,\quad (-7)^2=49,\quad (-5)^2=25,\quad (-3)^2=9,\quad (-1)^2=1 \] \[ 1^2=1,\quad 3^2=9,\quad 5^2=25,\quad 7^2=49,\quad 9^2=81 \]

Step 5: Add squared deviations. \[ 81+49+25+9+1+1+9+25+49+81=330 \]

Step 6: Find variance. \[ \sigma^2=\frac{330}{10} \] \[ \sigma^2=33 \] \[ \therefore \text{Correct Answer is (B)} \]