Subtopic: Mean, median, mode and standard deviation
Question Statement:
A dataset consists of $5$ observations: $3, 7, 8, 12, 15$. The sample variance $s^2$ (computed using divisor $n - 1$) is ________ (round off to one decimal place).
Answer Key:
21.5
Step-by-Step Solution & Explanation
Compute the sample mean:
$\bar{x} = \frac{3 + 7 + 8 + 12 + 15}{5} = \frac{45}{5} = 9$
Compute the deviations from the mean and their squares:
- $(3 - 9)^2 = (-6)^2 = 36$
- $(7 - 9)^2 = (-2)^2 = 4$
- $(8 - 9)^2 = (-1)^2 = 1$
- $(12 - 9)^2 = 3^2 = 9$
- $(15 - 9)^2 = 6^2 = 36$
Sum of squared deviations:
$\sum_{i=1}^5 (x_i - \bar{x})^2 = 36 + 4 + 1 + 9 + 36 = 86$
Sample variance with $n - 1 = 4$ degrees of freedom:
$s^2 = \frac{86}{4} = 21.5$