Statistics
Variance of frequency distribution
nta_pyq_2023_jan
Grade 11
Question:
If the variance of the frequency distribution
| $x_i$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|--------|---|---|----|----|---|---|---|
| Frequency $f_i$ | 3 | 6 | 16 | $\alpha$ | 9 | 5 | 6 |
is 3, then $\alpha$ is equal to ______.
Step-by-Step Solution
Key Concept: Use the variance formula $\sigma^2 = \frac{\sum f_i d_i^2}{\sum f_i} - \left(\frac{\sum f_i d_i}{\sum f_i}\right)^2$ with deviation $d_i = x_i - 5$ as the assumed mean. Note that $\sum f_i d_i = 0$ (verified from solution).
$\sum f_i d_i^2 = 27+24+16+0+9+20+54 = 150$. $\sum f_i d_i = 0$ (mean = 5). $\sigma^2 = \frac{150}{45+\alpha} = 3 \Rightarrow \alpha = 5$.
Correct Answer: 5