Statistics Question 8
Question 8 - 2024 (30 Jan Shift 2)
The variance $\sigma^{2}$ of the data
| $x_{i}$ | 0 | 1 | 5 | 6 | 10 | 12 | 17 |
|---|---|---|---|---|---|---|---|
| $f_{i}$ | 3 | 2 | 3 | 2 | 6 | 3 | 3 |
Is
Show Answer
Answer (29)
Solution
| $\mathrm{x}_{\mathrm{i}}$ | $\mathrm{f}_{\mathrm{i}}$ | $\mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}$ | $\mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}{ }^{2}$ |
|---|---|---|---|
| 0 | 3 | 0 | 0 |
| 1 | 2 | 2 | 2 |
| 5 | 3 | 15 | 75 |
| 6 | 2 | 12 | 72 |
| 10 | 6 | 60 | 600 |
| 12 | 3 | 36 | 432 |
| 17 | 3 | 51 | 867 |
| $\sum \mathrm{f}_{\mathrm{i}}=22$ | $\sum \mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}{ }^{2}=2048$ |
$\therefore \quad \sum \mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}=176$
So $\overline{\mathrm{x}}=\frac{\sum \mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}=\frac{176}{22}=8$
for $\sigma^{2}=\frac{1}{\mathrm{~N}} \sum \mathrm{f}{\mathrm{i}} \mathrm{x}{\mathrm{i}}^{2}-(\overline{\mathrm{x}})^{2}$
$=\frac{1}{22} \times 2048-(8)^{2}$
$=93.090964$
$=29.0909$
