Question: User: D. x̄ = 8.27
class interval f x fx x^2 fx^2 f(x−x̄)^2
1-3 9 2 18 4 36 353.73
4-6 21 5 105 25 525 224.45
7-9 22 8 176 64 1408 1.59
10-12 18 11 196 121 2178 134.23
13-15 1 14 14 194 196 32.84
16-18 3 17 51 289 867 228.68
19-21 3 20 60 400 1200 412.93
22-24 1 23 23 529 529 219
N = 78
Σfx = 645
Σf(x−x̄)^2 = 1605.346
σ² = Σf(x−x̄)^2 / N = 1605.346 / 78 = 20.581
therefore the population variance of the data is 20.581
s² = Σf(x−x̄)^2 / N−1 = 1605.346 / 77 = 20.847
therefore the sample variance of the data is 20.847
σ = √20.581 = 4.537
therefore the population standard deviation of the data is 4.537
s = √20.847 = 4.564
therefore the sample standard deviation of the data is 4.564
QD = (Q3 − Q1) / 2 = (10.68 − 5) / 2 = 2.79
therefore, the quartile deviation of the data is 2.79
PR = P90 − P10 = 13.10 − 3.10 = 10
therefore, the percentile range of the data is 10
1. **State the problem:** Calculate the population variance ($\sigma^2$), sample variance ($s^2$), population standard deviation ($\sigma$), sample standard deviation ($s$), quartile deviation (QD), and percentile range (PR) from the given frequency distribution data with mean $\bar{x} = 8.27$.
2. **Recall formulas:**
- Population variance: $$\sigma^2 = \frac{\sum f(x - \bar{x})^2}{N}$$
- Sample variance: $$s^2 = \frac{\sum f(x - \bar{x})^2}{N - 1}$$
- Population standard deviation: $$\sigma = \sqrt{\sigma^2}$$
- Sample standard deviation: $$s = \sqrt{s^2}$$
- Quartile deviation: $$QD = \frac{Q_3 - Q_1}{2}$$
- Percentile range: $$PR = P_{90} - P_{10}$$
3. **Given data:**
- $N = 78$
- $\sum f(x - \bar{x})^2 = 1605.346$
- $Q_3 = 10.68$, $Q_1 = 5$
- $P_{90} = 13.10$, $P_{10} = 3.10$
4. **Calculate population variance:**
$$\sigma^2 = \frac{1605.346}{78} = 20.581$$
5. **Calculate sample variance:**
$$s^2 = \frac{1605.346}{77} = 20.847$$
6. **Calculate population standard deviation:**
$$\sigma = \sqrt{20.581} = 4.537$$
7. **Calculate sample standard deviation:**
$$s = \sqrt{20.847} = 4.564$$
8. **Calculate quartile deviation:**
$$QD = \frac{10.68 - 5}{2} = \frac{5.68}{2} = 2.79$$
9. **Calculate percentile range:**
$$PR = 13.10 - 3.10 = 10$$
**Final answers:**
- Population variance: $20.581$
- Sample variance: $20.847$
- Population standard deviation: $4.537$
- Sample standard deviation: $4.564$
- Quartile deviation: $2.79$
- Percentile range: $10$