1. **State the problem:** We have monthly take-home salaries data and need to construct a frequency distribution using Sturge's rule, then find the mean, median, mode, mean absolute deviation (MAD), variance, and standard deviation. 2. **List the data:** $$526, 675, 552, 636, 580, 643, 600, 523, 727, 502, 686, 547, 481, 601, 573, 558, 586, 597, 545, 547, 630, 532, 659, 544, 507, 641, 585, 620, 613, 710, 509, 480, 487, 688, 604$$ 3. **Count the number of data points:** There are $n=35$ salaries. 4. **Find the range:** Minimum salary $=480$ Maximum salary $=727$ Range $=727 - 480 = 247$ 5. **Determine number of classes using Sturge's rule:** $$k = 1 + 3.322 \log_{10}(n) = 1 + 3.322 \log_{10}(35) \approx 1 + 3.322 \times 1.544 = 6.13 \approx 6$$ 6. **Calculate class width:** $$\text{class width} = \frac{\text{range}}{k} = \frac{247}{6} \approx 41.17 \approx 42$$ 7. **Construct frequency distribution classes:** Classes (start at 480): - 480–521 - 522–563 - 564–605 - 606–647 - 648–689 - 690–731 8. **Tally frequencies:** - 480–521: 480, 481, 487, 502, 507, 509, 523 (7 values) - 522–563: 523, 526, 532, 544, 545, 547, 547, 552, 558 (9 values) - 564–605: 573, 580, 585, 586, 597, 600, 601, 604 (8 values) - 606–647: 613, 620, 630, 636, 641, 643 (6 values) - 648–689: 659, 675, 686, 688 (4 values) - 690–731: 710, 727 (2 values) 9. **Calculate midpoints for each class:** - 480–521: $\frac{480+521}{2} = 500.5$ - 522–563: $\frac{522+563}{2} = 542.5$ - 564–605: $\frac{564+605}{2} = 584.5$ - 606–647: $\frac{606+647}{2} = 626.5$ - 648–689: $\frac{648+689}{2} = 668.5$ - 690–731: $\frac{690+731}{2} = 710.5$ 10. **Calculate mean:** $$\bar{x} = \frac{\sum f_i x_i}{n}$$ Where $f_i$ is frequency and $x_i$ is midpoint. $$\bar{x} = \frac{7\times500.5 + 9\times542.5 + 8\times584.5 + 6\times626.5 + 4\times668.5 + 2\times710.5}{35}$$ $$= \frac{3503.5 + 4882.5 + 4676 + 3759 + 2674 + 1421}{35} = \frac{20916}{35} = 597.6$$ 11. **Calculate median:** Median class is the class where cumulative frequency reaches $\frac{n}{2} = 17.5$. Cumulative frequencies: - 7 - 16 - 24 (median class: 564–605) Median formula: $$\text{Median} = L + \left(\frac{\frac{n}{2} - F}{f_m}\right) \times w$$ Where: - $L=563.5$ (lower boundary of median class) - $F=16$ (cumulative frequency before median class) - $f_m=8$ (frequency of median class) - $w=42$ (class width) $$\text{Median} = 563.5 + \left(\frac{17.5 - 16}{8}\right) \times 42 = 563.5 + 0.1875 \times 42 = 563.5 + 7.875 = 571.375$$ 12. **Calculate mode:** Mode class is the class with highest frequency: 522–563 (9 values). Mode formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times w$$ Where: - $L=521.5$ (lower boundary of modal class) - $f_1=9$ (frequency of modal class) - $f_0=7$ (frequency before modal class) - $f_2=8$ (frequency after modal class) - $w=42$ $$\text{Mode} = 521.5 + \frac{9 - 7}{2\times9 - 7 - 8} \times 42 = 521.5 + \frac{2}{18 - 15} \times 42 = 521.5 + \frac{2}{3} \times 42 = 521.5 + 28 = 549.5$$ 13. **Calculate Mean Absolute Deviation (MAD):** $$\text{MAD} = \frac{\sum f_i |x_i - \bar{x}|}{n}$$ Calculate $|x_i - \bar{x}|$ for each midpoint and multiply by frequency: - $|500.5 - 597.6| = 97.1 \times 7 = 679.7$ - $|542.5 - 597.6| = 55.1 \times 9 = 495.9$ - $|584.5 - 597.6| = 13.1 \times 8 = 104.8$ - $|626.5 - 597.6| = 28.9 \times 6 = 173.4$ - $|668.5 - 597.6| = 70.9 \times 4 = 283.6$ - $|710.5 - 597.6| = 112.9 \times 2 = 225.8$ Sum = 1963.2 $$\text{MAD} = \frac{1963.2}{35} = 56.1$$ 14. **Calculate variance:** $$\text{Variance} = \frac{\sum f_i (x_i - \bar{x})^2}{n}$$ Calculate $(x_i - \bar{x})^2 \times f_i$: - $(97.1)^2 \times 7 = 9409.4 \times 7 = 65865.8$ - $(55.1)^2 \times 9 = 3036.0 \times 9 = 27324.0$ - $(13.1)^2 \times 8 = 171.6 \times 8 = 1372.8$ - $(28.9)^2 \times 6 = 835.2 \times 6 = 5011.2$ - $(70.9)^2 \times 4 = 5026.8 \times 4 = 20107.2$ - $(112.9)^2 \times 2 = 12748.4 \times 2 = 25496.8$ Sum = 145177.8 $$\text{Variance} = \frac{145177.8}{35} = 4147.9$$ 15. **Calculate standard deviation:** $$\text{SD} = \sqrt{\text{Variance}} = \sqrt{4147.9} = 64.4$$ **Final answers:** - Mean $= 597.6$ - Median $= 571.4$ - Mode $= 549.5$ - MAD $= 56.1$ - Variance $= 4147.9$ - Standard deviation $= 64.4$