1. **Problem Statement:** Given a grouped frequency distribution of employee performance rates, we need to find: i) Modal performance rate ii) Mean deviation iii) Coefficient of variation iv) Quartile coefficient of dispersion v) Pearson's coefficient of skewness 2. **Data Table:** | Performance | Frequency (f) | |-------------|---------------| | 10-20 | 3 | | 20-30 | 6 | | 30-40 | 10 | | 40-50 | 15 | | 50-60 | 7 | | 60-70 | 12 | | 70-80 | 6 | 3. **Step i) Estimate Modal Performance Rate:** - Mode formula for grouped data: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ where: - $L$ = lower boundary of modal class - $f_1$ = frequency of modal class - $f_0$ = frequency of class before modal class - $f_2$ = frequency of class after modal class - $h$ = class width - Modal class is the class with highest frequency: 40-50 with $f_1=15$ - $L=40$, $f_0=10$ (30-40), $f_2=7$ (50-60), $h=10$ Calculate: $$\text{Mode} = 40 + \frac{(15 - 10)}{(2 \times 15 - 10 - 7)} \times 10 = 40 + \frac{5}{30 - 17} \times 10 = 40 + \frac{5}{13} \times 10 = 40 + 3.85 = 43.85$$ 4. **Step ii) Calculate Mean Deviation:** - Find class midpoints $x_i$: 10-20: 15, 20-30: 25, 30-40: 35, 40-50: 45, 50-60: 55, 60-70: 65, 70-80: 75 - Calculate mean $\bar{x}$: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{3\times15 + 6\times25 + 10\times35 + 15\times45 + 7\times55 + 12\times65 + 6\times75}{3+6+10+15+7+12+6}$$ $$= \frac{45 + 150 + 350 + 675 + 385 + 780 + 450}{59} = \frac{2835}{59} \approx 48.05$$ - Calculate mean deviation: $$\text{MD} = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$$ Calculate each $|x_i - \bar{x}|$ and multiply by $f_i$: - $3 \times |15 - 48.05| = 3 \times 33.05 = 99.15$ - $6 \times |25 - 48.05| = 6 \times 23.05 = 138.3$ - $10 \times |35 - 48.05| = 10 \times 13.05 = 130.5$ - $15 \times |45 - 48.05| = 15 \times 3.05 = 45.75$ - $7 \times |55 - 48.05| = 7 \times 6.95 = 48.65$ - $12 \times |65 - 48.05| = 12 \times 16.95 = 203.4$ - $6 \times |75 - 48.05| = 6 \times 26.95 = 161.7$ Sum: $$99.15 + 138.3 + 130.5 + 45.75 + 48.65 + 203.4 + 161.7 = 827.45$$ Mean deviation: $$\text{MD} = \frac{827.45}{59} \approx 14.03$$ 5. **Step iii) Coefficient of Variation (CV):** - Calculate standard deviation $\sigma$: $$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$$ Calculate each $(x_i - \bar{x})^2 f_i$: - $3 \times (15 - 48.05)^2 = 3 \times 1092.3 = 3276.9$ - $6 \times (25 - 48.05)^2 = 6 \times 531.3 = 3187.8$ - $10 \times (35 - 48.05)^2 = 10 \times 170.3 = 1703$ - $15 \times (45 - 48.05)^2 = 15 \times 9.3 = 139.5$ - $7 \times (55 - 48.05)^2 = 7 \times 48.3 = 338.1$ - $12 \times (65 - 48.05)^2 = 12 \times 287.3 = 3447.6$ - $6 \times (75 - 48.05)^2 = 6 \times 726.3 = 4357.8$ Sum: $$3276.9 + 3187.8 + 1703 + 139.5 + 338.1 + 3447.6 + 4357.8 = 16450.7$$ Standard deviation: $$\sigma = \sqrt{\frac{16450.7}{59}} = \sqrt{278.8} \approx 16.7$$ - Coefficient of variation: $$\text{CV} = \frac{\sigma}{\bar{x}} \times 100 = \frac{16.7}{48.05} \times 100 \approx 34.76\%$$ 6. **Step iv) Quartile Coefficient of Dispersion:** - Find $Q_1$ and $Q_3$ using cumulative frequency: Cumulative frequencies: 3, 9, 19, 34, 41, 53, 59 - $Q_1$ position: $\frac{1}{4} \times 59 = 14.75$th value (in 30-40 class) - $Q_3$ position: $\frac{3}{4} \times 59 = 44.25$th value (in 50-60 class) Calculate $Q_1$: - Lower boundary $L=30$, cumulative frequency before class $F=9$, frequency $f=10$, class width $h=10$ $$Q_1 = L + \frac{(\frac{n}{4} - F)}{f} \times h = 30 + \frac{(14.75 - 9)}{10} \times 10 = 30 + 5.75 = 35.75$$ Calculate $Q_3$: - Lower boundary $L=50$, cumulative frequency before class $F=41$, frequency $f=7$, class width $h=10$ $$Q_3 = 50 + \frac{(44.25 - 41)}{7} \times 10 = 50 + \frac{3.25}{7} \times 10 = 50 + 4.64 = 54.64$$ Quartile coefficient of dispersion: $$\text{QCD} = \frac{Q_3 - Q_1}{Q_3 + Q_1} = \frac{54.64 - 35.75}{54.64 + 35.75} = \frac{18.89}{90.39} \approx 0.209$$ 7. **Step v) Pearson's Coefficient of Skewness:** - Using mean, mode, and standard deviation: $$\text{Skewness} = \frac{\bar{x} - \text{Mode}}{\sigma} = \frac{48.05 - 43.85}{16.7} = \frac{4.2}{16.7} \approx 0.25$$ **Final answers:** - Modal performance rate $\approx 43.85$ - Mean deviation $\approx 14.03$ - Coefficient of variation $\approx 34.76\%$ - Quartile coefficient of dispersion $\approx 0.209$ - Pearson's coefficient of skewness $\approx 0.25$