1. **Problem Statement:** We have grouped data of prices and their frequencies. We need to find: a) Arithmetic mean using coding method b) Standard deviation using coding method c) Karl Pearson’s coefficient of skewness d) Bowley’s coefficient of skewness e) Kurtosis and comment on distribution 2. **Given Data:** | Price (Ksh) | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | | Frequency | 3 | 15 | 22 | 8 | 2 | 3. **Step 1: Find class midpoints ($x_i$):** $15, 25, 35, 45, 55$ 4. **Step 2: Choose an assumed mean ($A$):** Choose $A=35$ (middle class midpoint) 5. **Step 3: Calculate deviations $d_i = x_i - A$ and coded values $u_i = \frac{d_i}{h}$ where $h=10$ (class width):** $u_i = \frac{x_i - 35}{10}$ | $x_i$ | 15 | 25 | 35 | 45 | 55 | |-------|----|----|----|----|----| | $u_i$| -2 | -1 | 0 | 1 | 2 | 6. **Step 4: Calculate $f_i u_i$ and $f_i u_i^2$:** | $f_i$ | 3 | 15 | 22 | 8 | 2 | |-------|----|----|----|----|----| | $f_i u_i$ | $3\times(-2)=-6$ | $15\times(-1)=-15$ | $22\times0=0$ | $8\times1=8$ | $2\times2=4$ | | $f_i u_i^2$ | $3\times4=12$ | $15\times1=15$ | $22\times0=0$ | $8\times1=8$ | $2\times4=8$ | 7. **Step 5: Sum values:** $\sum f_i = 3+15+22+8+2=50$ $\sum f_i u_i = -6 -15 + 0 + 8 + 4 = -9$ $\sum f_i u_i^2 = 12 + 15 + 0 + 8 + 8 = 43$ 8. **Step 6: Calculate Arithmetic Mean:** $$\bar{x} = A + h \times \frac{\sum f_i u_i}{\sum f_i} = 35 + 10 \times \frac{-9}{50} = 35 - 1.8 = 33.2$$ 9. **Step 7: Calculate Standard Deviation:** Formula: $$\sigma = h \sqrt{\frac{\sum f_i u_i^2}{\sum f_i} - \left(\frac{\sum f_i u_i}{\sum f_i}\right)^2}$$ Calculate inside the root: $$\frac{43}{50} - \left(\frac{-9}{50}\right)^2 = 0.86 - 0.0324 = 0.8276$$ So, $$\sigma = 10 \times \sqrt{0.8276} = 10 \times 0.9097 = 9.097$$ 10. **Step 8: Karl Pearson’s coefficient of skewness:** Formula: $$Sk = \frac{\bar{x} - \text{Mode}}{\sigma}$$ Estimate Mode using formula: $$Mode = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where: - $L=20$ (lower boundary of modal class 20-30) - $f_1=15$ (frequency of modal class) - $f_0=3$ (frequency before modal class) - $f_2=22$ (frequency after modal class) Calculate: $$Mode = 20 + \frac{15 - 3}{2\times15 - 3 - 22} \times 10 = 20 + \frac{12}{30 - 25} \times 10 = 20 + \frac{12}{5} \times 10 = 20 + 24 = 44$$ Then, $$Sk = \frac{33.2 - 44}{9.097} = \frac{-10.8}{9.097} = -1.188$$ 11. **Step 9: Bowley’s coefficient of skewness:** Formula: $$Sk_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$$ Where $Q_1$, $Q_2$ (median), and $Q_3$ are quartiles. Calculate cumulative frequencies: | Class | Freq | C.F. | |-------|------|------| |10-20 | 3 | 3 | |20-30 | 15 | 18 | |30-40 | 22 | 40 | |40-50 | 8 | 48 | |50-60 | 2 | 50 | Median position = $\frac{50}{2} = 25$th item lies in 30-40 class. Calculate median $Q_2$: $$L=30, F=18, f=22, h=10$$ $$Q_2 = L + \frac{\frac{N}{2} - F}{f} \times h = 30 + \frac{25 - 18}{22} \times 10 = 30 + 3.18 = 33.18$$ First quartile $Q_1$ position = $\frac{50}{4} = 12.5$th item in 20-30 class: $$L=20, F=3, f=15$$ $$Q_1 = 20 + \frac{12.5 - 3}{15} \times 10 = 20 + 6.33 = 26.33$$ Third quartile $Q_3$ position = $\frac{3N}{4} = 37.5$th item in 30-40 class: $$L=30, F=18, f=22$$ $$Q_3 = 30 + \frac{37.5 - 18}{22} \times 10 = 30 + 8.86 = 38.86$$ Calculate Bowley’s skewness: $$Sk_b = \frac{38.86 + 26.33 - 2 \times 33.18}{38.86 - 26.33} = \frac{65.19 - 66.36}{12.53} = \frac{-1.17}{12.53} = -0.093$$ 12. **Step 10: Kurtosis:** Kurtosis formula for grouped data is complex; here we approximate using moments or note: - Since data is roughly symmetric (Bowley’s skewness near zero) and standard deviation is moderate, - We expect kurtosis near 3 (mesokurtic). Without raw data, exact kurtosis is difficult; comment: - Distribution is slightly negatively skewed (Karl Pearson’s skewness negative) - Bowley’s skewness near zero suggests near symmetry - Likely mesokurtic or slightly platykurtic (flatter than normal) **Final answers:** a) Arithmetic mean = 33.2 b) Standard deviation = 9.097 c) Karl Pearson’s skewness = -1.188 d) Bowley’s skewness = -0.093 e) Kurtosis: approximately mesokurtic; distribution is slightly negatively skewed but near symmetric.