1. **Problem Statement:** Find the midpoint, mean, median, mode, variance, standard deviation, and coefficients of mean, median, and mode for the frequency distribution given in QNO1. 2. **Given Data:** Weight intervals (kg): 0-9, 10-19, 20-29, 30-39, 40-49, 50-59 Frequencies (f): 3, 17, 36, 58, 27, 6 3. **Step 1: Find midpoints (x) of each class interval:** Midpoint formula: $$x = \frac{\text{lower limit} + \text{upper limit}}{2}$$ - 0-9: $\frac{0+9}{2} = 4.5$ - 10-19: $\frac{10+19}{2} = 14.5$ - 20-29: $\frac{20+29}{2} = 24.5$ - 30-39: $\frac{30+39}{2} = 34.5$ - 40-49: $\frac{40+49}{2} = 44.5$ - 50-59: $\frac{50+59}{2} = 54.5$ 4. **Step 2: Calculate total frequency (N):** $$N = 3 + 17 + 36 + 58 + 27 + 6 = 147$$ 5. **Step 3: Calculate mean ($\bar{x}$):** Formula: $$\bar{x} = \frac{\sum f x}{N}$$ Calculate $f x$ for each class: - $3 \times 4.5 = 13.5$ - $17 \times 14.5 = 246.5$ - $36 \times 24.5 = 882$ - $58 \times 34.5 = 2001$ - $27 \times 44.5 = 1201.5$ - $6 \times 54.5 = 327$ Sum: $13.5 + 246.5 + 882 + 2001 + 1201.5 + 327 = 4671.5$ Mean: $$\bar{x} = \frac{4671.5}{147} \approx 31.78$$ 6. **Step 4: Find median class:** Median class is where cumulative frequency crosses $\frac{N}{2} = 73.5$ Cumulative frequencies: - 3 - 3+17=20 - 20+36=56 - 56+58=114 (crosses 73.5, so median class is 30-39) 7. **Step 5: Calculate median:** Formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ Where: - $L = 30$ (lower limit of median class) - $F = 56$ (cumulative frequency before median class) - $f_m = 58$ (frequency of median class) - $h = 10$ (class width) Calculate: $$\text{Median} = 30 + \left(\frac{73.5 - 56}{58}\right) \times 10 = 30 + \left(\frac{17.5}{58}\right) \times 10 \approx 30 + 3.02 = 33.02$$ 8. **Step 6: Find mode class:** Mode class is the class with highest frequency: 58 in 30-39 9. **Step 7: Calculate mode:** Formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where: - $L = 30$ - $f_1 = 58$ (frequency of modal class) - $f_0 = 36$ (frequency before modal class) - $f_2 = 27$ (frequency after modal class) - $h = 10$ Calculate: $$\text{Mode} = 30 + \frac{58 - 36}{2 \times 58 - 36 - 27} \times 10 = 30 + \frac{22}{116 - 63} \times 10 = 30 + \frac{22}{53} \times 10 \approx 30 + 4.15 = 34.15$$ 10. **Step 8: Calculate variance ($\sigma^2$) and standard deviation ($\sigma$):** Formula for variance: $$\sigma^2 = \frac{\sum f x^2}{N} - \bar{x}^2$$ Calculate $x^2$ and $f x^2$: - $4.5^2 = 20.25$, $3 \times 20.25 = 60.75$ - $14.5^2 = 210.25$, $17 \times 210.25 = 3574.25$ - $24.5^2 = 600.25$, $36 \times 600.25 = 21609$ - $34.5^2 = 1190.25$, $58 \times 1190.25 = 69034.5$ - $44.5^2 = 1980.25$, $27 \times 1980.25 = 53466.75$ - $54.5^2 = 2970.25$, $6 \times 2970.25 = 17821.5$ Sum $f x^2 = 60.75 + 3574.25 + 21609 + 69034.5 + 53466.75 + 17821.5 = 165566.75$ Variance: $$\sigma^2 = \frac{165566.75}{147} - (31.78)^2 = 1126.6 - 1009.8 = 116.8$$ Standard deviation: $$\sigma = \sqrt{116.8} \approx 10.81$$ 11. **Step 9: Calculate coefficients:** - Coefficient of mean: $$\frac{\sigma}{\bar{x}} = \frac{10.81}{31.78} \approx 0.34$$ - Coefficient of median: $$\frac{\sigma}{\text{Median}} = \frac{10.81}{33.02} \approx 0.33$$ - Coefficient of mode: $$\frac{\sigma}{\text{Mode}} = \frac{10.81}{34.15} \approx 0.32$$ **Final answers:** - Midpoints: 4.5, 14.5, 24.5, 34.5, 44.5, 54.5 - Mean: 31.78 - Median: 33.02 - Mode: 34.15 - Variance: 116.8 - Standard Deviation: 10.81 - Coefficient of Mean: 0.34 - Coefficient of Median: 0.33 - Coefficient of Mode: 0.32