1. **State the problem:** We need to calculate the mean, median, lower quartile (Q1), and mode for the grouped data of weights of 120 articles. 2. **Given data:** Weight intervals (in gm): 0-10, 10-20, 20-30, 30-40, 40-50, 50-60 Number of articles (frequencies): 14, 17, 22, 26, 23, 18 3. **Calculate class midpoints (x):** $$5, 15, 25, 35, 45, 55$$ 4. **Calculate mean:** Formula: $$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate $$f_i x_i$$: $$14\times5=70, 17\times15=255, 22\times25=550, 26\times35=910, 23\times45=1035, 18\times55=990$$ Sum of frequencies $$=120$$ Sum of $$f_i x_i = 70+255+550+910+1035+990=3810$$ Mean $$= \frac{3810}{120} = 31.75$$ gm 5. **Calculate median:** Median class is where cumulative frequency $$\geq \frac{120}{2} = 60$$ Cumulative frequencies: 14, 31, 53, 79, 102, 120 Median class is 30-40 (4th class) Formula: $$\text{Median} = l + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ Where: $l=30$ (lower boundary of median class), $N=120$ (total frequency), $F=53$ (cumulative frequency before median class), $f=26$ (frequency of median class), $h=10$ (class width) Calculate: $$30 + \left(\frac{60 - 53}{26}\right) \times 10 = 30 + \left(\frac{7}{26}\right) \times 10 = 30 + 2.69 = 32.69$$ gm 6. **Calculate lower quartile (Q1):** Q1 position $$= \frac{N}{4} = 30$$ Cumulative frequencies: 14, 31, 53, ... Q1 class is 10-20 (2nd class) Using formula: $$Q1 = l + \left(\frac{\frac{N}{4} - F}{f}\right) \times h$$ Where: $l=10$, $F=14$, $f=17$, $h=10$ Calculate: $$10 + \left(\frac{30 - 14}{17}\right) \times 10 = 10 + \left(\frac{16}{17}\right) \times 10 = 10 + 9.41 = 19.41$$ gm 7. **Calculate mode:** Mode class is the class with highest frequency: 30-40 with 26 articles Formula: $$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ Where: $l=30$, $f_1=26$ (modal class frequency), $f_0=22$ (frequency before modal class), $f_2=23$ (frequency after modal class), $h=10$ Calculate: $$30 + \left(\frac{26 - 22}{2\times26 - 22 - 23}\right) \times 10 = 30 + \left(\frac{4}{52 - 45}\right) \times 10 = 30 + \left(\frac{4}{7}\right) \times 10 = 30 + 5.71 = 35.71$$ gm **Final answers:** - Mean = 31.75 gm - Median = 32.69 gm - Lower Quartile (Q1) = 19.41 gm - Mode = 35.71 gm