1. Problem 1: Frequency distribution of heights of 50 students. 1. a. Find the lower limit of the median class. - Total frequency $N=50$. - Median class is the class where cumulative frequency just exceeds $\frac{N}{2}=25$. - Cumulative frequencies: 2, 7, 17, 36, 46, 50. - Median class is $161-165$ (since cumulative frequency 36 > 25). - Lower limit of median class = 161. 1. b. Find the mean by short-cut method. - Let assumed mean $A=158$ (midpoint of $156-160$). - Calculate midpoints $x_i$: 148, 153, 158, 163, 168, 173. - Calculate deviations $d_i = x_i - A$: -10, -5, 0, 5, 10, 15. - Frequencies $f_i$: 2, 5, 10, 19, 10, 4. - Calculate $f_i d_i$: $2(-10)=-20$, $5(-5)=-25$, $10(0)=0$, $19(5)=95$, $10(10)=100$, $4(15)=60$. - Sum $\sum f_i d_i = -20 - 25 + 0 + 95 + 100 + 60 = 210$. - Mean $= A + \frac{\sum f_i d_i}{N} = 158 + \frac{210}{50} = 158 + 4.2 = 162.2$ cm. 1. c. Draw ogive curve. - Plot cumulative frequencies against upper class boundaries: (150,2), (155,7), (160,17), (165,36), (170,46), (175,50). - X-axis: height intervals, Y-axis: cumulative frequency. 2. Problem 2: Frequency distribution table. 2. a. Mid-point of mode class. - Frequencies: 3, 10, 18, 25, 8, 6. - Mode class is class with highest frequency = $48-53$. - Midpoint = $\frac{48+53}{2} = 50.5$. 2. b. Find median. - Total frequency $N=3+10+18+25+8+6=70$. - Median class is where cumulative frequency $> 35$. - Cumulative frequencies: 3, 13, 31, 56, 64, 70. - Median class: $48-53$. - Median formula: $$\text{Median} = l + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ where $l=48$, $F=31$, $f=25$, $h=6$. - Calculate: $$\text{Median} = 48 + \left(\frac{35 - 31}{25}\right) \times 6 = 48 + \frac{4}{25} \times 6 = 48 + 0.96 = 48.96$$ 2. c. Draw frequency polygon. - Midpoints: 32.5, 38.5, 44.5, 50.5, 56.5, 62.5. - Frequencies: 3, 10, 18, 25, 8, 6. - Plot points and connect with straight lines. 3. Problem 3: Marks of 50 students. 3. a. Find median of data 14, 9, 7, 10, 12, 11, 6, 13, 15, 18. - Sort data: 6,7,9,10,11,12,13,14,15,18. - Number of data points $n=10$ (even). - Median = average of 5th and 6th values: $$\frac{11 + 12}{2} = 11.5$$ 3. b. Find mode by frequency distribution with class interval 5. - Group marks into classes: e.g., 44-48, 49-53, ..., 75-79. - Count frequencies, mode is class with highest frequency. 3. c. Draw ogive curve. - Plot cumulative frequencies against class boundaries. 4. Problem 4: Marks of 30 students. 4. a. Mode formula for classified data: $$\text{Mode} = l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ where $l$ = lower limit of modal class, $f_1$ = frequency of modal class, $f_0$ = frequency of class before modal, $f_2$ = frequency of class after modal, $h$ = class width. 4. b. Frequency table with class intervals and mean calculation. 4. c. Draw frequency polygon. 5. Problem 5: Frequency distribution of marks of 40 students. 5. a. Mid value of post median class. - Median class found by cumulative frequency. - Post median class is class after median class. - Mid value = average of class limits. 5. b. Mean by short-cut method. 5. c. Draw ogive curve. 6. Problem 6: Frequency distribution of weights of 60 students. 6. a. Mid-value of mode class. 6. b. Median from table. 6. c. Draw histogram. Slug: "frequency distributions" Subject: "statistics" Desmos: {"latex":"","features":{"intercepts":true,"extrema":true}} q_count: 18