This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the calculations for the mean, median, mode, and standard deviation from the given data. First, let's set up the table with the necessary columns for calculations. The class intervals are 30-34, 35-39, 40-44, 45-49, 50-54. The class width (c) is 5 (e.g., 34 - 30 + 1 = 5). The assumed mean (A) is given as 34.2. | Class Interval | F (Frequency) | x (Midpoint) | d = x - A | u = (x - A)/(c) | Fu | Fu^2 | | :------------- | :------------: | :----------: | :----------: | :--------------------: | :----: | :----: | | 30-34 | 10 | 32 | 32 - 34.2 = -2.2 | (-2.2)/(5) = -0.44 | -4.4 | 1.936 | | 35-39 | 12 | 37 | 37 - 34.2 = 2.8 | (2.8)/(5) = 0.56 | 6.72 | 3.7632 | | 40-44 | 8 | 42 | 42 - 34.2 = 7.8 | (7.8)/(5) = 1.56 | 12.48 | 19.4688 | | 45-49 | 15 | 47 | 47 - 34.2 = 12.8 | (12.8)/(5) = 2.56 | 38.4 | 98.304 | | 50-54 | 5 | 52 | 52 - 34.2 = 17.8 | (17.8)/(5) = 3.56 | 17.8 | 63.368 | | Total | F = 50 | | | | Fu = 70.9 | Fu^2 = 186.84 | 1. Calculate Mean (x): Step 1: Use the formula for the assumed mean method. x = A + (( Fu)/( F)) × c Step 2: Substitute the values from the table. x = 34.2 + ((70.9)/(50)) × 5 Step 3: Simplify the expression. x = 34.2 + (70.9)/(10) x = 34.2 + 7.09 x = 41.29 The mean is 41.29. 2. Calculate Median: Step 1: Find the total frequency (N) and the median position. N = F = 50 Median position = (N)/(2) = (50)/(2) = 25^th value Step 2: Determine the median class using cumulative frequency (CF). | Class Interval | F | CF | | :------------- | :-: | :-: | | 30-34 | 10 | 10 | | 35-39 | 12 | 22 | | 40-44 | 8 | 30 | (The 25th value falls in this class) | 45-49 | 15 | 45 | | 50-54 | 5 | 50 | The median class is 40-44. Step 3: Identify the values for the median formula. Lower boundary of median class (L) = 39.5 Frequency of median class (f_m) = 8 Cumulative frequency of the class before the median class (CF_b) = 22 Class width (c) = 5 Step 4: Apply the median formula for grouped data. Median = L + ((N)/(2) - CF_bf_m) × c Median = 39.5 + ((25 - 22)/(8)) × 5 Median = 39.5 + ((3)/(8)) × 5 Median = 39.5 + (15)/(8) Median = 39.5 + 1.875 Median = 41.375 The median is 41.375. 3. Calculate Mode: Step 1: Identify the modal class (class with the highest frequency). The highest frequency is 15, which corresponds to the class 45-49. Modal class = 45-49. Step 2: Identify the values for the mode formula. Lower boundary of modal class (L) = 44.5 Frequency of modal class (f_m) = 15 Frequency of the class before the modal class (f_1) = 8 (frequency of 40-44) Frequency of the class after the modal class (f_2) = 5 (frequency of 50-54) Class width (c) = 5 Step 3: Apply the mode formula for grouped data. Mode = L + ((f_m - f_1)/((f_m - f_1) + (f_m - f_2))) × c Mode = 44.5 + ((15 - 8)/((15 - 8) + (15 - 5))) × 5 Mode = 44.5 + ((7)/(7 + 10)) × 5 Mode = 44.5 + ((7)/(17)) × 5 Mode = 44.5 + (35)/(17) Mode = 44.5 + 2.05882... Mode ≈ 46.56 The mode is 46.56. 4. Calculate Standard Deviation (): Step 1: Use the formula for standard deviation with the step deviation method. = c × sqrt(( Fu^2)/( F) - (( Fu)/( F))^2) Step 2: Substitute the values from the table. = 5 × sqrt((186.84)/(50) - ((70.9)/(50))^2) Step 3: Perform the calculations inside the square root. = 5 × sqrt(3.7368 - (1.418)^2) = 5 × sqrt(3.7368 - 2.010724) = 5 × sqrt(1.726076) Step 4: Calculate the square root and multiply by c. = 5 × 1.313794... ≈ 6.57 The standard deviation is 6.57. 3 done, 2 left today. You're making progress.