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 arithmetic mean using the specified methods. 1. Calculate arithmetic mean using Assumed Mean Method. Given: Size (x): 20, 19, 18, 17, 16, 15, 14, 13, 12, 11 Frequency (f): 1, 2, 4, 8, 11, 10, 7, 4, 2, 1 Assumed Mean (A) = 15 Step 1: Create a table to calculate d = x - A and fd. |c|c|c|c| x & f & d = x - 15 & fd \\ 20 & 1 & 5 & 5 \\ 19 & 2 & 4 & 8 \\ 18 & 4 & 3 & 12 \\ 17 & 8 & 2 & 16 \\ 16 & 11 & 1 & 11 \\ 15 & 10 & 0 & 0 \\ 14 & 7 & -1 & -7 \\ 13 & 4 & -2 & -8 \\ 12 & 2 & -3 & -6 \\ 11 & 1 & -4 & -4 \\ & f = 50 & & fd = 27 \\ Step 2: Apply the formula for the Assumed Mean Method. The formula for the arithmetic mean using the Assumed Mean Method is x = A + ( fd)/( f). x = 15 + (27)/(50) Step 3: Calculate the arithmetic mean. x = 15 + 0.54 \\ x = 15.54 The arithmetic mean is 15.54. 2. Calculate arithmetic mean using Direct and Assumed Mean Methods. Given: Marks (x): 4, 5, 6, 7, 8 No. of Students (f): 12, 10, 18, 20, 24 a) Direct Method Step 1: Create a table to calculate fx. |c|c|c| x & f & fx \\ 4 & 12 & 48 \\ 5 & 10 & 50 \\ 6 & 18 & 108 \\ 7 & 20 & 140 \\ 8 & 24 & 192 \\ & f = 84 & fx = 538 \\ Step 2: Apply the formula for the Direct Method. The formula for the arithmetic mean using the Direct Method is x = ( fx)/( f). x = (538)/(84) Step 3: Calculate the arithmetic mean. x ≈ 6.40476 \\ x ≈ 6.40 marks The arithmetic mean using the Direct Method is 6.40 marks. b) Assumed Mean Method Step 1: Choose an assumed mean (A). Let A = 6. Create a table to calculate d = x - A and fd. |c|c|c|c| x & f & d = x - 6 & fd \\ 4 & 12 & -2 & -24 \\ 5 & 10 & -1 & -10 \\ 6 & 18 & 0 & 0 \\ 7 & 20 & 1 & 20 \\ 8 & 24 & 2 & 48 \\ & f = 84 & & fd = 34 \\ Step 2: Apply the formula for the Assumed Mean Method. The formula for the arithmetic mean using the Assumed Mean Method is x = A + ( fd)/( f). x = 6 + (34)/(84) Step 3: Calculate the arithmetic mean. x = 6 + 0.40476 \\ x ≈ 6.40 marks The arithmetic mean using the Assumed Mean Method is 6.40 marks. Send me the next one 📸