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.
ScanSolve AI Answer
To find the mean deviation, we first need to calculate the mean (x) of the distribution. Step 1: Calculate the midpoint (x) for each class interval. The midpoint is calculated as lower limit + upper limit2. For 1-20: x_1 = (1+20)/(2) = 10.5 For 21-40: x_2 = (21+40)/(2) = 30.5 For 41-60: x_3 = (41+60)/(2) = 50.5 For 61-80: x_4 = (61+80)/(2) = 70.5 For 81-100: x_5 = (81+100)/(2) = 90.5 Step 2: Calculate fx for each class and the sum of frequencies ( f) and the sum of fx ( fx). | Marks (Class) | Frequency (f) | Midpoint (x) | fx | |x - x| | f|x - x| | | :------------ | :-------------- | :------------- | :--- | :-------------- | :--------------- | | 1-20 | 8 | 10.5 | 84 | | | | 21-40 | 14 | 30.5 | 427 | | | | 41-60 | 8 | 50.5 | 404 | | | | 61-80 | 48 | 70.5 | 3384 | | | | 81-100 | 12 | 90.5 | 1086 | | | | Total | f = 90 | | fx = 5385 | | | Step 3: Calculate the mean (x). The formula for the mean of grouped data is x = ( fx)/( f). x = (5385)/(90) = 59.8333... We will use x ≈ 59.83 for calculations, keeping more precision in mind. Step 4: Calculate the absolute deviation |x - x| for each midpoint. |10.5 - 59.8333| = |-49.3333| = 49.3333 |30.5 - 59.8333| = |-29.3333| = 29.3333 |50.5 - 59.8333| = |-9.3333| = 9.3333 |70.5 - 59.8333| = |10.6667| = 10.6667 |90.5 - 59.8333| = |30.6667| = 30.6667 Step 5: Calculate f|x - x| for each class and find their sum ( f|x - x|). | Marks (Class) | Frequency (f) | Midpoint (x) | fx | |x - x| | f|x - x| | | :------------ | :-------------- | :------------- | :--- | :-------------- | :--------------- | | 1-20 | 8 | 10.5 | 84 | 49.3333 | 8 × 49.3333 = 394.6664 | | 21-40 | 14 | 30.5 | 427 | 29.3333 | 14 × 29.3333 = 410.6662 | | 41-60 | 8 | 50.5 | 404 | 9.3333 | 8 × 9.3333 = 74.6664 | | 61-80 | 48 | 70.5 | 3384 | 10.6667 | 48 × 10.6667 = 512.0016 | | 81-100 | 12 | 90.5 | 1086 | 30.6667 | 12 × 30.6667 = 368.0004 | | Total | f = 90 | | fx = 5385 | | f|x - x| = 1760.001 | Step 6: Calculate the Mean Deviation (MD). The formula for mean deviation of grouped data is MD = f|x - x| f. MD = (1760.001)/(90) MD ≈ 19.55556 Rounding to two decimal places: MD ≈ 19.56 The mean deviation is 19.56. Last free one today — make it count tomorrow, or type /upgrade for unlimited.