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 calculate the standard deviation for grouped data, we first find the midpoint of each class, then calculate the mean, and finally apply the standard deviation formula. Step 1: Determine the midpoint (x_m) for each class interval and calculate fx_m and fx_m^2. We'll create an extended frequency table: | Marks (x) | Frequency (f) | Midpoint (x_m) | fx_m | x_m^2 | fx_m^2 | | :------------ | :-------------- | :--------------- | :----- | :------ | :------- | | 10 < x 20 | 3 | 15 | 45 | 225 | 675 | | 20 < x 30 | 7 | 25 | 175 | 625 | 4375 | | 30 < x 40 | 12 | 35 | 420 | 1225 | 14700 | | 40 < x 50 | 15 | 45 | 675 | 2025 | 30375 | | 50 < x 60 | 8 | 55 | 440 | 3025 | 24200 | | 60 < x 70 | 3 | 65 | 195 | 4225 | 12675 | | 70 < x 80 | 2 | 75 | 150 | 5625 | 11250 | Step 2: Calculate the sums for frequency, fx_m, and fx_m^2. f = 3+7+12+15+8+3+2 = 50 fx_m = 45+175+420+675+440+195+150 = 2100 fx_m^2 = 675+4375+14700+30375+24200+12675+11250 = 98250 Step 3: Calculate the mean (x). x = ( fx_m)/( f) = (2100)/(50) = 42 Step 4: Calculate the standard deviation (). The formula for standard deviation for grouped data is: = sqrt(( fx_m^2)/( f) - (( fx_m)/( f))^2) Substitute the calculated values: = sqrt((98250)/(50) - ((2100)/(50))^2) = sqrt(1965 - (42)^2) = sqrt(1965 - 1764) = sqrt(201) ≈ 14.1774 Rounding to three decimal places: ≈ 14.177 The standard deviation is 14.177. Send me the next one 📸