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 from a frequency distribution, we use the formula: = sqrt(( fx^2)/( f) - (( fx)/( f))^2) where x represents the number of hours, f represents the number of students (frequency), f is the total number of students, and fx is the sum of (hours × frequency). First, let's organize the data and calculate the necessary sums: | Number of hours (x) | Number of students (f) | fx | x^2 | fx^2 | | :--------------------: | :----------------------: | :--: | :---: | :----: | | 0 | 1 | 0 | 0 | 0 | | 1 | 3 | 3 | 1 | 3 | | 2 | 3 | 6 | 4 | 12 | | 3 | 3 | 9 | 9 | 27 | | 4 | 5 | 20 | 16 | 80 | | 5 | 6 | 30 | 25 | 150 | | 6 | 2 | 12 | 36 | 72 | | 7 | 1 | 7 | 49 | 49 | | Total | f = 24 | fx = 87 | | fx^2 = 393 | Step 1: Calculate the total number of students ( f). f = 1 + 3 + 3 + 3 + 5 + 6 + 2 + 1 = 24 Step 2: Calculate the sum of (hours × frequency) ( fx). fx = (0 × 1) + (1 × 3) + (2 × 3) + (3 × 3) + (4 × 5) + (5 × 6) + (6 × 2) + (7 × 1) fx = 0 + 3 + 6 + 9 + 20 + 30 + 12 + 7 = 87 Step 3: Calculate the sum of (hours squared × frequency) ( fx^2). fx^2 = (0^2 × 1) + (1^2 × 3) + (2^2 × 3) + (3^2 × 3) + (4^2 × 5) + (5^2 × 6) + (6^2 × 2) + (7^2 × 1) fx^2 = (0 × 1) + (1 × 3) + (4 × 3) + (9 × 3) + (16 × 5) + (25 × 6) + (36 × 2) + (49 × 1) fx^2 = 0 + 3 + 12 + 27 + 80 + 150 + 72 + 49 = 393 Step 4: Calculate the mean (x). x = ( fx)/( f) = (87)/(24) = 3.625 Step 5: Substitute the values into the standard deviation formula. = sqrt(( fx^2)/( f) - (( fx)/( f))^2) = sqrt((393)/(24) - ((87)/(24))^2) = sqrt(16.375 - (3.625)^2) = sqrt(16.375 - 13.140625) = sqrt(3.234375) ≈ 1.7984368 Rounding to three decimal places: ≈ 1.798 The standard deviation is 1.798. Last free one today — make it count tomorrow, or type /upgrade for unlimited.