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 solve this problem, we first need to organize the given data into a frequency distribution table using the specified class intervals (10-19, 20-29, etc.). The given data points are: 23, 45, 67, 34, 56, 78, 89, 12, 45, 67, 54, 32, 76, 88, 43, 21, 65, 98, 54, 37, 29, 84, 71, 63, 55, 49, 91, 33, 27, 58, 62, 74, 39, 81, 47, 53, 69, 25, 72, 44, 36, 59, 83, 66, 41, 57, 73, 52, 68, 35 There are N = 50 data points. Here is the frequency distribution table: | Class Interval | Class Mark (x) | Frequency (f) | fx | Cumulative Frequency (cf) | | :------------- | :------------- | :------------ | :------ | :------------------------ | | 10-19 | 14.5 | 1 | 14.5 | 1 | | 20-29 | 24.5 | 5 | 122.5 | 6 | | 30-39 | 34.5 | 7 | 241.5 | 13 | | 40-49 | 44.5 | 7 | 311.5 | 20 | | 50-59 | 54.5 | 9 | 490.5 | 29 | | 60-69 | 64.5 | 8 | 516.0 | 37 | | 70-79 | 74.5 | 6 | 447.0 | 43 | | 80-89 | 84.5 | 5 | 422.5 | 48 | | 90-99 | 94.5 | 2 | 189.0 | 50 | | Total | | f = 50 | fx = 2755 | | --- i) Mean Step 1: Calculate the mean (x) using the formula x = ( fx)/( f). x = (2755)/(50) = 55.1 The mean is 55.1. --- ii) Median class Step 2: Determine the median position and identify the median class. The median position is (N)/(2) = (50)/(2) = 25. Looking at the cumulative frequency column, the 25th value falls within the class where the cumulative frequency first exceeds 25. This is the 50-59 class (cf = 29). The median class is 50-59. --- iii) Modal class Step 3: Identify the modal class. The modal class is the class with the highest frequency. From the table, the highest frequency is 9, which corresponds to the 50-59 class. The modal class is 50-59. --- iv) Mean deviation Step 4: Calculate the mean deviation (MD) using the formula MD = f|x - x| f. First, we calculate |x - x| and f|x - x| for each class. x = 55.1. | Class Mark (x) | |x - x| | Frequency (f) | f|x - x| | | :------------- | :-------------- | :------------ | :--------------- | | 14.5 | 40.6 | 1 | 40.6 | | 24.5 | 30.6 | 5 | 153.0 | | 34.5 | 20.6 | 7 | 144.2 | | 44.5 | 10.6 | 7 | 74.2 | | 54.5 | 0.6 | 9 | 5.4 | | 64.5 | 9.4 | 8 | 75.2 | | 74.5 | 19.4 | 6 | 116.4 | | 84.5 | 29.4 | 5 | 147.0 | | 94.5 | 39.4 | 2 | 78.8 | | Total | | f = 50 | f|x - x| = 834.8 | MD = (834.8)/(50) = 16.696 ≈ 16.70 The mean deviation is 16.70. --- v) Semi-interquartile range Step 5: Calculate the semi-interquartile range (SIQR) using the formula SIQR = (Q_3 - Q_1)/(2). First, find the first quartile (Q_1) and third quartile (Q_3). Q_1 position: (N)/(4) = (50)/(4) = 12.5^th value. This falls in the 30-39 class. Q_3 position: (3N)/(4) = (3 × 50)/(4) = 37.5^th value. This falls in the 70-79 class. The formula for a quartile is Q_k = L + ((kN)/(4) - cf_bf_m)c, where L is the lower class boundary, cf_b is the cumulative frequency before the quartile class, f_m is the frequency of the quartile class, and c is the class width. For Q_1: L = 29.5, cf_b = 6, f_m = 7, c = 10. Q_1 = 29.5 + ((12.5 - 6)/(7)) × 10 = 29.5 + ((6.5)/(7)) × 10 ≈ 29.5 + 9.2857 = 38.7857 For Q_3: L = 69.5, cf_b = 37, f_m = 6, c = 10. Q_3 = 69.5 + ((37.5 - 37)/(6)) × 10 = 69.5 + ((0.5)/(6)) × 10 ≈ 69.5 + 0.8333 = 70.3333 Now, calculate SIQR: SIQR = (Q_3 - Q_1)/(2) = (70.3333 - 38.7857)/(2) = (31.5476)/(2) ≈ 15.7738 The semi-interquartile range is 15.77. --- vi) Standard deviation Step 6: Calculate the standard deviation (s) using the formula s = sqrt( f(x - x))^2 f. First, we calculate (x - x)^2 and f(x - x)^2 for each class. x = 55.1. | Class Mark (x) | (x - x) | (x - x)^2 | Frequency (f) | f(x - x)^2 | | :------------- | :-------------- | :---------------- | :------------ | :----------------- | | 14.5 | -40.6 | 1648.36 | 1 | 1648.36 | | 24.5 | -30.6 | 936.36 | 5 | 4681.80 | | 34.5 | -20.6 | 424.36 | 7 | 2970.52 | | 44.5 | -10.6 | 112.36 | 7 | 786.52 | | 54.5 | -0.6 | 0.36 | 9 | 3.24 | | 64.5 | 9.4 | 88.36 | 8 | 706.88 | | 74.5 | 19.4 | 376.36 | 6 | 2258.16 | | 84.5 | 29.4 | 864.36 | 5 | 4321.80 | | 94.5 | 39.4 | 1552.36 | 2 | 3104.72 | | Total | | | f = 50 | f(x - x)^2 = 20481.00 | s = sqrt((20481.00)/(50)) = sqrt(409.62) ≈ 20.2391 The standard deviation is 20.24. Drop the next question!