Example 1 Consider the following grade Distribution points from a class with 28 students: 1品 70,73,74,76,76,76,76,77,77,77, 8 78,78,79,79,83,83,84,85,86,87, 4 89,89,90,90,92,92,93,97 2 ■ where the numbers have 0 727782879297 been arranged in numerical order.Grouping the results Cumulative together in six intervals yields Interval number Cumulative number of students number 30252015 70-74: 3 3 75-79: 11 14 80-84: 3 17 50 85-89: 5 90-94: 5 器 727782879297 95-99: 1 28 Shanghai Jiao Tong University 11
Shanghai Jiao Tong University 11 Example 1 n Consider the following grade points from a class with 28 students: n 70, 73, 74, 76, 76, 76, 76, 77, 77, 77, 78, 78,79, 79,83, 83, 84, 85, 86, 87, 89, 89, 90, 90, 92, 92, 93, 97 n where the numbers have been arranged in numerical order. Grouping the results together in six intervals yields
Discrete Distributions Mean: 元= ∑ N = Mode:most frequent value Median:"middle value" Variance: i=1 N Standard Deviation: Range=max-min Shanghai Jiao Tong University 12
Shanghai Jiao Tong University 12 Discrete Distributions
Variance The N-1 in the denominator often causes some confusion.It can be shown that using N-1 instead of N gives a better estimate of the true variance when a finite sample is used to estimate the properties of a large population. Shanghai Jiao Tong University 13
Shanghai Jiao Tong University 13 Variance
The solution of the example For the data given in the last 12 example: 70,73,74,76,76,76,76,77,77 6 77,78,78,79,79,83,83,84,85,86, 4 2 87,89,89,90,90,92,92,93,97 0 727782879297 N=28 The mean is 82.35 The median is 81.00 The variance is 51.20 The standard deviation is 7.15 The range is 97-70=27 Shanghai Jiao Tong University 14
Shanghai Jiao Tong University 14 The solution of the example
Exercise in class (1) ( Given the following number: ■51,53,54,56,56,56,56,57,57,57,58,59,63,63,64,65， 66,67,69,69,70,72,72,72,73,76 (a)Plot a histogram of this data using six intervals. (b)Plot the cumulative distribution also. Shanghai Jiao Tong University 15
Shanghai Jiao Tong University 15 Exercise in class (1) n Given the following number: n 51,53,54,56,56,56,56,57,57,57,58,59,63,63,64,65, 66,67,69,69,70,72,72,72,73,76 (a) Plot a histogram of this data using six intervals. (b) Plot the cumulative distribution also