Statistics 312 6 Descriptive Summary Measures Measures of Central Location The Arithmetic Mean x N for a population x x n for a sample Ex The data represent the number of textbooks purchased by a sample of seven students 10 4 7 5 7 8 9 x 10 4 7 5 7 8 9 50 7 14 7 7 1 Statistics 312 6 Descriptive Summary Measures 2 The mean is affected by any outliers and skews Because the mean is nonresistant there are alternative measures that are more resistant to outliers and skews The Median The median is a measure of central tendency that occupies the middle position of data placed in order of magnitude If n is odd the median is the middle number of the data placed in order of magnitude It occupies the n 1 2 th position If n is even the median is the average of the middle two numbers of the data placed in order of magnitude It is the average of the numbers in the n 2 2 th n 2 th and positions Ex Reordering the sample of books 4 5 7 7 8 9 10 The median is 7 If there were an eighth person who purchased 12 books the median would be 7 5 The Mode The mode by definition is the most frequently occurring value in a series Statistics 312 6 Descriptive Summary Measures 3 Mode 58 The kth percentile Pk is such that no more than k percent of the data are less than Pk and no more than 100 k percent are greater than Pk Usually used with large data sets The first quartile Q1 is the point that separates the lower 25 percent of the values from the upper 75 percent value corresponding to the n 1 4 ordered observation The third quartile Q3 is the point that separates the upper 25 percent of the values from the lower 75 percent value corresponding to the 3 n 1 4 ordered observation Ex Books 4 5 7 7 8 9 10 n 1 4 2 so Q1 5 3 n 1 4 6 soQ3 9 If position 5 average two nearest values else if not integer round Statistics 312 6 Descriptive Summary Measures 4 Statistics 312 6 Descriptive Summary Measures Measures of Variability Range Maximum Minimum Ex Books 4 5 7 7 8 9 10 Range 10 4 6 Range 104 14 90 Interquartile Range IQR Q3 Q1 Ex The sample of books IQR 77 47 30 Q1 5 Q3 9 IQR 9 5 4 5 Statistics 312 6 Descriptive Summary Measures The Standard Deviation The population standard deviation is the square root of the average of the squared deviations of the individual data items about their mean x 2 N And when a sample data set is used the standard deviation is found with s x x 2 n 1 The value under the square root is called the variance Ex X X 11 4 X 11 4 2 3 8 4 70 56 8 3 4 11 56 9 2 4 5 76 17 5 6 31 36 20 8 6 73 96 57 2 s x x n 1 s 48 3 6 95 193 20 2 193 20 4 48 3 6 Statistics 312 6 Descriptive Summary Measures Read pp 104 117 Prob 3 1 3 2a skip 4 6 b 3 5a skip 4 6 b 3 9a skip 4 6 7