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Unformatted text preview: Topic 2. Descriptive Statistics: Part II Cyr Emile MLAN, Ph.D. mlan@stat.uconn.edu Descriptive Statistics p. 1/43 Introduction Text Reference : Introduction to Probability and Statistics for Engineers and Scientists, Chapter 2. Reading Assignment : Sections 2.32.6, September 810 It is often of interest to summarize a data by a set of wellchosen numbers. That is, numbers that might serve to characterize the data set and convey some of its salient features. We call any of these summarizing number a statistic . We present few of these summary measures. Descriptive Statistics p. 2/43 Numerical Measures for Univariate Quantitative Variable Let x 1 , x 2 , ... , x n denote n measurements a quantitative variable x . Measures of Location Measures of location describe the center" of a set of data values or a distribution such as those displayed in a stemandleaf or a histogram. The most common of these are sample mean, sample median, and sample mode . Descriptive Statistics p. 3/43 Measures of Location Definition 1.1 : The sample mean of a sample of n measurements x 1 , x 2 , ... , x n is defined by x = 1 n n summationdisplay i =1 x i . The corresponding population mean is denoted by . x is the arithmetic average of the n numerical values x 1 , x 2 , ... , x n . When the data is put in a grouped form, the sample mean can then be regarded as a weighted average of the distinct values, the weight being equal to the fraction of times each observation appears in the data values. Descriptive Statistics p. 4/43 Measures of Location Example 1.7 : U.S. Masters Golf Tournament The winning scores in the U.S. Masters golf tournament in the years from 1982 to 1991 were as follows: 284 280 277 282 279 285 281 283 278 277 Find the sample mean of these winning scores. Solution : The sample mean of the winning scores in the U.S. Masters golf tournament in the years 1982 to 1991 is x = 284 + 280 + 277 + 282 + 279 + 285 + 281 + 283 + 278 + 277 10 = 280 . 6 Descriptive Statistics p. 5/43 Measures of Location Example 1.8 : Example 1.5 revisited What is the average zink intake from our 40 patients with rheumatoid arthritis? Solution : x = (8 . 0 + 12 . 9 + + 8 . 8 + 10 . 7) / 40 = 10 . 675 The average zink intake from our 40 patients with rheumatoid arthritis is 10.675 mg/MJ. Descriptive Statistics p. 6/43 Measures of Location Example 1.9 : Shoe Sizes The following table represents the relative frequency distribution of shoe size of 84 males that entered in a shoe store. Determine the average shoe size of these 84 consumers. Shoe Size Frequency Relative frequency 5 3 0.0357 6 1 0.0119 7 13 0.1548 8 16 0.1904 9 21 0.2500 10 19 0.2262 11 11 0.1310 Descriptive Statistics p. 7/43 Measures of Location Solution : x = 5(3) + 6(1) + 7(13) + 8(16) + 9(21) + 10(19) + 11(11) 84 , = 740 84 = 8 . 8095 ....
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