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Lecture2standard - Statistics 511 Statistical Methods Dr...

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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Lecture 2: Measures of Location and Variability Devore: Section 1.3-1.4 Oct, 2011 Page 1
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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Sample mean The sample mean of the n numbers x 1 , . . . , x n is ¯ x = x 1 + x 2 + . . . + x n n Alternative notation: ¯ x = n i =1 x i n The following dataset shows the crack length in μ m for smooth bar tensile specimens subjected to stress corrosion tests. The data is x 1 = 16 . 1 , x 2 = 9 . 6 , x 3 = 24 . 9 , ..., x 21 = 28 . 5 . Oct, 2011 Page 2
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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 The mean is clearly ¯ x = 444 . 8 21 = 21 . 18 . Physical interpretation of the mean: the balance point for a system of weights. R commands: with(xmp01.12, mean(crackLength)) and with(xmp01.12, sum(crackLength)) Lack of robustness is the main shortcoming of the mean as a measure of center. In the example before, there is a value x 14 = 45 . 0 which is an outlier. Without it, we have ¯ x = 399 . 8 20 = 19 . 99 . Oct, 2011 Page 3
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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Sample median The sample median ˜ x is the middle value in the set of data that has been arranged in ascending order. For an even number of data points the median is the average of the middle two. More precisely, suppose the number of observations n is odd. Then, the median ˜ x is the observation number n +1 2 .
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