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Section 3 Operation Management Statistics

# Section 3 Operation Management Statistics - Comparisons of...

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Comparisons of Statistics Professor Harvey A. Singer School of Management George Mason University I Inferences about the Data Distribution. 1 Skewness vs. Symmetry. Skewness refers to the lack of symmetry of distribution of data about its center. 1.1 A simple skewness factor. The quantity x x x ˆ - is a non-dimensional measure of the lack of symmetry or skewness, where x is the sample mean and x ˆ is the sample median. It compares the offset between the sample mean and median to the sample mean. This skewness factor contains an algebraic sign. The algebraic sign indicates the direction of skewing. The magnitude of the skewing factor indicates the severity of the skewing. A skewness factor with a value of zero indicates no skewing: the mean and median coincide. When the median is less in magnitude than the mean, x x ˆ and the median lies to the left of the mean on the number line. As a result, the term in the numerator is positive and so too is the skewness factor. In this case, the distribution is positively skewed so that the “hump” is on the left and the long tail is on the right; the distribution is right-skewed. The interpretation is that most of the data is less in value than the sample mean. More of the sample data lie below (to the left) of the mean than above it (to the right). When the median is greater in magnitude than the mean, x x ˆ < and the median lies to the right of the mean on a number line. As a result, the term in the numerator is negative and so too is the skewness factor. In this case, the data distribution is negatively skewed so that the “hump” is on the right and the long tail is on the left; the distribution is left- skewed. The interpretation is that most of the data is greater in value than the sample mean. More of the sample data lie above (to the right) of the mean than below it (to the left). © 1999 by Harvey A. Singer 1

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The higher the magnitude the more severely skewed and the more asymmetric is the distribution. In summary, = < - right skewed if skew no symmetric if left skewed if x x x 0 ) ( 0 0 ˆ 1.2 An alternative skewness factor. An alternative non-dimensional measure of skewing is simply the ratio of the median to mean, viz x x ˆ is another dimensionless measure of the skew or lack of symmetry of a distribution. Here - = - < skewed left if skew no symmetric if skewed right if x x 1 ) ( 1 1 ˆ 1.3 Pearson skewness factor. Yet another dimensionless measure of skewness is the Pearson skewness factor, defined as ( 29 s x x ˆ 3 - The same rules and interpretations apply as for the simple skewness factor in section 1.1 above. 2
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Section 3 Operation Management Statistics - Comparisons of...

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