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STATISTICS_PLS - Department of Mechanical Engineering...

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Department of Mechanical Engineering STATISTICS ES 140 Section 5 Fall 2006
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Department of Mechanical Engineering Theory – Measures of Center Mode -the most frequent data point in a set Median -the middle score in the set Mean, Average, or Arithmetic Mean -the sum of the scores in the set divided by the number of scores Example: Set A: 12,34,36,42,52,54,68,72,81,93 Set B: 152,154,155,155,156,158,159,161,163,164
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Department of Mechanical Engineering Mode The most frequent data point in a set (Hint: some sets have no mode) Set A: 12,34,36,42,52,54,68,72,81,93 Set B: 152,154,155,155,156,158,159,161,163,164
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Department of Mechanical Engineering Median The middle score in the set (Hint: if there are an even number of terms, use a value halfway between middle terms) Set A: 12,34,36,42,52,54,68,72,81,93 Set B: 152,154,155,155,156,158,159,161,163,164
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Department of Mechanical Engineering Mean Average, or Arithmetic Mean -the sum of the scores divided by the number of scores Set A: 12,34,36,42,52,54,68,72,81,93 Set B: 152,154,155,155,156,158,159,161,163,164 Σ Set A =544 #scores=10 average A =54.4 Σ Set B =1577 #scores=10 average B =157.7
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Department of Mechanical Engineering Comparison of Measures SET MODE MEDIAN MEAN A None 53 54.4 B 155 157 157.7 Note: The different “center” values for a given set may not always be close to each other. e.g. SET C: 1, 3, 3, 59, 68, 76, 1000 C 3 59 172.86
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Department of Mechanical Engineering Effect of outlying data The most important difference between the mean and the median is shown below. Observe what happens to a set of five scores when the largest one is made considerably larger: Set D: 12, 13, 23, 32, 43 Mean = 24.6 Median = 23 Set D Altered: 12, 13, 23, 32, 143 Mean = 44.6 Median = 23
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Department of Mechanical Engineering Measures of Variability Range Variance Standard Deviation
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Department of Mechanical Engineering Range The difference between the smallest and the largest value Set E: 80 85 79 90 95 98 92
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Department of Mechanical Engineering Variance ( 29 ( 29 value Data data the of Mean points data of Number 1 Variance 1 2 2 = = = - - = = i n i i x n n x s μ μ Set E: 80 85 79 90 95 98 92
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Department of Mechanical Engineering Variance Calculations ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 62 . 53 6 43 . 88 92 43 . 88 98 43 . 88 95 43 . 88 90 43 . 88 79 43 . 88 85 43 . 88 80 1 ) ( 43 . 88 7 92 98 95 90 79 85 80 2 2 2 2 2 2 2 2 1 2 2 1 = - + - + - + - + - + - + - = - - = = + + + + + + = = = = D n i i D n i i D s n x s n x μ μ
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Department of Mechanical Engineering Standard Deviation ( 29 ( 29 value Data data the of Mean points data of Number 1 Deviation Standard 1 2 = = = - - = = i n i i x n n x s μ μ Set E: 80 85 79 90 95 98 92 32 . 7 62 . 53 2 = = = D D s s
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Department of Mechanical Engineering Interpretation of Standard Deviation If the values in a set are more "spread out" (i.e. heterogeneous), the value of the standard deviation is larger. For example, if the standard deviation of 6th grade students’ reading grades is 1.68 and the standard deviation of their math scores is 0.94, you would know that the students are more varied in reading performance than in math performance.
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