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# handout_1_2 - 1.2 Describing Distributions with Numbers...

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1.2 Describing Distributions with Numbers Measures of Center The Mean x The mean x of a set of observations is equal to the sum of their values divided by the number of observations. If the n observations are x 1 , x 2 , …, x n , their mean is: n n 2 1 x x x + + + or, more compactly: n 1 i x The (capital Greek sigma) means “add them all up.” The weakness of the mean is that it is not a resistant measure of center: it is not resistant to the influence of a few extreme observations. In the example below, note that decreasing the minimum value significantly has a large impact on the mean. x : 80, 90, 90, 100 x = 90 y : 20, 90, 90, 100 y = 75 The Median M The median M is the midpoint of a distribution. To find the median of a distribution: 1) Arrange all the observations in order from smallest to largest. 2) If the number of observations n is odd, the median is the center observation. The location of the median is ( n +1)/2 observations up from the bottom of the list. 3) If the number of observations n is even, the median is the mean of the two center observations. The location of the median is ( n +1)/2 observations up from the bottom of the list.

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a : 1, 4, 8, 14, 20 [ n +1]/2= 6/2= 3 The third observation is the median. The median is 8. b : 1, 4, 8, 14, 20, 21 [ n +1]/2= 7/2= 3.5 The median is the average of the third and fourth observations. The median is 11. c : 1, 4, 8, 14, 20, 100 [ n +1]/2=7/2= 3.5 The median is the average of the third and fourth observations. The median is 11. The strength of the median is that it is a resistant measure of center: it is resistant to the influence of a few extreme observations. In the example above, note that increasing the maximum value significantly from b to c did not change the median at all. The Mode The mode is the number that occurs most frequently in a set of data values. x : 80, 90, 90, 100 The mode is 90. Measures of Spread While we care about measures of center, we also care about measures of spread. For instance, median income might be the same in country A and country B . If, however, everyone has the same income in country A while there is extreme inequality in country B , then country A and country B are very different countries.
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handout_1_2 - 1.2 Describing Distributions with Numbers...

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