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One-Variable Data Analysis 67
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The median of a ordered dataset is the middle value in the set If the dataset has an odd number of values, the median is a member of the set and is the middle value If there are 3 values, the median is the second value If there are 5, it is the third, etc If the dataset has an even number of values, the median is the mean of the two middle numbers If there are 4 values, the median is the mean of the second and third values In general, if there are n values in the ordered dataset, the median is at the you will find the median at the n +1 position If you have 28 terms in order, 2
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28 + 1 = 145th position (that is, between the 14th and 2 n +1 as the value of the median rather than as the 15th terms) Be careful not to interpret 2 location of the median example: Consider once again the data in the previous example from Babe Ruth s career What was the median number of home runs per year he hit during his major league career solution: First, put the numbers in order from smallest to largest: 0, 2, 3, 4, 6, 11, 22, 25, 29, 34, 35, 41, 41, 46, 46, 46, 47, 49, 54, 54, 59, 60 There are 22 scores, so the median is found at the 115th position, between the 11th and 12th scores (35 and 41) So the median is 35 + 41 = 38 2 The 1-Var Stats procedure, described in the previous Calculator Tip box, will, if you scroll down to the second screen of output, give you the median (as part of the entire five-number summary of the data: minimum, lower quartile; median, upper quartile; maximum)
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Resistant
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Although the mean and median are both measures of center, the choice of which to use depends on the shape of the distribution If the distribution is symmetric and mound shaped, the mean and median will be close However, if the distribution has outliers or is strongly skewed, the median is probably the better choice to describe the center This is because it is a resistant statistic, one whose numerical value is not dramatically affected by extreme values, while the mean is not resistant example: A group of five teachers in a small school have salaries of \$32,700, \$32,700, \$38,500, \$41,600, and \$44,500 The mean and median salaries for these teachers are \$38,160 and \$38,500, respectively Suppose the highest paid teacher gets sick, and the school superintendent volunteers to substitute for her The superintendent s salary is \$174,300 If you replace the \$44,500 salary with the \$174,300 one, the median doesn t change at all (it s still \$38,500), but the new mean is \$64,120 almost everybody is below average if, by average, you mean mean It s sort of like Lake Wobegon, where all of the children are expected to be above average
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68 U Step 4 Review the Knowledge You Need to Score High
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example: For the graph given below, would you expect the mean or median to be larger Why
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solution: You would expect the median to be larger than the mean Because the graph is skewed to the left, and the mean is not resistant, you would expect the mean to be pulled to the left (in fact, the dataset from which this graph was drawn from has a mean of 54 and a median of 6, as expected, given the skewness)
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Measures of Spread
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Simply knowing about the center of a distribution doesn t tell you all you might want to know about the distribution One group of 20 people earning \$20,000 each will have the same mean and median as a group of 20 where 10 people earn \$10,000 and 10 people earn \$30,000 These two sets of 20 numbers differ not in terms of their center but in terms of their spread, or variability Just as there were measures of center based on the mean and the median, we also have measures of spread based on the mean and the median
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