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Chapter 4: Central Tendency How do we quantify the ‘middle’ of a distribution of numbers? Three ways: The mode, median and the mean Mode (Mo): The score that occurs with the greatest frequency The modal color of M&M’s is brown. Note that the mode is appropriate for nominal scores Example: the mode of this sample of 7 numbers: 5,3,1,6,2,8,3 is 3.

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Median (Mdn): The value that divides the distribution into halves. Same as P 50 We calculate the median by calculating P 50. Ice Dancing, compulsory dance scores, 2010 Winter Olympics Rank order Subtract 1/2Divide by 23Multiply by 100 The percentile score for P 50 is 84.58 89.621413.50.5959 85.361312.50.5454 84.581211.50.5050 83.891110.50.4646 83.12109.50.4141 Score Calculating the median is easy (forget the book’s formula on page 67): If there are an odd number of scores, then the median is the middle number. If there are an even number, then the median is the average of the middle two. Example: Find the median of this sample of 7 numbers: 5,3,1,6,2,8,3 Reorder: 1,2,3,3,5,6,8. The median is 3.

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mean: the sum of all scores divided by the total number of scores. Using mathematical symbols: x : mean of a population of scores (pronounced ‘mu of x’) : mean of a sample of scores (pronounced ‘x bar’) n: the number of scores in a sample N: the number of scores in a population X i: the ith number in a sample’ : means to add (sigma) Which we shorten to: The sample mean is: The population mean is:

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Example: find the mean of this sample of 7 numbers: 5,3,1,6,2,8,3 X 1 = 5, X 2 = 3, X 3 = 1, X 4 = 6, X 5 = 2, X 6 = 8, X 7 = 3

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-The mean is affected by all numbers in the sample (unlike the mode or median) From the example above: -Multiplying all scores by a number multiplies the mean by that same number -Adding a number to all scores adds the same number to the mean. Fun facts about the mean -The mean is the ‘balance’ point of a distribution. 12345678 -The mean is the value that makes

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The median is not sensitive to outliers: Median of 1,2,4,5, and 10000 is 4. The mean is 2002.4 325375425475525575625675725775 0 5 10 15 20 25 30 35 40 GRE quant scores Frequency Median=705 Mean=683.9 For a skewed distribution, the mean is pulled toward the tail compared to the median More stuff about the median and mean If a distribution is ‘symmetric’ (like a normal distribution), then the mean and median are the same.

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Bottom 98% of US household income (2005) Medians are usually reported for income because of the positive skew in the distribution http://www.visualizingeconomics.com/wp-content/uploads/2005_income_distribution.gif

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Reaction times are a classic example of a skewed distribution in psychology

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3.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Measures of Center.

3.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Measures of Center.

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