Central Tendency

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Central Tendency
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Variables have distributions

• A variable is something that changes or has
  different values (e.g., anger).
• A distribution is a collection of measures,
  usually across people.
• Distributions of numbers can be summarized with
  numbers (called statistics or parameters).

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Central Tendency refers to the Middle of the
Distribution
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Middle of the Distribution
(Common Statistics)

• Mode
• Most common score
• Median
• Top from bottom 50 percent
• Mean
• Arithmetic mean or average

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Mode

• The most frequently occurring score. Can have
  bimodal and multimodal distributions. Modal
  psychology student is female. Modal number of
  pubs from grad school is zero.

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Median

• Score that separates top 50 from bottom 50
• Even number of scores, median is half way between
  two middle scores.
• 1 2 3 4 5 6 7 8 Median is 4.5
• Odd number of scores, median is the middle number
• 1 2 3 4 5 6 7 Median is 4

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Mean

• Sum of scores divided by the number of people.
  Population mean is (mu) and sample mean is
  (X-bar).
• We calculate the sample mean by
• We calculate the population mean by

Raw score is X. N is number of people. Sigma
(Greek symbol like big E) is summation sign. Add
up scores and divide by the number of people.
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Computation of Mean
X (scores) Sum 246 12
2 Mean 12 / 3 4
4
6

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Deviations from the Mean

• Deviation defined.
• Deviations sum to zero.
• Raw scores
• Deviation scores

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8 9 10
7 8 9 10 11
0
-1 0 1
-2 -1 0 1 2
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Comparison of stats (1)

• Mode
• Good for nominal variables
• Good if you need to know most frequent
  observation
• Quick and easy

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Comparison of stats (2)

• Median
• Good for bad (skewed) distributions
• Good for distributions with arbitrary ceiling or
  floor
• Often used with distributions of money

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Comparison of stats (3)

• Mean
• Used for inference as well as description best
  estimator of the parameter
• Based on all data in the distribution
• Generally preferred except for bad
  distribution.
• Most commonly used statistic for central
  tendency.

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Effects of Distribution Shape
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Review

• What is central tendency?
• Mode
• Median
• Mean

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Computation

• Consider the following scores 1, 2, 2, 3, 3, 3,
  4, 5
• For the above set of scores, what is N?
• Cannot be determined
• 2
• 3
• 8

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Computation

• Consider the following scores 1, 2, 2, 3, 3, 3,
  4, 5
• For the above set of scores, what is the
  percentage (relative frequency) of 2s?
• 2
• 10
• 20
• 25

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Computation

• Consider the following scores 1, 2, 2, 3, 3, 3,
  4, 5
• For the above set of scores, what is the mode?
• 2
• 3
• 4
• 5

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Discussion Questions

• Name a variable where it would be better to find
  the median than the mean.
• Why is it misleading to say that the average
  person has 1.2 brothers? Why might it be useful
  or helpful to say it anyway?