Measures of Central Tendency

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Measures of Central Tendency

• CJ 526 Statistical Analysis in Criminal Justice

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Introduction

• Central Tendency

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Characteristics of a Measure of Central Tendency

• Single number that represents the entire set of
  data (average)

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Alternate Names

• Also known as _____ value
• Average
• Typical
• Usual
• Representative
• Normal
• Expected

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Three Measures of Central Tendency

• Mode
• Median
• Mean

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The Mode

• Score or qualitative category that occurs with
  the greatest frequency
• Always used with nominal data we find the most
  frequently occurring category

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Mode

• Example of modal category
• Sample of 25 married 30 single 22 divorced
• Married is the modal category
• Determined by inspection not by computation
  counting up the number of times a value occurs

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Example of Finding the Mode

• X 8 6 7 9 10 6
• Mode 6
• Y 1 8 12 3 8 5 6
• Mode 8
• Can have more than one mode
• 1 2 2 8 10 5 5 6
• Mode 2 and 5

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The Median

• The point in a distribution that divides it into
  two equal halves

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• Symbolized by Md

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Finding the Median

• Arrange the scores in ascending or descending
  numerical order

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Finding the Median -- continued

• If there is an even number of scores the median
  corresponds to a value halfway between the two
  middle scores

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Example of Finding the Median

• X 6 6 7 8 9 10 11
• Median 8
• Y 1 3 5 6 8 12
• Median 5.5

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The Mean

• The sum of the scores divided by the number of
  scores

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Formula for finding the Mean

• Symbolized by M or X-bar

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Characteristics of the Mean

• The mean may not necessarily be an actual score
  in a distribution

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Deviation Score

• Measure of how far away a given score is from the
  mean
• x X - M

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Example of Finding the Mean

• X 8 6 7 11 3
• Sum 35
• N 5
• M 7

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Selecting a Measure of Central Tendency

• Choice depends on

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Nature of the Variable

• Nominal -- Mode

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Shape of the Distribution

• Symmetrical Mean
• Not symmetricalthe median will be better
• Any time there are extreme scores the median will
  be better

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Example

• Median income if someone loses their job an
  income of 0this would pull the average down
• Median housing values an unusually nice house
  or poor house would affect the average
• Better to use the median

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Intended Use of Statistic

• Descriptive -- Mode Median or Mean

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Central Tendency and the Shape of a Distribution

• Symmetrical
• Unimodal Mo Md M