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Measures of Central Tendency

Measures of Central Tendency. CJ 526 Statistical Analysis in Criminal Justice. Introduction. Central Tendency. Characteristics of a Measure of Central Tendency. Single number that represents the entire set of data (average). Alternate Names. Also known as _____ value Average Typical

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Measures of Central Tendency

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  1. Measures of Central Tendency CJ 526 Statistical Analysis in Criminal Justice

  2. Introduction • Central Tendency

  3. Characteristics of a Measure of Central Tendency • Single number that represents the entire set of data (average)

  4. Alternate Names • Also known as _____ value • Average • Typical • Usual • Representative • Normal • Expected

  5. Three Measures of Central Tendency • Mode • Median • Mean

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

  7. 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

  8. 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

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

  10. Symbolized by Md

  11. Finding the Median • Arrange the scores in ascending or descending numerical order

  12. Finding the Median -- continued • If there is an even number of scores, the median corresponds to a value halfway between the two middle scores

  13. 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

  14. The Mean • The sum of the scores divided by the number of scores

  15. Formula for finding the Mean • Symbolized by M or “X-bar”

  16. Characteristics of the Mean • The mean may not necessarily be an actual score in a distribution

  17. Deviation Score • Measure of how far away a given score is from the mean • x = X - M

  18. Example of Finding the Mean • X: 8, 6, 7, 11, 3 • Sum = 35 • N = 5 • M = 7

  19. Selecting a Measure of Central Tendency • Choice depends on

  20. Nature of the Variable • Nominal -- Mode

  21. Shape of the Distribution • Symmetrical – Mean • Not symmetrical—the median will be better • Any time there are extreme scores the median will be better

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

  23. Intended Use of Statistic • Descriptive -- Mode, Median, or Mean

  24. Central Tendency and the Shape of a Distribution • Symmetrical • Unimodal: Mo = Md = M

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