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Measures of Central Tendency

Learn about central tendency in statistical analysis, including mode, median, and mean. Discover how to calculate and interpret these measures accurately. Explore the relevance of each measure based on data distribution shapes.

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

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

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

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

  6. 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

  7. 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

  8. Example Subject # Test Score • 82 • 90 • 84 • 83 • 95 Mode = ?

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

  10. Finding the Median • Arrange the scores in ascending or descending numerical order • If there is an odd number of scores, the Md is the middle score

  11. Finding the Median -- continued 3. If there is an even number of scores, the median corresponds to a value halfway between the two middle scores

  12. 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

  13. The Mean • The sum of the scores divided by the number of scores • The arithmetic average

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

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

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

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

  18. Selecting a Measure of Central Tendency • Choice depends on • Measurement level of data • If the data is nominal, the mode must be used • The mode can also be used for other levels of measurement

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

  20. 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

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

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