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Measures of Central Tendency

Learn about mode, median, and mean in statistics, including how to calculate them and their significance in data analysis. Discover concepts such as sigma notation, resistant measures, and trimmed mean.

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Measures of Central Tendency

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  1. Measures of Central Tendency • Mode • Median • Mean

  2. The Mode the value or property that occurs most frequently in the data

  3. Find the mode: 6, 7, 2, 3, 4, 6, 2, 6 The mode is 6.

  4. Find the mode: 6, 7, 2, 3, 4, 5, 9, 8 There is no mode for this data.

  5. The Median the central value of an ordered distribution

  6. To find the median of raw data: • Order the data from smallest to largest. • For an odd number of data values, the median is the middle value. • For an even number of data values, the median is found by dividing the sum of the two middle values by two.

  7. Find the median: Data: 5, 2, 7, 1, 4, 3, 2 Rearrange: 1, 2, 2, 3, 4, 5, 7 The median is 3.

  8. Find the median: Data: 31, 57, 12, 22, 43, 50 Rearrange: 12, 22, 31, 43, 50, 57 The median is the average of the middle two values =

  9. The Mean The mean of a collection of data is found by: • summing all the entries • dividing by the number of entries

  10. Find the mean: 6, 7, 2, 3, 4, 5, 2, 8

  11. Sigma Notation The symbol  means “sum the following.” is the Greek letter (capital) sigma.

  12. Sample mean “x bar” Population mean Greek letter (mu) Notations for mean

  13. Number of entries in a set of data • If the data represents a sample, the number of entries = n. • If the data represents an entire population, the number of entries = N.

  14. Sample mean

  15. Population mean

  16. Resistant Measure a measure that is not influenced by extremely high or low data values

  17. Mean Median The mean is less resistant. It can be made arbitrarily large by increasing the size of one value. Which is less resistant?

  18. Trimmed Mean a measure of center that is more resistant than the mean but is still sensitive to specific data values

  19. To calculate a (5 or 10%) trimmed mean • Order the data from smallest to largest. • Delete the bottom 5 or 10% of the data. • Delete the same percent from the top of the data. • Compute the mean of the remaining 80 or 90% of the data.

  20. 15, 17, 18, 20, 20, 25, 30, 32, 36, 60 Delete the top and bottom 10% New data list: 17, 18, 20, 20, 25, 30, 32, 36 10% trimmed mean = Compute a 10% trimmed mean:

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