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Chapter 3 : Averages and Variation

Chapter 3 : Averages and Variation. Section 2 : Measures of Dispersion. Reflects the amount of spread or variability in a collection of data. Example : Find the mean and median for the following sets of data. 71 73 74 76 77 79 Mean & Median = 75 46 63 70 80 91 100

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Chapter 3 : Averages and Variation

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  1. Chapter 3: Averages and Variation Section 2: Measures of Dispersion

  2. Reflects the amount of spread or variability in a collection of data. Example: Find the mean and median for the following sets of data. 71 73 74 76 77 79 Mean & Median = 75 46 63 70 80 91 100 Mean & Median = 75 Measure of Dispersion

  3. A measure of central tendency is incapable of detecting differences in the spread or variability in a collection of data values.

  4. Range • The difference between the highest and lowest data values. • Range = Highest – Lowest

  5. 1 2 3 4 5 6 7 8 9 10 11 Example: Find the mean and range for the following sets of data. Number of Books Read by History Students Mean = 6 Range = 10

  6. 1 2 3 4 5 6 7 8 9 10 11 Number of Books Read by Sociology Student Mean = 6 Range = 10

  7. A A A B B B X X 38 25 34 24 26 23 24 22 20 21 20 19 16 18 14 17 6 16 2 15 The sum of the deviations from the mean always equals zero.

  8. Variance • The average of the sum of the squared deviation scores. • Population Variance = • Sample Variance s2 =

  9. Standard Deviation • Square root of the variance • Typical distance from the mean for the data values • Population Standard Deviation = • Sample Standard Deviation s=

  10. Example • Find the population variance for the following data values. • 6 11 5 1 6 6 7 5 7 6 ***First find the population mean =

  11. Population Variance Sample (cont) x x – (x – )2 6 0 0 11 5 25 5 -1 1 1 -5 25 6 0 0 6 0 0 7 1 1 5 -1 1 7 1 1 6 0 0 54

  12. Population Variance Sample (cont) • = = = 5.4 • Using the previous example, the population standard deviation would be found by: = = 2.32

  13. Example Sample Variance • Find the sample variance and sample standard deviation for the following data values. 4 3 7 4 2 • First find the sample mean = = = 4

  14. Sample Variance Example (Cont) Use a table to calculate the sample variance. x x – (x – )2 2 -2 4 4 0 0 3 -1 1 7 3 9 4 0 0 14

  15. Sample Variance Example (Cont) • s2 = = = 3.5 • s = = 1.87 Take the square root of the sample variance to find the sample standard deviation.

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