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

3. Measure 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 –Mean & Median = 75

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

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

6. Example: Find the mean and range for the following sets of data. Number of Books Read by History Students Mean = 6 Range = 10 1234567891011

7. Number of Books Read by Sociology Student Mean = 6 Range = 10 1234567891011

8. AAABBB 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.

9. Variance The average of the sum of the squared deviation scores. Population Variance = Sample Variance s 2 =

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

11. 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 =

12. Population Variance Sample (cont) 11 5 25 5 -1 1 1 -5 25 6 0 0 7 1 1 5 -1 1 7 1 1 6 0 0 x x – (x – ) 2 60 0 54

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

14. 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

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

16. Sample Variance Example (Cont) s 2 = = = 3.5 s = = 1.87 Take the square root of the sample variance to find the sample standard deviation.