1. Measures of Variation

2. Variation Variation describes how widely data values are spread out about the center of a distribution.

3. Larger variation The variation or dispersion in a set of values refers to how spread out the values are from each other. · The variation is small when the values are close together. · There is no variation if the values are the same. · Same Center Smaller variation Larger variation

4. Big Bank (three line wait times): 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0 Mean______ Median_______ Mode_________ Best Bank (one line with three tellers wait times): 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8 Mean______ Median_______ Mode_________ All the measures of ‘center’ for each bank are essentially equal…… Example: Why does Variation matter?? 7.2 7.7 7.2 7.8

5. Why Variation Matters 6-B

6. Range: a measure of variation The Range of a set of data is the difference between the highest and the lowest values: Range = highest value (max) − lowest value (min) EXAMPLE: Big Bank range = 11 − 4.1 = 6.9 min Best Bank range = 7.8 − 6.6 = 1.2 min **Big Bank has a greater variance of wait times.

7. BUT: The Range can be a misleading measure of variation!! Measure of variation Difference between the largest and the smallest observations: Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5

8. a measure of variation of the scores about the mean (average deviation from the mean) Standard Deviation

9. Describing Spread: Standard Deviation Roughly speaking, standard deviation is the average distance values fall from the mean.

10. Comparing Standard Deviations Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s =.9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C

11. Calculated Standard Deviation is a measure of Variation in data Sample Data SetMeanStandard Deviation 100, 100, 100, 100, 1001000 90, 90, 100, 110, 11010010 30, 90, 100, 110, 17010050 90, 90, 100, 110, 32014299.85

12. Sample Standard Deviation Formula  ( x - x ) 2 n - 1 S =S =

13. Symbols for MEAN sample mean population mean

14. Population Standard Deviation 2  ( x - µ ) N  =

15. Symbols for Standard Deviation Sample Population   x x  n s S x x  n-1 Book Some graphics calculators Some non-graphics calculators Textbook Some graphics calculators Some non-graphics calculators

16. Measures of Variation Variance standard deviation squared s  2 2 } use square key on calculator Notation

17. Sample Variance Population Variance  ( x - x ) 2 n - 1 s 2 =  (x - µ)2 (x - µ)2 N  2 =

18. Round-off Rule for measures of variation Carry one more decimal place than is present in the original set of values. Round only the final answer, never in the middle of a calculation.