1. Example - Fax Here are the number of pages faxed by each fax sent from our Math and Stats department since April 24 th, in the order that they occurred. 5, 1, 2, 7, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 7, 37, 2, 8, 2, 25

2. Example - Fax Here are the number of pages faxed by each fax sent from our Math and Stats department since April 24 th, in the order that they occurred. 5, 1, 2, 7, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 7, 37, 2, 8, 2, 25 Find the 40 th percentile, Q U, Q L, M and IQR.

3. How to Find Percentiles 1) Rank the n points of data from lowest to highest 2) Pick a percentile ranking you want to find. Say p. 3) Compute –If L is not a whole number then go halfway between the whole number below and above –

4. Example - Fax 1) Rank the n points of data from lowest to highest 5, 1, 2, 7, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 7, 37, 2, 8, 2, 25 Find the 40 th percentile.

5. Example - Fax 1) Rank the n points of data from lowest to highest 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Find the 40 th percentile.

6. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 2) Pick a percentile ranking you want to find. 40% Find the 40 th percentile.

7. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 2) Pick a percentile ranking you want to find. 40% 3) Compute

8. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 2) Pick a percentile ranking you want to find. 40% 3) Compute

9. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 3) Compute Half way between the 11 th and 12 th number.

10. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 3) Compute Half way between the 11 th and 12 th number. Answer: 2

11. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 To compute Q U and Q L, M. Find the Median, divide the data into two equal parts and then the Medians of these.

12. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 M = 3 Q U = 6.5 Q L = 2 IQR=6.5-2=4.5.

13. Percentiles Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not affected at all by outliers.

14. Percentiles Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not effected at all by outliers. To use the IQR as a measure of variance we need to find the Five Number Summary of the data and then construct a Box Plot.

15. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 To compute Q U and Q L, M. Find the Median, divide the data into two equal parts and then the Medians of these. An example and more specific instructions will be done on the board.

16. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 M = 3 Q U = 6.5 Q L = 2 IQR=6.5-2=4.5

17. Percentiles Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not effected at all by outliers.

18. Percentiles Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not effected at all by outliers. To use the IQR as a measure of variance we need to find the Five Number Summary of the data and then construct a Box Plot.

19. Five Number Summary and Outliers The Five Number Summary of a data set consists of five numbers, –MIN, Q L, M, Q U, Max

20. Five Number Summary and Outliers The Five Number Summary of a data set consists of five numbers, –MIN, Q L, M, Q U, Max Suspected Outliers lie –Above 1.5 IQRs but below 3 IQRs from the Upper Quartile –Below 1.5 IQRs but above 3 IQRs from the Lower Quartile Highly Suspected Outliers lie –Above 3 IQRs from the Upper Quartile –Below 3 IQRs from the Lower Quartile.

21. Five Number Summary and Outliers The Inner Fences are: – data between the Upper Quartile and 1.5 IQRs above the Upper Quartile and –data between the Lower Quartile and 1.5 IQRs below the Lower Quartile The Outer Fences are: – data between 1.5 IQRs above the Upper Quartile and 3 IQRs above the Upper Quartile and –data between 1.5 IQRs Lower Quartile and 3 IQRs below the Lower Quartile

22. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Min=1 Q L = 2 M = 3 Q U = 6.5 Max = 37 IQR=6.5-2=4.5

23. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Min=1Inner Fence Limits Q L = 22-(1.5)*4.5=-4.75 M = 36+(1.5)*4.5=12.75 Q U = 6.5 Max = 37. IQR=6.5-2=4.5.

24. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Min=1Inner Fence Limits Q L = 22-(1.5)*4.5=-4.75 M = 36+(1.5)*4.5=12.75 Q U = 6.5 Max = 37. IQR=6.5-2=4.5

25. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Min=1Inner Fence Limits Q L = 22-(1.5)*4.5=-4.75 M = 36+(1.5)*4.5=12.75 Q U = 6.5Outer Fence Limits Max = 372-3*4.5=-11.5 IQR=6.5-2=4.56+(3)*4.5=19.5

26. Example - Fax 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 10, 13, 25, 37 Min=1Inner Fence Limits Q L = 22-(1.5)*4.5=-4.75 M = 36+(1.5)*4.5=12.75 Q U = 6.5Outer Fence Limits Max = 372-3*4.5=-11.5 IQR=6.5-2=4.56+(3)*4.5=19.5

27. Definition: Boxplot A boxplot is a graph of lines (from lowest point inside the lower inner fence to highest point in the upper inner fence) and boxes (from Lower Quartile to Upper quartile) indicating the position of the median. Outliers Median Highest data Point less than the upper inner fence Lower Quartile Upper Quartile * Lowest data Point more than the lower inner fence

28. 2.109 a. 85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196

29. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 Min 85 Max 196 Q u = 172 Q L = 151 M = 165

30. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 Min 85Inner Fence Limits: Max 196172+1.5(21)=203.5 Q u = 172151-1.5(21)=119.5 Q L = 151 M = 165 IQR =21

31. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 Min 85Inner Fence Limits: Max 196172+1.5(21)=203.5 Q u = 172151-1.5(21)=119.5 Q L = 151 M = 165 IQR =21

32. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 Min 85Inner Fence Limits: Max 196172+1.5(21)=203.5 Q u = 172151-1.5(21)=119.5 Q L = 151 Outer Fence Limits: M = 165 172+3(21)=235 IQR =21 151-3(21)=88

33. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 Min 85Inner Fence Limits: Max 196172+1.5(21)=203.5 Q u = 172151-1.5(21)=119.5 Q L = 151 Outer Fence Limits: M = 165 172+3(21)=235 IQR =21 151-3(21)=88

34. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 80100120140 180 160200

35. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 80100120140 180 160200

36. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 80100120140 180 160200 *

37. 2.109 a.85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 80100120140 180 160200 * * Suspected Outlier Highly Suspected Outlier