Example - Fax

1. Example - Fax Here are the number of pages faxed by each fax sent from our Math and Stats department since April 24th, 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 24th, 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 40th percentile, QU , QL , 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 40th 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 40th 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 40th 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 11th and 12th 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 11th and 12th 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 QU and QL , 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 QU = 6.5 QL = 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 QU and QL , 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 QU = 6.5 QL = 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, QL , M, QU, Max

20. Five Number Summary and Outliers • The Five Number Summary of a data set consists of five numbers, • MIN, QL , M, QU, 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 QL= 2 M = 3 QU = 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=1 Inner Fence Limits QL= 2 2-(1.5)*4.5=-4.75 M = 3 6+(1.5)*4.5=12.75 QU = 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=1 Inner Fence Limits QL= 2 2-(1.5)*4.5=-4.75 M = 3 6+(1.5)*4.5=12.75 QU = 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=1 Inner Fence Limits QL= 2 2-(1.5)*4.5=-4.75 M = 3 6+(1.5)*4.5=12.75 QU = 6.5Outer Fence Limits Max = 37 2-3*4.5=-11.5 IQR=6.5-2=4.5 6+(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=1 Inner Fence Limits QL= 2 2-(1.5)*4.5=-4.75 M = 3 6+(1.5)*4.5=12.75 QU = 6.5Outer Fence Limits Max = 37 2-3*4.5=-11.5 IQR=6.5-2=4.5 6+(3)*4.5=19.5

27. Definition: Boxplot A boxplotis 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. * Lowest data Point more than the lower inner fence Highest data Point less than the upper inner fence Median Upper Quartile Lower Quartile Outliers

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 • 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 Qu = 172 QL= 151 M = 165

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

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

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

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

34. 2.109 • 85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 180 80 100 120 140 160 200

35. 2.109 • 85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 180 80 100 120 140 160 200

36. 2.109 • 85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 * 180 80 100 120 140 160 200

37. 2.109 • 85 100 121 142 145 157 158 159 161 163 165 166 170 171 171 172 172 173 184 187 196 * Suspected Outlier Highly Suspected Outlier * 180 80 100 120 140 160 200