1. M08-Numerical Summaries 2 1  Department of ISM, University of Alabama, 1995-2003 Lesson Objectives  Learn what percentiles are and how to calculate quartiles.  Learn to find the five number summary.  Learn how to construct and use Boxplots.

2. M08-Numerical Summaries 2 2  Department of ISM, University of Alabama, 1995-2003 If x = the 100p th percentile, then at least 100p% of data is   x  at least 100(1-p)% of data is   x. Sample 100p th percentile: Example: You are told you scored 47; then you hear “47” is at the 82 nd percentile. 82% of the sample have scores   47, AND 18% have scores  47.

3. M08-Numerical Summaries 2 3  Department of ISM, University of Alabama, 1995-2003 1. Order the data. 2. Calculate np. 3a. If np is NOT an integer, Finding 100p th percentile: n = 25, p = 1/3 np = 8.333, round up; find the obs. in this position. 9 th position will be the 33.333 %tile.

4. M08-Numerical Summaries 2 4  Department of ISM, University of Alabama, 1995-2003 1. Order the data. 2. Calculate np. 3b. If np IS an integer, say k, Finding 100p th percentile: n = 25, p =.40 np = _____, then avg the k th and (k+1) th ordered values. average of ______ & ____ positions will be the 40 th %tile.

5. M08-Numerical Summaries 2 5  Department of ISM, University of Alabama, 1995-2003 1. Maximum 2. 3 rd Quartile, Q 3 = 75th p’tile 3. Median 4. 1 st Quartile, Q 1 = 25th p’tile 5. Minimum Five Number Summary

6. M08-Numerical Summaries 2 6  Department of ISM, University of Alabama, 1995-2003 1st Quartile (25th percentile) : at least 25% of the data values lie at or below it. 3rd Quartile (75th percentile) : at least 75% of the data values lie at or below it. Quartiles:

7. M08-Numerical Summaries 2 7  Department of ISM, University of Alabama, 1995-2003 Method 1: Percentile method Q 1 located at position (n+1)*1/4 Q 2 located at position (n+1)*2/4 Q 3 located at position (n+1)*3/4 n Q1 Q2 Q3 5 8 11

8. M08-Numerical Summaries 2 8  Department of ISM, University of Alabama, 1995-2003 Step 1: Order the data: 12, 14, 16, 18, 19, 21, 22, 25, 27 Max = Q 3 = Median = Q 1 = Min = Example 6

9. M08-Numerical Summaries 2 9  Department of ISM, University of Alabama, 1995-2003 median of observations below the median’s position. Q 3 = median of observations above the median’s position. Q 1 = Method 2: Median method

10. M08-Numerical Summaries 2 10  Department of ISM, University of Alabama, 1995-2003 Ordered data: 12, 14, 16, 18, 19, 21, 22, 25, 27 Max = Q 3 = Median = Q 1 = Min = Example 6

11. M08-Numerical Summaries 2 11  Department of ISM, University of Alabama, 1995-2003 IQR = Q - Q 13  IQR is the range of the middle 50% of the data.  Observations more than 1.5 IQR’s beyond quartiles are considered outliers. 4. Interquartile Range (IQR)

12. M08-Numerical Summaries 2 12  Department of ISM, University of Alabama, 1995-2003 Which summary statistics should I use? Shape? Location? Variation?

13. M08-Numerical Summaries 2 13  Department of ISM, University of Alabama, 1995-2003 Boxplot A graphically display of the five number summary (also called a box-and-whiskers plot)

14. M08-Numerical Summaries 2 14  Department of ISM, University of Alabama, 1995-2003 Ordered data: 12, 14, 16, 18, 19, 21, 22, 25, 27 Max = Q 3 = Median = Q 1 = Min = 27.0 19.0 12.0 Q 1 = 15.0 Q 3 = 23.5 23.5 15.0 IQR = 8.5 Example 6

15. M08-Numerical Summaries 2 15  Department of ISM, University of Alabama, 1995-2003 Ordered data: 12, 14, 16, 18, 19, 21, 22, 25, 27 Example 6A What if.... 19, 19, 19, Ordered data: 12, 14, 16, 18, 19, 21, 22, 25, 27 Example 6B What if.... X

16. M08-Numerical Summaries 2 16  Department of ISM, University of Alabama, 1995-2003 28 22 24 12 14 16 18 20 26 Note: Middle 50% of data are within the range of the box Note: Middle 50% of data are within the range of the box Max = Q 3 = Median = Q 1 = Min = 27.0 19.0 12.0 23.5 15.0 IQR = 8.5

17. M08-Numerical Summaries 2 17  Department of ISM, University of Alabama, 1995-2003 Use side-by-side boxplots to display two variables when one is quantitative, and one is categorical. Useful tool for comparing distributions.

18. M08-Numerical Summaries 2 18  Department of ISM, University of Alabama, 1995-2003 AB C Part Suppliers; who is best? 15.000 14.980 15.020 15. 040 14.960

19. M08-Numerical Summaries 2 19  Department of ISM, University of Alabama, 1995-2003 Modified Boxplot Observations more than 1.5 IQR’s beyond quartiles are considered outliers.  Useful in detecting outliers:  More accurate picture of data.  Available in Minitab (boxplot); not in Excel.

20. M08-Numerical Summaries 2 20  Department of ISM, University of Alabama, 1995-2003 13, 24, 26, 26, 27, 28, 36, 46 Maximum = 3rd Quartile = Median = 1st Quartile = Minimum = 26.5 46.0 26.5 13.0 25.032.0 25.0 IQR = 7.0 Example 7 1.5 IQR = 1.5 7.0 = 10.5

21. M08-Numerical Summaries 2 21  Department of ISM, University of Alabama, 1995-2003 Q 3 = 32.0 Q 1 = 25.0 Q - 1.5 IQR = 14.5 1 Q + 1.5 IQR = 42.5 3 * Note: Whiskers go to the most extreme value within the limits, not to the limits. Note: Whiskers go to the most extreme value within the limits, not to the limits. 48 44 40 36 32 28 24 20 16 12 * 1.5 IQR Data: 13, 24, 26, 26, 27, 28, 36, 46

22. M08-Numerical Summaries 2 22  Department of ISM, University of Alabama, 1995-2003 * 48 44 40 36 32 28 24 20 16 12 * Data: 13, 24, 26, 26, 27, 28, 36, 46 Finished Box Plot

23. Formula Sheet Example Q1Q1 Q3Q3 M MaxMin  Q 3 +1.5 IQRQ 1 -1.5 IQR Lines extend to the smallest & largest obs. inside of limits. Modified Box Plot: Box Plot: Plot each obs. that is beyond the “outlier limits” on each end. Note: For this problem, no data are below the lower “outlier limit”. 1.5 IQR

24. M08-Numerical Summaries 2 24  Department of ISM, University of Alabama, 1995-2003 Match each of the following descriptions to one of the following histograms. 1. Scores on an EASY Math exam. 2. Heights of a group of students. 3. Number of medals won by medal winning countries in the 1996 Winter Olympics. 4. SAT scores for some college students. 5. Last digit in SSN for 100 people.

25. M08-Numerical Summaries 2 25  Department of ISM, University of Alabama, 1995-2003 A B C D E Match descriptions to a Histograms. 1. Scores on an EASY Math exam. 2. Heights of a group of students. 3. Number of medals won by medal winning countries in the 1996 Winter Olympics. 4. SAT scores for some college students. 5. Last digit in SSN for 100 people.

26. M08-Numerical Summaries 2 26  Department of ISM, University of Alabama, 1995-2003 Match each of the following Boxplots (1,2,3,4,5) to one of the Histograms (A-E) above.

27. M08-Numerical Summaries 2 27  Department of ISM, University of Alabama, 1995-2003 12345

28. M08-Numerical Summaries 2 28  Department of ISM, University of Alabama, 1995-2003 Descriptive Statistics Variable N Mean Median Range A 100 50.6 51.0 20.0 B 100 49.9 50.1 42.6 C 100 49.9 50.6 12.9 D 100 54.1 32.9 415.4 E 100 50.4 49.8 32.9

29. M08-Numerical Summaries 2 29  Department of ISM, University of Alabama, 1995-2003 Descriptive Statistics Variable N Mean Median Range A 100 50.6 51.0 20.0 B 100 49.9 50.1 42.6 C 100 49.9 50.6 12.9 D 100 54.1 32.9 415.4 E 100 50.4 49.8 32.9 1 3 2 4 5

30. M08-Numerical Summaries 2 30  Department of ISM, University of Alabama, 1995-2003