1. Lecture 16 Sec. 5.3.1 – 5.3.3 Tue, Feb 12, 2008 The Five-Number Summary

2. A five-number summary of a sample or population consists of five numbers that divide the sample or population into four equal parts. These numbers are called the quartiles.  0 th Quartile = minimum.  1 st Quartile = Q1.  2 nd Quartile = median.  3 rd Quartile = Q3.  4 th Quartile = maximum.

3. Example If the distribution were uniform from 0 to 10, what would be the five-number summary? 156789234 0 10

4. Example If the distribution were uniform from 0 to 10, what would be the five-number summary? 156789234 0 10 50% Median

5. Example If the distribution were uniform from 0 to 10, what would be the five-number summary? 156789234 0 10 25% MedianQ3Q1

6. Example Where would the median and quartiles be in this symmetric non-uniform distribution? 1234567

7. Example Where would the median and quartiles be in this symmetric non-uniform distribution? 1234567

8. Example Where would the median and quartiles be in this symmetric non-uniform distribution? 1234567 Median

9. Example Where would the median and quartiles be in this symmetric non-uniform distribution? 1234567 MedianQ3Q1

10. Percentiles – Textbook’s Method The p th percentile – A value that separates the lower p% of a sample or population from the upper (100 – p)%.  p% or more of the values fall at or below the p th percentile, and  (100 – p)% or more of the values fall at or above the p th percentile.

11. Finding Quartiles of Data To find the quartiles, first find the median (2 nd quartile). Then the 1 st quartile is the “median” of all the numbers that are listed before the 2 nd quartile. The 3 rd quartile is the “median” of all the numbers that are listed after the 2 nd quartile.

12. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

13. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32 Median

14. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32 Median Find “median”

15. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32 Median Q3Q1

16. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32 MedianQ3Q1MinMax

17. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33

18. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33 Median 19.5

19. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33 Median 19.5 Q3 27.5 Q1 12.5

20. Example Find the quartiles of the sample 5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33 Median 19.5 Q3 27.5 Q1 12.5 MinMax

21. The Interquartile Range The interquartile range (IQR) is the difference between Q3 and Q1. The IQR is a commonly used measure of spread, or variability. Like the median, it is not affected by extreme outliers.

22. IQR The IQR of 22, 28, 31, 40, 42, 56, 78, 88, 97 is IQR = Q3 – Q1 = 78 – 31 = 47.

23. IQR Find the IQR for the sample  5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240. Are the data skewed?

24. Salaries of School Board Chairmen County/CitySalaryCounty/CitySalary Henrico$20,000Powhatan$4,800 Chesterfield18,711Colonial Hgts5,100 Richmond11,000Goochland5,500 Hanover11,000Hopewell4,500 Petersburg8,500Charles City4,500 Sussex7,000Cumberland3,600 Caroline5,000Prince George3,750 New Kent6,500King & Queen3,000 Dinwiddie5,120King William2,400 Louisa4,921West Point0

25. Five-Number Summaries and Stem-and-Leaf Displays StemLeaf 13 189 21334 255789 3034 38 GPA Data