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

2. The Five-Number Summary
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.
0th Quartile = minimum.
1st Quartile = Q1.
2nd Quartile = median.
3rd Quartile = Q3.
4th Quartile = maximum.

3. Example
If the distribution were uniform from 0 to 10, what would be the five-number summary? 1 5 6 7 8 9 2 3 4 10

4. Example
If the distribution were uniform from 0 to 10, what would be the five-number summary? 1 5 6 7 8 9 2 3 4 10
50%
50%
Median

5. Example
If the distribution were uniform from 0 to 10, what would be the five-number summary? 1 5 6 7 8 9 2 3 4 10
25%
25%
25%
25% Q1
Median Q3

6. Example
Where would the median and quartiles be in this symmetric non-uniform distribution? 1 2 3 4 5 6 7

7. Example
Where would the median and quartiles be in this symmetric non-uniform distribution? 1 2 3 4 5 6 7

8. Example
Where would the median and quartiles be in this symmetric non-uniform distribution? 1 2 3 4 5 6 7
Median

9. Example
Where would the median and quartiles be in this symmetric non-uniform distribution? 1 2 3 4 5 6 7 Q1
Median Q3

10. Percentiles – Textbook’s Method
The pth 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 pth percentile, and
(100 – p)% or more of the values fall at or above the pth percentile.

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

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

16. Example Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
Min Q1
Median Q3
Max

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 Q1
12.5
Median
19.5 Q3
27.5

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

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 Are the data skewed?
5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.
Are the data skewed?

24. Salaries of School Board Chairmen
County/City
Salary
Henrico
$20,000
Powhatan
$4,800
Chesterfield
18,711
Colonial Hgts
5,100
Richmond
11,000
Goochland
5,500
Hanover
Hopewell
4,500
Petersburg
8,500
Charles City
Sussex
7,000
Cumberland
3,600
Caroline
5,000
Prince George
3,750
New Kent
6,500
King & Queen
3,000
Dinwiddie
5,120
King William
2,400
Louisa
4,921
West Point

25. Five-Number Summaries and Stem-and-Leaf Displays1 3 89 2
1334
55789
034 8
GPA Data