1. Measuring Variation – The Five-Number Summary
Lecture 18
Sec – 5.3.3
Mon, Feb 19, 2007

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 the following non-uniform distribution? 1 2 3 4 5 6 7

7. Example The five-number summary is
Minimum = 0,
Q1 = 2.5,
Median = 5,
Q3 = 7.5,
Maximum = 10.
If the distribution is not uniform, or if we are dealing with a list of numbers, the answer will not be so clear.

8. Quartiles – TI-38’s Method
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 median.
The 3rd quartile is the “median” of all the numbers that are listed after the median.

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

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

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

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

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

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

15. 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

16. 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

17. 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.

18. Percentiles – Textbook’s Method
Find the 25th percentile of the following sample:
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32.

19. Percentiles – Textbook’s Method
Value
% at or below
% at or above 5 8 10 15 17 19 20 24 25 30 32

20. Percentiles – Textbook’s Method
Value
% at or below
% at or above 5 9% 8
18% 10
27% 15
36% 17
45% 19
55% 20
64% 24
73% 25
82% 30
91% 32
100%

21. Percentiles – Textbook’s Method
Value
% at or below
% at or above 5 9%
100% 8
18%
91% 10
27%
82% 15
36%
73% 17
45%
64% 19
55% 20 24 25 30 32

22. Percentiles – Textbook’s Method
Value
% at or below
% at or above 5 9%
100% 8
18%
91% 10
27%
82% 15
36%
73% 17
45%
64% 19
55% 20 24 25 30 32

23. Percentiles – Textbook’s Method
Value
% at or below
% at or above 5 9%
100% 8
18%
91% 10
27%
82% 15
36%
73% 17
45%
64% 19
55% 20 24 25 30 32
Min Q1
Median Q3
Max

24. Percentiles – Excel’s Formula
To find position, or rank, of the pth percentile, compute the value

25. Excel’s Percentile Formula
This gives the position (r = rank) of the pth percentile.
Round r to the nearest whole number.
The number in that position is the pth percentile.

26. Excel’s Percentile Formula
Special case: If r is a “half-integer,” for example 10.5, then take the average of the numbers in positions r and r + 1, just as we did for the median when n was even.
Microsoft Excel will interpolate whenever r is not a whole number.
Therefore, by rounding, our answers may differ from Excel.

27. Example Use Excel’s formula to find a 5-number summary of
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32.
FiveNumberSummary.xls

28. The Principle
Excel’s formula is based on the gaps between the numbers, not the numbers themselves.

29. Example Find the quartiles for the sample
5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.

30. Excel’s Percentile Formula
The formula may be reversed to find the percentile rank of a number, given its position, or rank, in the sample.
The formula is

31. Example
In the sample
22, 28, 31, 40, 42, 56, 78, 88, 97
what percentile rank is associated with 40?

32. Example For the sample what is the percentile rank of 45?
5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240,
what is the percentile rank of 45?

33. 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.

34. Example
The IQR of
22, 28, 31, 40, 42, 56, 78, 88, 97
is IQR = Q3 – Q1 = 78 – 31 = 47.
Find the IQR for the sample
5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.

35. Two Homework Problems
For the % on-time-arrival data (p. 252), use the formula, with rounding, to find
The 10th percentile.
The 43rd percentile.
The 69th percentile.
The 95th percentile.
Use the formula to find the percentile percentages, with rounding, of the following % on-time arrivals.
76.0, 81.1, 85.8, 90.3.