Download - The Five-Number Summary
Lecture 16
Sec. 5.3.1 – 5.3.3
Tue, Feb 12, 2008
The Five-Number Summary
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.
Example
If the distribution were uniform from 0 to 10, what would be the five-number summary?
1 5 6 7 8 92 3 40 10
Example
If the distribution were uniform from 0 to 10, what would be the five-number summary?
1 5 6 7 8 92 3 40 10
50% 50%
Median
Example
If the distribution were uniform from 0 to 10, what would be the five-number summary?
1 5 6 7 8 92 3 40 10
25% 25% 25% 25%
Median Q3Q1
Example
Where would the median and quartiles be in this symmetric non-uniform distribution?
1 2 3 4 5 6 7
Example
Where would the median and quartiles be in this symmetric non-uniform distribution?
1 2 3 4 5 6 7
Example
Where would the median and quartiles be in this symmetric non-uniform distribution?
1 2 3 4 5 6 7
Median
Example
Where would the median and quartiles be in this symmetric non-uniform distribution?
1 2 3 4 5 6 7
Median Q3Q1
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.
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.
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
Median
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
MedianFind “median” Find “median”
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
Median Q3Q1
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32
Median Q3Q1Min Max
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33
Median19.5
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33
Median19.5
Q327.5
Q112.5
Example
Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33
Median19.5
Q327.5
Q112.5
Min Max
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.
IQR
The IQR of
22, 28, 31, 40, 42, 56, 78, 88, 97
is IQR = Q3 – Q1 = 78 – 31 = 47.
IQR
Find the IQR for the sample5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.
Are the data skewed?
Salaries of School Board Chairmen
County/City Salary County/City Salary
Henrico $20,000 Powhatan $4,800
Chesterfield 18,711 Colonial Hgts 5,100
Richmond 11,000 Goochland 5,500
Hanover 11,000 Hopewell 4,500
Petersburg 8,500 Charles City 4,500
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 0
Five-Number Summaries and Stem-and-Leaf Displays
Stem Leaf
1 3
1 89
2 1334
2 55789
3 034
3 8
GPA Data