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

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 9 2 3 4 10

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

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

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 Q1 Median Q3

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 Find “median” Median Find “median”

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

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

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

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

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

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

Five-Number Summaries and Stem-and-Leaf Displays 1 3 89 2 1334 55789 034 8 GPA Data