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