The five number summary
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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 .

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The Five-Number Summary

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The five number summary

The Five-Number Summary

Lecture 16

Sec. 5.3.1 – 5.3.3

Tue, Feb 12, 2008


The five number summary1

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

10

1

2

3

4

5

6

7

8

9

0

Example

  • If the distribution were uniform from 0 to 10, what would be the five-number summary?


Example1

10

1

2

3

4

5

6

7

8

9

0

Example

  • If the distribution were uniform from 0 to 10, what would be the five-number summary?

50%

50%

Median


Example2

10

1

2

3

4

5

6

7

8

9

0

Example

  • If the distribution were uniform from 0 to 10, what would be the five-number summary?

25%

25%

25%

25%

Q1

Median

Q3


Example3

Example

  • Where would the median and quartiles be in this symmetric non-uniform distribution?

1

2

3

4

5

6

7


Example4

Example

  • Where would the median and quartiles be in this symmetric non-uniform distribution?

1

2

3

4

5

6

7


Example5

Example

  • Where would the median and quartiles be in this symmetric non-uniform distribution?

1

2

3

4

5

6

7

Median


Example6

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

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

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.


Example7

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32


Example8

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Median


Example9

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Find “median”

Find “median”

Median


Example10

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Median

Q1

Q3


Example11

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Min

Q1

Median

Q3

Max


Example12

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33


Example13

Example

  • Find the quartiles of the sample

    5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33

Median

19.5


Example14

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


Example15

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

  • 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.


The five number summary

IQR

  • The IQR of

    22, 28, 31, 40, 42, 56, 78, 88, 97

    is IQR = Q3 – Q1 = 78 – 31 = 47.


The five number summary

IQR

  • Find the IQR for the sample

    • 5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.

  • Are the data skewed?


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