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

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.


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?


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


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


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”

Find “median”

Median


Example

  • Find the quartiles of the sample

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

Median

Q1

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

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

  • Are the data skewed?


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