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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

- 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

- 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

- 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

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

25%

25%

25%

25%

Q1

Median

Q3

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

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

Median

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

1

2

3

4

5

6

7

Q1

Median

Q3

- 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.

- 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.

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Median

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Find “median”

Find “median”

Median

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Median

Q1

Q3

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32

Min

Q1

Median

Q3

Max

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33

- Find the quartiles of the sample
5, 8, 10, 15, 17, 19, 20, 24, 25, 30, 32, 33

Median

19.5

- 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

- 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 (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 IQR of
22, 28, 31, 40, 42, 56, 78, 88, 97

is IQR = Q3 – Q1 = 78 – 31 = 47.

- Find the IQR for the sample
- 5, 20, 30, 45, 60, 80, 100, 140, 175, 200, 240.

- Are the data skewed?

GPA Data