5 number summary

You only need the five values listed above (min, Q1, Q2, Q3, and max) in order to draw your box-and-whisker plot. This set of five values has been given the name "the five-number summary".

Examples

Example 1

Earlier you were asked to compute the five number summary for 0, 0, 1, 2, 63, 61, 27, 13. It helps to order the data.

0, 0, 1, 2, 13, 27, 61, 63

• Since there are 8 observations, the median is the average of the 4th and 5th observations: 2+132=7.5
• The lowest observation is 0.
• The highest observation is 63.
• The middle of the lower half is 0+12=0.5
• The middle of the upper half is 27+612=44

The five number summary is 0, 0.5, 7.5, 44, 63

Example 2

Create a set of data that meets the following five number summary:

{2, 5, 9, 18, 20}

Suppose there are 8 data points. The lowest point must be 2 and the highest point must be 20. The middle two points must average to be 9 so they could be 8 and 10. The second and third points must average to be 5 so they could be 4 and 6. The sixth and seventh points need to average to be 18 so they could be 18 and 18. Here is one possible answer:

2, 4, 6, 8, 10, 18, 18, 20

Example 3

Compute the five number summary for the following data:

 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10, 15

There are 20 observations.

• Lower : 1
• Q1:2+32=2.5
• Q2:4+52=4.5
• Q3:6+72=6.5
• Upper : 15

Example 4

Compute the five number summary for the following data:

4, 8, 11, 11, 12, 14, 16, 20, 21, 25

There are 10 observations total in this set of data. 

• Lowest value (minimum) : 4
• Q1:11 (Note that this is the median of the first half of the data - 4, 8, 11, 11, 12)
• Q2:12+142=13 (Note that this is the median of the full set of data)
• Q3:20 (Note that this is the median of the second half of the data - 14, 16, 20, 21, 25)
• Upper value (maximum) : 25

The five number summary is 4, 11, 13, 20, 25.

Example 5

Compute the five number summary for the following data:

3, 7, 10, 14, 19, 19, 23, 27, 29

There are 9 observations total. To calculate Q1 and Q3, you should include the median in both the lower half and upper half calculations.

• Lowest value (minimum) : 3
• Q1:10 (this is the median of 3, 7, 10, 14, 19)
• Q2:19
• Q3:23 (this is the median of 19, 19, 23, 27, 29)
• Upper value (maximum) : 29

The five number summary is 3, 10, 19, 23, 29.

Assignment: 

Sources: https://www.purplemath.com/modules/boxwhisk2.htm

https://www.ck12.org/c/statistics/five-number-summary/lesson/Five-Number-Summary-PCALC/