When given a long list of numbers, it is useful to summarize the data. One way to summarize the data is to give the lowest number, the highest number and the middle number. In addition to these three numbers it is also useful to give the median of the lower half of the data and the median of the upper half of the data. These five numbers give a very concise summary of the data.

What is the five number summary of the following data?

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

### The Five Number Summary

Suppose you have ordered data with \begin{align*}m\end{align*} observations. The rank of each observation is shown by its index.The **rank** of an observation is the number of observations that are less than or equal to the value of that observation.

\begin{align*}y_1 \le y_2 \le y_3 \le \cdots \le y_m\end{align*}

In data sets that are large enough, you can divide the numbers into four parts called **quartiles**. The quartiles of interest are the first quartile, \begin{align*}Q1\end{align*}, the second quartile, \begin{align*}Q2\end{align*}, and the third quartile \begin{align*}Q3\end{align*}. The **second quartile**, \begin{align*}Q2\end{align*}, is defined to be the median of the data. The **first quartile**, \begin{align*}Q1\end{align*}, is defined to be the median of the lower half of the data. The **third quartile**, \begin{align*}Q3\end{align*}, is similarly defined to be the median of the upper half of the data.

These three numbers in addition to the minimum and maximum values are the five number summary. Note that there are variations of the five number summary that you can study in a statistics course.

Take the following data:

2, 7, 17, 19, 25, 26, 26, 32

There are 8 observations total in this set of data.

- Lowest value (minimum) : 2
- \begin{align*}Q1:\frac{7+17}{2}=12\end{align*} (Note that this is the median of the first half of the data - 2, 7, 17, 19)
- \begin{align*}Q2:\frac{19+25}{2}=22\end{align*} (Note that this is the median of the full set of data)
- \begin{align*}Q3: 26\end{align*} (Note that this is the median of the second half of the data - 25, 26, 26, 32)
- Upper value (maximum) : 32

The 5 number summary is 2, 12, 22, 26, 32.

### 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 \begin{align*}4^{th}\end{align*} and \begin{align*}5^{th}\end{align*} observations: \begin{align*}\frac{2+13}{2}=7.5\end{align*}
- The lowest observation is 0.
- The highest observation is 63.
- The middle of the lower half is \begin{align*}\frac{0+1}{2}=0.5\end{align*}
- The middle of the upper half is \begin{align*}\frac{27+61}{2}=44\end{align*}

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
- \begin{align*}Q1 : \frac{2+3}{2}=2.5\end{align*}
- \begin{align*}Q2 : \frac{4+5}{2}=4.5\end{align*}
- \begin{align*}Q3 : \frac{6+7}{2}=6.5\end{align*}
- 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
- \begin{align*}Q1 : 11\end{align*}
*(Note that this is the median of the first half of the data - 4, 8, 11, 11, 12)* - \begin{align*}Q2 : \frac{12+14}{2}=13\end{align*}
*(Note that this is the median of the full set of data)* - \begin{align*}Q3 : 20\end{align*}
*(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 \begin{align*}Q1\end{align*} and \begin{align*}Q3\end{align*}, you should include the median in both the lower half and upper half calculations.

- Lowest value (minimum) : 3
- \begin{align*}Q1 : 10\end{align*}
*(this is the median of 3, 7, 10, 14, 19)* - \begin{align*}Q2 : 19\end{align*}
- \begin{align*}Q3 : 23\end{align*}
*(this is the median of 19, 19, 23, 27, 29)* - Upper value (maximum) : 29

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

### Review

Compute the five number summary for each of the following sets of data:

- 0.16, 0.08, 0.27, 0.20, 0.22, 0.32, 0.25, 0.18, 0.28, 0.27
- 77, 79, 80, 86, 87, 87, 94, 99
- 79, 53, 82, 91, 87, 98, 80, 93
- 91, 85, 76, 86, 96, 51, 68, 92, 85, 72, 66, 88, 93, 82, 84
- 335, 233, 185, 392, 235, 518, 281, 208, 318
- 38, 33, 41, 37, 54, 39, 38, 71, 49, 48, 42, 38
- 3, 7, 8, 5, 12, 14, 21, 13, 18
- 6, 22, 11, 25, 16, 26, 28, 37, 37, 38, 33, 40, 34, 39, 23, 11, 48, 49, 8, 26, 18, 17, 27, 14
- 9, 10, 12, 13, 10, 14, 8, 10, 12, 6, 8, 11, 12, 12, 9, 11, 10, 15, 10, 8, 8, 12, 10, 14, 10, 9, 7, 5, 11, 15, 8, 9, 17, 12, 12, 13, 7, 14, 6, 17, 11, 15, 10, 13, 9, 7, 12, 13, 10, 12
- 49, 57, 53, 54, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
- 18, 20, 24, 21, 5, 23, 19, 22
- 900, 840, 880, 880, 800, 860, 720, 720, 620, 860, 970, 950, 890, 810, 810, 820, 800, 770, 850, 740, 900, 1070, 930, 850, 950, 980, 980, 880, 960, 940, 960, 940, 880, 800, 850, 880, 760, 740, 750, 760, 890, 840, 780, 810, 760, 810, 790, 810, 820, 850
- 13, 15, 19, 14, 26, 17, 12, 42, 18
- 25, 33, 55, 32, 17, 19, 15, 18, 21
- 149, 123, 126, 122, 129, 120

### Review (Answers)

To see the Review answers, open this PDF file and look for section 15.3.