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# Five Number Summary

## Quartiles or medians and the minimum and maximum values.

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Five Number Summary

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

#### Guidance

Suppose you have ordered data with m\begin{align*}m\end{align*} observations. The rank of each observation is shown by its index.

y1y2y3ym\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, Q1\begin{align*}Q1\end{align*}, the second quartile, Q2\begin{align*}Q2\end{align*}, and the third quartile Q3\begin{align*}Q3\end{align*}. The second quartile, Q2\begin{align*}Q2\end{align*}, is defined to be the median of the data. The first quartile, Q1\begin{align*}Q1\end{align*}, is defined to be the median of the lower half of the data. The third quartile, Q3\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.

Example A

Compute the five number summary for the following data.

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

Solution: There are 8 observations total.

• Lowest value (minimum) : 2
• Q1:7+172=12\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)
• Q2:19+252=22\begin{align*}Q2:\frac{19+25}{2}=22\end{align*} (Note that this is the median of the full set of data)
• Q3:26\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

Example B

Compute the five number summary for the following data:

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

Solution: There are 10 observations total.

• Lowest value (minimum) : 4
• Q1:11\begin{align*}Q1 : 11\end{align*} (Note that this is the median of the first half of the data - 4, 8, 11, 11, 12)
• Q2:12+142=13\begin{align*}Q2 : \frac{12+14}{2}=13\end{align*} (Note that this is the median of the full set of data)
• Q3:20\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

Example C

Compute the five number summary for the following data:

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

Solution: There are 9 observations total. To calculate Q1\begin{align*}Q1\end{align*} and Q3\begin{align*}Q3\end{align*}, you should include the median in both the lower half and upper half calculations.

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

Concept Problem Revisited

To compute the five number summary, 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\begin{align*}4^{th}\end{align*} and 5th\begin{align*}5^{th}\end{align*} observations: 2+132=7.5\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 0+12=0.5\begin{align*}\frac{0+1}{2}=0.5\end{align*}
• The middle of the upper half is 27+612=44\begin{align*}\frac{27+61}{2}=44\end{align*}

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

#### Vocabulary

The rank of an observation is the number of observations that are less than or equal to the value of that observation.

Data is divided into four parts by the first quartile (Q1)\begin{align*}(Q1)\end{align*}, second quartile (Q2)\begin{align*}(Q2)\end{align*} and third quartile (Q3)\begin{align*}(Q3)\end{align*}. The second quartile is also known as the median.

#### Guided Practice

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

{2, 5, 9, 18, 20}

2. 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

3. Compute the five number summary for the following data:

1, 4, 96, 356, 2557, 9881, 14420, 20100

1. 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

2. There are 20 observations.

• Lower : 1
• Q1:2+32=2.5\begin{align*}Q1 : \frac{2+3}{2}=2.5\end{align*}
• Q2:4+52=4.5\begin{align*}Q2 : \frac{4+5}{2}=4.5\end{align*}
• Q3:6+72=6.5\begin{align*}Q3 : \frac{6+7}{2}=6.5\end{align*}
• Upper : 15

3. There are 8 observations.

• Lower : 1
• Q1:4+962=50\begin{align*}Q1 : \frac{4+96}{2}=50\end{align*}
• Q2:356+25572=1456.5\begin{align*}Q2 : \frac{356+2557}{2}=1456.5\end{align*}
• Q3:9881+144202=12150.5\begin{align*}Q3: \frac{9881+14420}{2}=12150.5\end{align*}
• Upper: 20100

#### Practice

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

1. 0.16, 0.08, 0.27, 0.20, 0.22, 0.32, 0.25, 0.18, 0.28, 0.27
2. 77, 79, 80, 86, 87, 87, 94, 99
3. 79, 53, 82, 91, 87, 98, 80, 93
4. 91, 85, 76, 86, 96, 51, 68, 92, 85, 72, 66, 88, 93, 82, 84
5. 335, 233, 185, 392, 235, 518, 281, 208, 318
6. 38, 33, 41, 37, 54, 39, 38, 71, 49, 48, 42, 38
7. 3, 7, 8, 5, 12, 14, 21, 13, 18
8. 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. 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
10. 49, 57, 53, 54, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
11. 18, 20, 24, 21, 5, 23, 19, 22
12. 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. 13, 15, 19, 14, 26, 17, 12, 42, 18
14. 25, 33, 55, 32, 17, 19, 15, 18, 21
15. 149, 123, 126, 122, 129, 120

### Vocabulary Language: English

first quartile

first quartile

The first quartile, also known as $Q_1$, is the median of the lower half of the data.
five number summary

five number summary

The five number summary of a set of data is the minimum, first quartile, second quartile, third quartile, and maximum.
Lower quartile

Lower quartile

The lower quartile, also known as $Q_1$, is the median of the lower half of the data.
Maximum

Maximum

The largest number in a data set.
Median

Median

The median of a data set is the middle value of an organized data set.
Minimum

Minimum

The minimum is the smallest value in a data set.
Quartile

Quartile

A quartile is each of four equal groups that a data set can be divided into.
rank

rank

The rank of an observation is the number of observations that are less than or equal to the value of that observation.
second quartile

second quartile

The second quartile, also known as $Q_2$, is the median of the data.
third quartile

third quartile

The third quartile, also known as $Q_3$, is the median of the upper half of the data.
Upper Quartile

Upper Quartile

The upper quartile, also known as $Q_3$, is the median of the upper half of the data.