Chapter 13 · Statistics and data displays

The Five-Number Summary

Learn the five numbers that sum up a data set: minimum, first quartile, median, third quartile and maximum. See how they split ordered data into four parts.

11 min read

A data set is a collection of numbers that someone has gathered. Imagine one that holds $200$ test scores. You cannot keep $200$ scores in your head. You already know single numbers that describe a data set, such as the median (its middle) and the range (the greatest value minus the least value). But one number alone cannot show where the data starts, where it ends, and how it is spread out in between.

A five-number summary does that job with just five numbers. Picture the data values lined up in order from least to greatest. Two of the five numbers mark the ends of the line. The other three are cut points that split the line into four parts, with about the same number of values in each part. You will soon see why it is about and not always exactly.

To build the summary, first put the data values in order from least to greatest. If a value appears more than once, list it every time. The five numbers are:

The minimum: the least value.
The first quartile, written $Q_1$ : the median of the lower half of the ordered list.
The median: the middle of the ordered list. With an odd count of values, it is the one middle value. With an even count, it is halfway between the two middle values: add them and divide by $2$ .
The third quartile, written $Q_3$ : the median of the upper half of the ordered list.
The maximum: the greatest value.

The lower half is the part of the ordered list before the middle. The upper half is the part after the middle. With an even count, the list splits cleanly into two halves. With an odd count, the median is the middle value itself, and in this course you always leave it out of both halves. Other rules exist and can give slightly different quartiles, so use this one every time.

Key concept

The five-number summary

A five-number summary describes a data set with five numbers: the minimum, the first quartile $Q_1$ , the median, the third quartile $Q_3$ and the maximum.

The minimum and the maximum mark the ends of the ordered data. $Q_1$ , the median and $Q_3$ cut it into four parts that each hold about one fourth of the values. It is not always exactly one fourth, for example when the list is short or values repeat.

$Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half. When the count of values is odd, you leave the median out of both halves.

The five numbers are always in order: each one is greater than or equal to the one before it. The symbol $\le$ means is less than or equal to.

\text{minimum} \le Q_1 \le \text{median} \le Q_3 \le \text{maximum}

Worked example

An even number of values

Eight students wrote down how many minutes they read last night: $20, 45, 12, 26, 14, 30, 18, 22$ . Find the five-number summary of these reading times.

Put the 8 reading times in order from least to greatest. Every number in the summary depends on where a value sits in the ordered list, so ordering comes first. Count to check that all 8 values are still there.
$12, 14, 18, 20, 22, 26, 30, 45$
Read the two ends of the ordered list. The first value is the least, so it is the minimum. The last value is the greatest, so it is the maximum.
$\text{minimum} = 12 \qquad \text{maximum} = 45$
Find the median. The count is 8, which is even, so there are two middle values: the 4th and the 5th, which are 20 and 22. The median is halfway between them, so add them and divide by 2.
$\text{median} = (20 + 22) \div 2 = 42 \div 2 = 21$
Split the ordered list into halves. The count is even, so the list splits cleanly with no value left over. The first 4 values are the lower half and the last 4 values are the upper half.
$\text{lower half: } 12, 14, 18, 20 \qquad \text{upper half: } 22, 26, 30, 45$
Find the first quartile, which is the median of the lower half. The lower half has 4 values, an even count, so take the number halfway between its two middle values, 14 and 18.
$Q_1 = (14 + 18) \div 2 = 32 \div 2 = 16$
Find the third quartile, which is the median of the upper half. Use the same method on its two middle values, 26 and 30.
$Q_3 = (26 + 30) \div 2 = 56 \div 2 = 28$

Answer

The five-number summary of the reading times is minimum $12$ , $Q_1 = 16$ , median $21$ , $Q_3 = 28$ and maximum $45$ , all in minutes.

Now look at what those five numbers show. Below, the ordered reading times appear again with a bar at each cut point. Notice that $16$ , $21$ and $28$ are not reading times anyone wrote down. A cut point can fall between two data values.

The three bars split the list into four sections. The table under the list has one row for each section. Its columns give the section's two edges, the number of reading times that fall in the section, and the section's width. The width is the upper edge minus the lower edge, so its unit is minutes.

12, 14 \; \underset{Q_1 = 16}{\Big|} \; 18, 20 \; \underset{\text{median} = 21}{\Big|} \; 22, 26 \; \underset{Q_3 = 28}{\Big|} \; 30, 45

The four sections of the reading times

Section	Edges, in minutes	Number of reading times	Width, in minutes
Minimum to $Q_1$	$12$ to $16$	$2$	$16 - 12 = 4$
$Q_1$ to median	$16$ to $21$	$2$	$21 - 16 = 5$
Median to $Q_3$	$21$ to $28$	$2$	$28 - 21 = 7$
$Q_3$ to maximum	$28$ to $45$	$2$	$45 - 28 = 17$

Read down the last two columns. Every section holds $2$ of the $8$ reading times. In this data set that is exactly one fourth each, which does not always happen, as the second example will show. The widths, though, are $4$ , $5$ , $7$ and $17$ minutes. Equal counts do not mean equal widths.

A narrow section means its values sit in a short stretch of the number line. A wide section means that part of the data covers a longer stretch. Here the bottom section is only $4$ minutes wide and holds $12$ and $14$ . The top section is $17$ minutes wide and holds $30$ and $45$ . Same count, very different spread.

The widths also link to a measure you know. Placed end to end, the four sections reach from the minimum to the maximum. So the widths add up to the range. Check: $4 + 5 + 7 + 17 = 33$ , and $45 - 12 = 33$ minutes.

A distance is not a count. In the reading times, $Q_1 = 16$ and $Q_3 = 28$ . Four reading times ( $18$ , $20$ , $22$ and $26$ ) lie between the quartiles, so it is tempting to say the interquartile range (IQR) is $4$ .

That answer counts values, but the IQR is a distance: $Q_3$ minus $Q_1$ . So you subtract: $28 - 16 = 12$ . The IQR is $12$ minutes, not $4$ values. That is the sum of the two middle widths, $5 + 7$ , since the two middle sections reach from $Q_1$ to $Q_3$ .

Counting fails because a count cannot measure spread. With this course's rule, every list of $8$ different values has four values between its quartiles. The count stays $4$ whether the quartiles are close together or far apart. The distance $Q_3 - Q_1$ changes, and that is what shows spread.

Worked example

An odd number of values

Nine students took a quiz. Their scores, in points, were $14, 9, 18, 11, 20, 8, 15, 12, 16$ . Find the five-number summary of the scores.

Put the 9 scores in order from least to greatest, because the summary depends on each value's position in the ordered list. The first value, 8, is the minimum. The last value, 20, is the maximum.
$8, 9, 11, 12, 14, 15, 16, 18, 20$
Find the median. The count is 9, which is odd, so one value sits exactly in the middle. It is the 5th value, with 4 values before it and 4 values after it.
$\text{median} = 14$
Split the list into halves. The count is odd, so use this course's rule: set the median, 14, aside so that it is in neither half. The lower half is the 4 values before it. The upper half is the 4 values after it.
$\text{lower half: } 8, 9, 11, 12 \qquad \text{upper half: } 15, 16, 18, 20$
Find the first quartile, the median of the lower half. That half has 4 values, an even count, so take the number halfway between its two middle values, 9 and 11.
$Q_1 = (9 + 11) \div 2 = 20 \div 2 = 10$
Find the third quartile, the median of the upper half, in the same way. Its two middle values are 16 and 18.
$Q_3 = (16 + 18) \div 2 = 34 \div 2 = 17$

Answer

The five-number summary of the quiz scores is minimum $8$ , $Q_1 = 10$ , median $14$ , $Q_3 = 17$ and maximum $20$ , all in points.

Check the order first: $8 \le 10 \le 14 \le 17 \le 20$ , just as the key idea says.

Now count the scores in each section. The section from $8$ to $10$ holds $8$ and $9$ . The section from $10$ to $14$ holds $11$ and $12$ . The section from $14$ to $17$ holds $15$ and $16$ . The section from $17$ to $20$ holds $18$ and $20$ . The ninth score is the median, $14$ . It sits right on the edge between two sections.

So each section has $2$ of the $9$ scores to itself. That is about one fourth, but not exactly, because $9$ values cannot be shared into four equal groups. The reading times split into exact fourths because $8$ values share evenly into four groups of $2$ and no value repeated. A repeated value can also land right on a cut point and blur the count. So you say each section holds about one fourth of the values, and the idea works best when a data set is not tiny.

Takeaway. A five-number summary is the minimum, $Q_1$ , the median, $Q_3$ and the maximum of a data set. To find it, order the values, find the median, then find the median of each half. When the count of values is odd, leave the median out of both halves.

The five numbers mark the ends of the ordered data and cut it into four sections. Each section holds about one fourth of the values. It is not always exactly one fourth, for example in a short list or when values repeat. The distances between the numbers, such as the range and the IQR ( $Q_3 - Q_1$ ), show how spread out the data is.

Next, you will use a five-number summary to draw and read a box plot.

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Create and Interpret Box Plots

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