Create and Interpret Box Plots | Grade 6 Math | LevelKite

A box plot is a picture of a data set's five-number summary, drawn along a number line. Recall the five values: the minimum, the lower quartile, the median, the upper quartile and the maximum. In this lesson, $Q_1$ stands for the lower quartile and $Q_3$ stands for the upper quartile.

A box plot has three parts:

a box that runs from $Q_1$ to $Q_3$ ,
a line inside the box at the median, and
two whiskers, which are lines that reach out from the box. The lower whisker joins the minimum to $Q_1$ . The upper whisker joins $Q_3$ to the maximum.

With one look, you can see where the data starts, where its middle values sit and where it ends.

Worked example

Make a box plot of 11 quiz scores

Eleven students took a quiz worth 20 points. Their scores, in points, are 15, 9, 18, 20, 12, 6, 17, 14, 19, 11 and 16. Make a box plot of the scores.

Put the scores in order from least to greatest, because the median and quartiles depend on each score's place in the ordered list. The first score is the minimum, 6. The last score is the maximum, 20.
$6,\ 9,\ 11,\ 12,\ 14,\ 15,\ 16,\ 17,\ 18,\ 19,\ 20$
There are 11 scores, an odd number, so the median is the one middle score. It is the 6th score, with 5 scores on each side of it.
$\text{median} = 15$
Because 11 is odd, leave the median, 15, out of both halves. The lower quartile is the median of the lower half, 6, 9, 11, 12, 14. The upper quartile is the median of the upper half, 16, 17, 18, 19, 20.
$Q_1 = 11, \quad Q_3 = 18$
Choose a scale, which is a number line with evenly spaced, labeled marks. It must reach the maximum, 20 points, so that every score fits. A number line marked every 1 point up to 20 does this.
Place the five values along the number line. The box runs from the lower quartile, 11, to the upper quartile, 18, with a line inside it at the median, 15. The lower whisker joins 6 to 11, and the upper whisker joins 18 to 20. The finished plot is just below.
$\text{minimum} = 6,\ Q_1 = 11,\ \text{median} = 15,\ Q_3 = 18,\ \text{maximum} = 20$

Answer

The box plot has a box from 11 to 18 points with a median line at 15 points, and whiskers that reach 6 and 20 points.

Box plotMinimum 6 · Lower quartile 11 · Median 15 · Upper quartile 18 · Maximum 20 points

Quiz scores: the box runs from 11 to 18 points, the median line is at 15, and the whiskers reach 6 and 20.

Check the plot against the summary, from the smallest value to the largest. The lower whisker starts at the minimum, 6 points. The box starts at $Q_1 = 11$ . The line inside the box is the median, 15. The box ends at $Q_3 = 18$ . The upper whisker ends at the maximum, 20.

To read any box plot, work the other way. The whisker ends give the minimum and the maximum. The box ends give $Q_1$ and $Q_3$ . The line inside the box gives the median. The plot marks only these five values. Nothing on it shows the other six scores, such as 9 and 17.

Worked example

Make a box plot of 12 homework times

Twelve students recorded how many minutes they spent on homework one evening. Their times, in minutes, are 37, 24, 60, 30, 20, 41, 33, 26, 48, 39, 23 and 38. Make a box plot of the times.

Order the times from least to greatest first, just as with the quiz scores. The minimum is 20 minutes and the maximum is 60 minutes.
$20,\ 23,\ 24,\ 26,\ 30,\ 33,\ 37,\ 38,\ 39,\ 41,\ 48,\ 60$
There are 12 times, an even number, so two times share the middle: the 6th, 33, and the 7th, 37. The median is their mean, the number halfway between them.
$\text{median} = \frac{33 + 37}{2} = \frac{70}{2} = 35$
Because 12 is even, no time is left out. The lower half is the first 6 times, from 20 to 33. Its two middle times are 24 and 26, so the lower quartile is their mean.
$Q_1 = \frac{24 + 26}{2} = \frac{50}{2} = 25$
The upper half is the last 6 times, from 37 to 60. Its two middle times are 39 and 41, so the upper quartile is their mean.
$Q_3 = \frac{39 + 41}{2} = \frac{80}{2} = 40$
Choose a scale that reaches the maximum, 60 minutes, so that every time fits. Marks every 1 minute would need 60 steps, which would be crowded. Marks every 5 minutes reach 60 in 12 steps.
$60 \div 5 = 12$
Place the five values. The box runs from 25 to 40 minutes, with the median line at 35. The lower whisker joins 20 to 25, and the upper whisker joins 40 to 60. The finished plot is just below.
$\text{minimum} = 20,\ Q_1 = 25,\ \text{median} = 35,\ Q_3 = 40,\ \text{maximum} = 60$

Answer

The box plot has a box from 25 to 40 minutes with a median line at 35 minutes, and whiskers that reach 20 and 60 minutes.

Box plotMinimum 20 · Lower quartile 25 · Median 35 · Upper quartile 40 · Maximum 60 minutes

Homework times: the box runs from 25 to 40 minutes, the median line is at 35, and the whiskers reach 20 and 60.

Key concept

Parts show shares, and lengths show spread

The median splits the ordered data into two halves, and each quartile splits one half in two. So the box's two ends and its median line cut the data into four parts: the lower whisker, the two parts of the box and the upper whisker. Each part holds about a quarter of the data, and the box holds about half. This is not always exact: a list may not split evenly into four, and values can repeat or sit right on a line.

Lengths show spread. The range is the distance between the whisker ends. The interquartile range (IQR) is the length of the box. Both are distances in the data's units, not counts of values. A longer part means its values are more spread out, not that it holds more values.

\text{range} = \text{maximum} - \text{minimum}, \qquad \text{IQR} = Q_3 - Q_1

Worked example

Read the homework box plot

Use only the homework box plot above. Read its five values. Then find the range and the IQR, and say what they mean for the students' homework times.

Read the whisker ends. The lower whisker ends at the minimum, and the upper whisker ends at the maximum.
$\text{minimum} = 20, \quad \text{maximum} = 60$
Read the box. Its two ends are the lower and upper quartiles, and the line inside it is the median.
$Q_1 = 25, \quad \text{median} = 35, \quad Q_3 = 40$
Find the range, the distance between the whisker ends, by subtracting the minimum from the maximum. In context, the longest homework time was 40 minutes more than the shortest.
$\text{range} = 60 - 20 = 40$
Find the IQR, the length of the box, by subtracting the lower quartile from the upper quartile. In context, about the middle half of the times fall within a 15-minute stretch, from 25 to 40 minutes. The 15 is a number of minutes, not a number of students.
$\text{IQR} = 40 - 25 = 15$
Describe the shares. The box holds about half of the data, so about half of the students worked between 25 and 40 minutes. The upper whisker holds about a quarter, so about a quarter of the students worked between 40 and 60 minutes.

Answer

The five values are 20, 25, 35, 40 and 60 minutes. The range is 40 minutes and the IQR is 15 minutes, so about half of the students spent between 25 and 40 minutes on homework.

The plot alone shows neither each time nor how many students there are. To check the shares, go back to the listed data. The table sorts the 12 ordered homework times into the four parts of the plot.

The ordered homework times, sorted into the four parts of the box plot

Part of the plot	Runs from (minutes)	Length (minutes)	Times in this part
Lower whisker	20 to 25	$25 - 20 = 5$	20, 23, 24
Lower part of the box	25 to 35	$35 - 25 = 10$	26, 30, 33
Upper part of the box	35 to 40	$40 - 35 = 5$	37, 38, 39
Upper whisker	40 to 60	$60 - 40 = 20$	41, 48, 60

The homework shares came out exact: a quarter in each part and half in the box. That happened because 12 splits evenly into four groups of 3, and no time sits on one of the box's lines. It will not always happen. In the quiz list, the scores 11, 15 and 18 sit right on the box's lines. The other 8 scores split 2 to each part: 6 and 9, 12 and 14, 16 and 17, and 19 and 20. Eleven scores cannot form four equal groups, so each part holds close to a quarter, not exactly a quarter.

Takeaway: To create a box plot, order the data and find its five-number summary, leaving the median out of both halves when the number of values is odd. Then choose a scale that reaches the maximum and place the five values: a box from $Q_1$ to $Q_3$ , a line at the median, and whiskers to the minimum and the maximum. To interpret a box plot, read those five values back and give the range and the IQR in the data's units, such as minutes. Each of the four parts holds about a quarter of the data, and a longer part means more spread, not more values.

Next, you will use box plots to compare data sets and to spot mistakes in box plots.