D. Data · Data, Statistics and Mathematical Modelling

Box Plots: Drawing, Reading and Comparing

D1.2 A1About 70 minutes

Learning goals

I can draw a box plot from a five-number summary, including a modified box plot with outliers.
I can read a box plot and describe what percent of the data is in each part.
I can compare two data sets using side-by-side box plots, commenting on centre, spread, shape and outliers.

Words to know

box plot
A graph of the five-number summary: a box from Q1 to Q3 with a line at the median and whiskers to the minimum and maximum.

whisker
A line from the box to the minimum or maximum (or to the last value that is not an outlier).

modified box plot
A box plot that shows outliers as separate dots.

consistent
Having little spread: the values are close together.

dominant hand
The hand you usually write or throw with.

Five numbers, one picture

In the last lesson you found five-number summaries. A list of five numbers is useful, but a picture lets you see the spread at a glance and compare two groups side by side.

A box plot (also called a box-and-whisker plot) draws the five-number summary on a number line. It is one of the most common graphs in science and statistics, because it shows the centre, the spread and any outliers all at once.

How to draw a box plot

Find the five-number summary: minimum, Q1, median, Q3, maximum.
Draw a number line with an even scale that covers the minimum to the maximum. Label it with the variable and units.
Draw a box from Q1 to Q3.
Draw a vertical line inside the box at the median.
Draw a whisker (line) from the box out to the minimum and another out to the maximum.
Give the graph a title.

The parts of a box plot, using the maple sap data from the last lesson (36, 40, 44.5, 49, 55). Each of the four sections holds about 25% of the data, even though the sections have different lengths.

Long section does not mean more data

Each whisker and each half of the box contains about the same number of values (about 25%). A long section means those values are spread out; a short section means they are bunched together.

Worked example

Example 1: Draw a box plot

Eleven Grade 9 students recorded how many minutes they spent outdoors on a snowy Saturday: 18, 22, 25, 27, 30, 31, 35, 38, 40, 44, 52. Find the five-number summary and describe the box plot you would draw.

n = 11, already in order. Median = 6th value = 31.
Lower half: 18, 22, 25, 27, 30, so Q1 = 25. Upper half: 35, 38, 40, 44, 52, so Q3 = 40.
Minimum = 18, maximum = 52. IQR = 40 − 25 = 15.
Outlier check: 1.5 × 15 = 22.5. Fences: 25 − 22.5 = 2.5 and 40 + 22.5 = 62.5. No outliers.
Number line from 15 to 55 in steps of 5. Box from 25 to 40, median line at 31, whiskers to 18 and 52.

Five-number summary 18, 25, 31, 40, 52. The right whisker (40 to 52) is longer than the left whisker (18 to 25), so the top quarter of times are more spread out.

The box plot for Example 1. The median line is a little left of the centre of the box, and the right whisker is the longest part.

Box plots with outliers (modified box plots)

If there is an outlier (beyond 1.5 × IQR from the box), most tools draw it as a separate dot. The whisker then stops at the most extreme value that is not an outlier. This is called a modified box plot. It stops one unusual value from stretching the whisker and hiding the real spread.

Worked example

Example 2: Compare two groups

In the ruler-drop test, a partner holds a ruler and lets it go without warning; you catch it and read the distance it fell (a smaller distance means a faster reaction). Lee did 15 trials with each hand.

Dominant hand (cm): 9, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 21, 24

Other hand (cm): 11, 13, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 25, 27, 34

Make box plots and compare.

Dominant: n = 15, median = 8th value = 15. Lower half (7 values) 9, 11, 12, 12, 13, 14, 14: Q1 = 12. Upper half 16, 16, 17, 18, 19, 21, 24: Q3 = 18. Summary: 9, 12, 15, 18, 24. IQR = 6.
Fences for dominant: 12 − 9 = 3 and 18 + 9 = 27. No outliers.
Other hand: median = 19. Lower half 11, 13, 15, 16, 17, 18, 18: Q1 = 16. Upper half 20, 21, 22, 23, 25, 27, 34: Q3 = 23. Summary: 11, 16, 19, 23, 34. IQR = 7.
Fences for other hand: 16 − 10.5 = 5.5 and 23 + 10.5 = 33.5. 34 is an outlier, so the whisker stops at 27 and 34 is drawn as a dot.
Compare centres: median 15 cm vs 19 cm. Compare spreads: IQR 6 cm vs 7 cm, similar.

Lee's dominant hand is faster: its median catch distance is 4 cm less. The spreads are similar (IQR 6 vs 7), and the other hand had one unusually slow catch (34 cm).

Side-by-side box plots on the same scale make comparison easy. The dominant-hand box and median sit further left (faster). The open circle at 34 is an outlier, so that whisker stops at 27.

Reading a box plot

From a box plot you can say things like:

"About half the trials were caught within 15 cm" (the median).
"About 75% of the trials were caught within 18 cm" (Q3).
"The middle 50% of trials were between 12 and 18 cm" (the box).
"About 25% of the trials were 18 cm or more" (the right whisker).

When you compare box plots, always comment on centre (medians), spread (IQRs or ranges), shape (which parts are long or short) and outliers, and finish with a conclusion in context.

Just for fun

A box plot walks into a party. Someone asks how it's feeling. It says, "Honestly? About 25% great, 50% okay, and one outlier over in the corner."

Hands-on activity

Ruler-drop reaction test

You need: a 30 cm ruler, a partner (a family member), paper and a pencil.

Your partner holds the ruler vertically at the 30 cm end, with the 0 cm end just between your open thumb and index finger (not touching).
Without warning, your partner lets go. Catch it as fast as you can. Record the number (cm) at the top of your thumb.
Do 15 trials with your dominant hand and 15 with your other hand. Rest your hand between trials.
Find the five-number summary and IQR for each hand, check for outliers, and draw both box plots on the same number line.
Write three sentences comparing the hands: one about centre, one about spread, one conclusion.

Extension: the time (in seconds) the ruler fell is about t = √(2d ÷ 9.8), where d is the distance in metres. A catch at 15 cm (0.15 m) is about √(0.3 ÷ 9.8) ≈ 0.17 s.

What to look for: a correct summary for each hand, both box plots on one scale, and a comparison that names the median and IQR.

Your turn

Find the five-number summary for: 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 8, 9, 12, 15, 22 (the number of books 16 students read over the summer).

For the data in question 1, are there any outliers? Where would the right whisker end on a modified box plot?

A box plot has minimum 40, Q1 52, median 60, Q3 64, maximum 90. About what percent of the values are between 52 and 64?

For the box plot in question 3, about what percent of values are greater than 60?

For the same box plot, which part of the data is most spread out? Explain.

Two classes' test marks: Class A median 74, IQR 10; Class B median 74, IQR 25. Write a comparison.

More practice: warm-up

What does the line inside the box show?

What do the ends of the box show?

A box plot's whiskers end at 8 and 47. What is the range?

A box goes from 20 to 35. What is the IQR?

How is an outlier usually shown on a box plot?

More practice: on your own

Make a five-number summary and describe the box plot for: 31, 35, 36, 38, 40, 41, 43, 45, 47, 50, 58, 63, 70.

Two hockey teams' goals per game: Team North summary 0, 2, 3, 4, 7; Team South summary 1, 2, 4, 5, 6. Which team scores more in a typical game? Which is more consistent? Explain.

A box plot of daily high temperatures in March has Q1 = −2 °C and Q3 = 6 °C. Explain what this tells you.

Explain why you should draw two box plots on the same number line when comparing them.

A data set's box plot has a very short left whisker and a very long right whisker. Is the mean likely to be greater or less than the median? Why?

Challenge

Sketch possible data: create a set of 8 values whose box plot has a box from 10 to 20, median 12, and whiskers to 5 and 40.

Can two data sets have identical box plots but different means? Give an example with 7 values each.

The two-hand ruler data had medians 15 cm and 19 cm. Using t = √(2d ÷ 9.8) with d in metres, find the difference in median reaction time in seconds.

Check your understanding

Quiz

On a box plot, what does the box show?

A box plot has Q1 = 14 and Q3 = 26. About what percent of the data is less than 14?

The right whisker of a box plot is twice as long as the left whisker. What does this mean?

Q1 = 30, Q3 = 40. On a modified box plot, which value would be drawn as a separate dot?

Box plot A has median 60 and IQR 8. Box plot B has median 70 and IQR 30. Which statement is true?

Why are box plots especially useful for comparing groups?

For parents and teachers

Big idea: D1.2 asks for box plots by name. Students should be able to build one, read percents from it, and compare two groups in context.

How to help: Do the ruler-drop test together; it is quick and gives real data. When your teen compares the hands, ask "How do you know?" after each claim so they point to the median or IQR.

Common difficulties:

Thinking a longer section contains more data (it contains the same 25%, just more spread out).
Drawing two box plots on different scales.
Forgetting to finish a comparison in context ("so my dominant hand reacts faster").

Safety: in the ruler-drop test, keep fingers open and relaxed; nothing is thrown.