Comparing groups with box plots

What you will be able to do

• compare groups by their medians, IQRs, ranges, shapes and outliers;
• describe how much two groups overlap from their quartiles;
• explain what a box plot hides, and when to draw a histogram as well.

Plants with and without fertiliser

A school garden club grows 26 bean plants in a made-up trial. Thirteen get fertiliser and 13 do not. After four weeks the club measures each plant, in cm.

• Without fertiliser: $14, 16, 17, 18, 19$, $19, 20, 21, 22, 22$, $23, 25, 31$.
• With fertiliser: $17, 20, 22, 23, 24$, $25, 26, 26, 27, 28$, $30, 31, 33$.

The five-number summaries are 14, 17.5, 20, 22.5 and 31 cm without fertiliser, and 17, 22.5, 26, 29 and 33 cm with it. Without fertiliser, the upper fence is $22.5 + 1.5 \times 5 = 30$ cm, so the plant of 31 cm is an outlier.

How to compare

Compare the groups on four things, and always in the context of the data.

1. Centre. The median with fertiliser is 26 cm, against 20 cm without. Typically, fertilised plants were about 6 cm taller.
2. Spread. The IQRs are 6.5 cm with fertiliser and 5 cm without, so the fertilised plants varied a little more.
3. Shape. Without fertiliser, the box and the whiskers are fairly balanced around the median. With fertiliser, the median sits a little left of the middle of its box.
4. Unusual values. One unfertilised plant, at 31 cm, is an outlier. It might be worth checking that it got no fertiliser by mistake.

How much do the groups overlap?

The quartiles let you make careful statements about overlap.

• The median with fertiliser, 26 cm, is above $Q_3$ without fertiliser, 22.5 cm. So at least half of the fertilised plants were taller than about three quarters of the unfertilised plants.
• The two boxes just touch: $Q_3$ without fertiliser and $Q_1$ with it are both 22.5 cm. The whiskers overlap a lot. Some plants without fertiliser were taller than some plants with it.

So the fertiliser group did better as a group, but not plant by plant. This trial is also small. You can learn how to test whether a difference this large could be luck in the last chapter of this collection.

What a box plot hides

A box plot shows only five numbers. Two very different data sets can share them. Look at the two data sets below. Each has 20 values with the same five-number summary: 0, 10, 20, 30 and 40. Set A is roughly flat. Set B has two humps and no values at all between 14 and 26. Their box plots are identical.

So use box plots to compare groups quickly, and draw a histogram as well when the shape matters. A box plot never shows two peaks or a gap inside the box.

You should find that the orange median sits above the blue $Q_3$ once the shift is more than 2.5 cm. With no shift and a stretch of 1, the two box plots lie on top of each other. A stretch of 2 doubles the orange IQR to 10 cm.

Practise with new numbers

Try it in Python

The program prints the five-number summary of each group with the rule of this chapter, and then draws side-by-side box plots with matplotlib.

Common mistakes

• Comparing only the medians. Say something about spread and shape too.
• Saying "every plant with fertiliser was taller". Box plots compare groups, not individuals; check the overlap.
• Comparing groups on different scales. Draw box plots on one shared axis.
• Trusting a box plot for shape. It cannot show two peaks or gaps; draw a histogram.

Remember these

• Compare centre (medians), spread (IQRs), shape and outliers, in the context of the data.
• If one median is above the other group's $Q_3$, at least half of one group beats about three quarters of the other.
• Box plots on one axis make comparison easy, but they hide the detailed shape.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Group P has median 48 and IQR 10. Group Q has median 55 and IQR 22. Compare the centres and the spreads.

2. The median of group Y equals $Q_1$ of group X. About what share of group X lies above the median of Y?

3. Two classes have identical box plots. Must their histograms look the same?

4. Why should side-by-side box plots share one axis?

References

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), section on box plots.