Drawing and reading stem-and-leaf diagrams, box plots and histograms

A list like 72, 64, 91, 58, 80, 72... tells you almost nothing at a glance. A good diagram lets you see three things straight away:

• where the middle is (what a typical value looks like),
• how spread out the values are,
• the shape: are the values bunched at one end, evenly spread, or is there a lonely value far away from the rest?

Each diagram in this lesson is good at different things, so you need to know how to draw and read all of them. We will use one small data set for the first two diagrams so you can compare them directly.

The data: the resting pulse rates (beats per minute) of 15 students: 72, 64, 91, 58, 80, 72, 67, 85, 62, 75, 96, 71, 64, 83, 78.

Split every number into two parts:

• the stem: the leading digit or digits (here, the tens digit),
• the leaf: the last digit (here, the units digit).

So 72 becomes stem 7, leaf 2. Write the stems once, in a column, smallest at the top. Then write each leaf on the row of its stem. Finally, put the leaves on each row in order and add a key so a reader knows what the numbers mean.

The point of the diagram is that it keeps every original value (you can read them all back) while the rows of leaves act like bars on a chart, showing the shape.

To compare two groups, share one column of stems between them. One group's leaves go to the right, as usual. The other group's leaves go to the left, and they read outwards from the stem, so the smallest leaf is nearest the stem.

For example, here are the pulse rates of 10 students after exercise (left) next to the 15 resting rates (right):

Key: 4 | 8 | 0 means 84 for "after exercise" and 80 for "resting".

On the left, row 9 holds 91, 93, 95, 98: the leaf nearest the stem is the smallest. The picture makes the comparison obvious: the "after exercise" leaves sit lower down the stems, so those pulse rates are higher. A full comparison would also say how spread out each group is.

A box-and-whisker plot (or box plot) shows just five numbers, but they are the five that describe the middle and the spread best:

1. the smallest value (minimum),
2. the lower quartile $Q_1$: a quarter of the data is below it,
3. the median: half the data is below it,
4. the upper quartile $Q_3$: three quarters of the data is below it,
5. the largest value (maximum).

How to find the quartiles: put the data in order and find the median (the middle value). The lower quartile is the median of the lower half of the data, and the upper quartile is the median of the upper half. If the number of values is odd, leave the median itself out of both halves.

How to draw it: draw a scale. Draw a box from $Q_1$ to $Q_3$ with a line across it at the median. Then draw whiskers out from the box to the minimum and the maximum.

How to read it: the box holds the middle half of the data, and its width is the interquartile range, $\text{IQR} = Q_3 - Q_1$. A wide box means the middle of the data is spread out. If the median line is not in the centre of the box, or one whisker is much longer than the other, the data are lopsided (skewed).

An outlier is a value that is unusually far from the rest. A question may give you a rule for deciding, and a common one is: a value is an outlier if it is more than $1.5 \times \text{IQR}$ beyond the box, that is, below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$.

For the pulse data, $1.5 \times 19 = 28.5$, so the limits are $64 - 28.5 = 35.5$ and $83 + 28.5 = 111.5$. Every value lies between them, so there are no outliers. If there were one, you would draw the whisker only to the last value that is not an outlier and mark the outlier with a cross.