Choosing the right data display

By the end of this lesson

By the end you can look at a set of raw data and choose an appropriate statistical diagram, then explain why that choice is better (or worse) than the alternatives.

What the question asks

This objective is about choosing: deciding which display suits the data you are given, and justifying that decision. (Drawing each display is covered in the next lessons.)

Typical wording:

The table shows the heights of 200 students. State, with a reason, which of a histogram, a box-and-whisker plot, or a stem-and-leaf diagram would be the most appropriate way to display these data.

You need to know the strengths and weaknesses of each common display.

Definition

Raw data

The original, unprocessed observations exactly as they were collected (e.g. a list of 50 exam scores, or the daily rainfall measurements for a month).

Words you need

Discrete data can only take separate values, usually counts (number of siblings: 0, 1, 2, ...).
Continuous data can take any value in a range, usually measurements (height, time, temperature).
The shape of a distribution is how the values are spread out when you draw them.
Symmetric: the two sides look like mirror images.
Skewed: one side has a long tail. Positive skew has the long tail on the right (high values); negative skew on the left.
Bimodal: two separate peaks.
An outlier is a value far away from the rest of the data.
The median is the middle value when the data are put in order; the quartiles cut the ordered data into quarters.
The $k$ th percentile is the value below which $k\%$ of the data lies.

What each display looks like

Stem-and-leaf diagram: each value is split into a stem (e.g. the tens digit) and a leaf (the units digit). Leaves are listed in order beside their stem, with a key such as 2 | 4 means 24.
Box-and-whisker plot: a box from $Q_1$ to $Q_3$ with a line at the median. Whiskers run out to the lowest and highest values.
Histogram: touching bars over the classes of grouped continuous data. The area of each bar represents its frequency.
Cumulative frequency graph: running totals plotted at the upper class boundaries and joined up.

The main displays and when to use them

Display	Best for	Not ideal for
Stem-and-leaf	Small to moderate data sets (≈ 15–50 values) where you want to keep every individual value visible	Large data sets (hundreds of values); comparing more than two groups
Box-and-whisker plot	Comparing two or more distributions side-by-side; showing median, quartiles, outliers, spread at a glance	Seeing the exact shape of the distribution or individual data points
Histogram	Large continuous data sets; showing the overall shape (skew, number of peaks, gaps)	Small data sets; discrete data with few distinct values
Cumulative frequency graph	Estimating medians, quartiles, percentiles; finding the proportion above/below a value	Showing the shape of the distribution directly

A back-to-back stem-and-leaf is a variant that puts two data sets on the same stem (one on the left, one on the right) for direct comparison.

Why it works

Think about what the reader needs to see.

If they need to spot every value → stem-and-leaf.
If they need to compare centres and spreads of several groups → box plots.
If they need to see the shape of a big continuous data set → histogram.
If they need to answer "what proportion is above X?" → cumulative frequency graph.

Worked example: Example 1: choosing for a small data set

A teacher records the marks (out of 50) of 15 students on a quiz: 23, 31, 18, 27, 35, 42, 29, 33, 26, 38, 21, 30, 45, 34, 28

Which display would be most appropriate: a stem-and-leaf diagram, a histogram, or a box plot? Give one advantage and one disadvantage of your choice.

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

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