5.1 - Choosing a data display
A list preserves every observation, but a well-chosen display reveals patterns much faster. In this part of 5.1, you will choose a display for a purpose, organise values in stem-and-leaf diagrams, and construct and interpret histograms. The central question is what each picture preserves and what it hides.
Raw data and representation
Two classes record their journey times to school. Would a list of every time, a picture of the distribution, or a single summary make the fairest comparison? The answer depends on what you want to discover.
Raw data are the individual observations before they are summarised. For example, journey times of , , and minutes retain more detail than “four journeys took less than minutes”. Grouping makes a large dataset manageable, but loses the exact observations.
Types of data
Categorical data describe groups, such as bus, walk or cycle. Numerical data record counts or measurements. A discrete variable takes separate possible values, such as the number of passengers; a continuous variable can take any value within an interval, such as a journey time. A continuous measurement remains continuous even when rounded for recording.
Choose a display by matching the data and the question. A frequency is a count of observations, rather than the value being measured.
| Display | Useful for | Advantage and limitation |
|---|---|---|
| Bar chart or pie chart | Frequencies of categories | Compares category sizes or shares; does not show individual numerical measurements. Separate bars represent separate categories. |
| Stem-and-leaf diagram | A modest number of numerical observations | Shows shape and retains every value; becomes crowded for a very large dataset. |
| Histogram | Numerical observations grouped into intervals | Shows the distribution and handles unequal intervals; exact individual values are lost. |
| Cumulative frequency graph | Running totals of numerical observations | Estimates counts below a value and percentiles; grouped estimates do not recover the raw data. |
| Box-and-whisker plot | Comparing centres and spreads of numerical datasets | Compact and easy to compare on a common scale; conceals much of the distribution's shape. |
| Time-series graph | Measurements recorded in time order | Shows change over time; rearranging observations by size would destroy that information. |
For a small set of exact journey times, a stem-and-leaf diagram lets you recover the actual times. For hundreds of times already grouped into intervals, use a histogram to examine where they cluster. The next lesson develops cumulative frequency graphs and box-and-whisker plots.
Key idea
A useful display answers a particular question. Ask whether you need individual observations, distribution shape, a comparison of summaries, or change over time before choosing it.
Make that choice for a dataset whose individual observations are no longer available.
Stem-and-leaf diagrams
A stem-and-leaf diagram splits each observation into a leading part and a final digit. Unlike a histogram, it retains the raw numerical values.
For minutes, choose stem and leaf . The same written pair could mean or in another dataset, so the key is essential. Put stems in increasing order, include repeated leaves, and order leaves along each row. Include an empty stem if it is needed to show a gap.
Stem-and-leaf key
A key states the value and units represented by a stem and a leaf. For example, “ represents minutes” fixes the scale of the whole diagram.
A useful check is to count the leaves: there must be one per observation, including repeats.
Worked example 1: retaining every observation
Question. Construct a stem-and-leaf diagram for these journey times, in minutes:
Plan. Use tens as stems, sort the units digits, and provide a key.
| Stem | Leaves |
|---|---|
Answer. The diagram above, with key: represents minutes. For example, the repeated leaves in the first row record two journeys of minutes.
Check. There are leaves. Reading the diagram back gives the same twelve observations, with minimum and maximum minutes.
Now add a second group using the same stems and key. The only structural change is that one group goes to the left of the stem. Its leaves increase as you read outwards from the stem, so they appear in descending order from left to right.
Worked example 2: a back-to-back comparison
Question. Compare group A from Example 1 with group B, whose times in minutes are
Construct a back-to-back stem-and-leaf diagram and state how many times in each group are at least minutes.
Plan. Put A to the left and B to the right of common tens stems. Read the row with stem to answer the count.
| Group A leaves | Stem | Group B leaves |
|---|---|---|
Key: represents minutes for A and minutes for B.
Answer. A has times at least minutes; B has . The common stems allow a direct comparison without discarding any observations.
Check. A still has leaves; B has . The values and confirm the counts directly from the lists.
The row labels alone do not determine the scale of the data.
Continue this lesson.
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