Representing and Interpreting Data — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the main ways of displaying data: bar charts, pie charts, pictograms, line graphs and stem-and-leaf diagrams. By the end of this guide you should be able to draw each of them accurately and read values back off them.

You should also be able to choose the right display for a given set of data, calculate pie chart angles, find averages from a stem-and-leaf diagram, compare two sets of data shown together, and recognise a misleading graph.

The organising idea is that each display answers a different question, and choosing badly makes data harder to read rather than easier. A bar chart compares sizes between categories. A pie chart shows proportions of a whole. A line graph shows change over time. A stem-and-leaf diagram is the only one of them that keeps the original values, which is why it is the only one you can calculate an exact median from. Knowing what each display is for tells you both which to draw and what a question is likely to ask about it.

Key terms and definitions

Frequency — how many times something occurs.

Category — a named group, such as a colour or a mode of transport.

Bar chart — equal-width bars with gaps, whose heights show frequency.

Pie chart — a circle split into sectors showing each category's share of the whole.

Pictogram — a display using symbols, each standing for a fixed number of items.

Key — the note explaining what a symbol or a stem-and-leaf digit represents.

Line graph — points joined by lines, showing how a quantity changes over time.

Stem-and-leaf diagram — a display splitting each value into a stem and a leaf, keeping the actual data.

Trend — the general direction of change across a line graph.

Core concepts

Bar charts

Bars are drawn with equal width and gaps between them, because the categories are separate things with nothing in between. Height shows frequency, read off the vertical axis.

The gaps matter: a bar chart shows categories, whereas a histogram shows continuous data and its bars touch. Questions do ask for that distinction.

A dual bar chart places two bars side by side for each category, which compares two groups directly. A composite (stacked) bar chart stacks them instead, which shows the combined total clearly but makes the individual parts harder to compare.

Always label both axes and give the chart a title. Marks are awarded for those separately from the bars.

Pie charts

A pie chart shows proportions, not amounts. The whole circle is 360°, and each category takes a sector in proportion to its frequency.

To find each angle: angle = (frequency ÷ total frequency) × 360°.

So with a total of 40 people and 10 choosing one option, that sector is (10 ÷ 40) × 360° = 90°.

Check the angles add to 360° before drawing. If they do not, an arithmetic error has crept in, and finding it now is much easier than later.

Reading one runs the process backwards: a sector's fraction of 360° gives its share of the total. A 60° sector is one sixth of the circle, so it represents one sixth of the data.

A common trap: two pie charts can have identical sectors while representing very different numbers of people, because a pie chart shows only proportions. You cannot compare actual quantities between two pie charts unless you are told both totals.

Pictograms

A pictogram uses a symbol to stand for a fixed number of items, and the key says how many. Part-symbols represent fractions of that number, so half a symbol with a key of 10 means 5.

Always read the key before reading any value. A pictogram whose key says one symbol equals 20 looks identical to one where it equals 2.

When drawing, keep the symbols the same size and line them up, so that comparing rows is straightforward.

Line graphs

A line graph plots points and joins them, usually with time on the horizontal axis. It suits data that changes continuously, such as temperature or population.

The trend is read from the overall shape — rising, falling or steady — and short-term wobbles should not be confused with it.

Values between plotted points can be estimated from the line, though for data that only exists at fixed moments this is an approximation rather than a fact.

Two lines on the same axes compare two sets of data over the same period, which is usually clearer than two separate graphs.

Stem-and-leaf diagrams

Each value is split into a stem (usually the tens) and a leaf (the units), so 47 has a stem of 4 and a leaf of 7.

A key is essential and is worth a mark on its own — without it, a stem of 4 and leaf of 7 could mean 47, 4.7 or 470.

The leaves are written in order, smallest first, within each stem. An ordered diagram is what makes the statistics easy to find.

Because the original values survive, you can read off exact statistics: the mode is the most frequent value, the median is found by counting to the middle, and the range is the largest value minus the smallest. No other display in this topic allows that.

A back-to-back stem-and-leaf diagram shares one stem between two sets of data, with one set's leaves running leftwards and the other's rightwards. It makes two distributions easy to compare. Note that the leaves on the left are read outwards from the stem, so they increase from right to left.

Choosing the right display

Match the display to the question being asked.

Use a bar chart to compare sizes across categories, a pie chart to show shares of a whole, a line graph to show change over time, a pictogram for a simple visual comparison, and a stem-and-leaf diagram when the actual values need to be kept.

Questions that ask you to comment on a choice want a reason in context: "a pie chart is suitable because the question is about the proportion of the class choosing each option".

Misleading graphs

Displays can mislead without containing a single wrong number, and exam questions ask you to spot how.

A vertical axis that does not start at zero exaggerates differences between bars. A missing or uneven scale makes values impossible to read. Unequal bar widths give some categories more visual weight than they deserve. A pictogram with different-sized symbols does the same.

When criticising a graph, name the specific fault and say what effect it has: "the vertical axis starts at 80, so the difference between the bars looks much larger than it is".

Worked examples

Example 1: Calculating pie chart angles

Thirty students chose a favourite sport: 12 football, 9 netball, 6 rugby and 3 tennis. Find the angle for each sector.

Each angle is the frequency over the total, multiplied by 360°.

Football: (12 ÷ 30) × 360° = 144°. Netball: (9 ÷ 30) × 360° = 108°. Rugby: (6 ÷ 30) × 360° = 72°. Tennis: (3 ÷ 30) × 360° = 36°.

Check: 144 + 108 + 72 + 36 = 360°. ✓ Doing that addition before drawing catches any error while it is still cheap to fix.

Example 2: Reading a stem-and-leaf diagram

A stem-and-leaf diagram has stems 2, 3 and 4, with leaves 3, 7 against stem 2; leaves 1, 4, 4, 8 against stem 3; and leaves 0, 5 against stem 4. The key says 2 | 3 means 23. Find the median and the range.

Listing the values in order: 23, 27, 31, 34, 34, 38, 40, 45.

There are 8 values, so the median lies between the 4th and 5th: both are 34, so the median is 34.

The range is the largest minus the smallest: 45 − 23 = 22.

Both answers are exact, because the diagram preserved every original value.

Example 3: Criticising a display

A bar chart compares two shops' sales. Its vertical axis runs from 90 to 110. Shop A's bar reaches 95 and Shop B's reaches 105. Explain why the chart is misleading.

The axis does not start at zero, so only a narrow slice of each bar is shown.

Shop B's bar appears roughly three times the height of Shop A's, which suggests three times the sales.

In fact the figures are 95 and 105, a difference of about 10%.

The fault is the truncated vertical axis, and its effect is to exaggerate the difference between the two shops.

Common mistakes and how to avoid them

Drawing bar chart bars touching. Categories are separate, so leave gaps. Touching bars belong to a histogram.

Pie chart angles not totalling 360°. Add them up before drawing.

Comparing quantities between two pie charts. They show proportions only; without the totals, no comparison of actual amounts is possible.

Reading a pictogram without checking the key. The key sets the whole scale.

Omitting the key on a stem-and-leaf diagram. It is worth a mark on its own.

Leaving stem-and-leaf leaves unordered. Order them, or the median cannot be counted off reliably.

Saying a graph is "wrong" without naming the fault. Identify the specific feature and the effect it has.

Forgetting axis labels and a title. These carry marks separately from the data.

Exam technique for "Representing and Interpreting Data"

Label both axes, add a title, and include a key wherever one is needed. These are the easiest marks in the topic and the most frequently dropped.

Calculate all pie chart angles first and check they total 360° before touching a protractor.

Order the leaves in a stem-and-leaf diagram as you write them, rather than tidying up afterwards.

Quote actual figures when interpreting a display. "Football was most popular, chosen by 12 of the 30 students" earns more than "football was most popular".

When comparing two sets of data, make two separate statements — one about the typical value and one about the spread — and put both in context.

When asked to criticise a graph, name the feature, then say what it makes the reader believe.

Quick revision summary

Each display answers a different question. Bar charts compare sizes; pie charts show proportions; line graphs show change over time; stem-and-leaf diagrams keep the actual values.

Bar charts have equal-width bars with gaps, because categories are separate. Touching bars mean a histogram.

Pie chart angle = (frequency ÷ total) × 360°, and the angles must total 360°. A pie chart shows proportions only, so two of them cannot be compared for actual amounts without their totals.

Pictograms depend entirely on the key — read it before reading any value.

Stem-and-leaf diagrams need a key and ordered leaves, and because the original values survive they give an exact median, mode and range.

Graphs mislead through a truncated axis, a missing scale, or unequal bar widths. Name the fault and its effect.

Always label axes, give a title, and quote real figures when interpreting.