Choosing Statistical Diagrams and Summary Measures Revision Notes | GCSE AQA Higher Maths

1. Why Choosing the Right Representation Matters

Imagine presenting the heights of 200 students using a list of 200 numbers. Nobody would spot the pattern. The right statistical diagram turns raw data into a story — but only if you match the diagram to the data type.

examiners regularly ask you to choose a representation and justify that choice. That means you need to know not just how to draw each diagram, but when each one is appropriate and why the alternatives would be wrong.

2. The Three Data Types — Your Starting Point

Every decision about representation starts with one question: what kind of data do you have? There are three categories you need to distinguish at GCSE.

Continuous data is measured, not counted, so it can take any value within a range. Height, temperature, and reaction time are all continuous — there is always a value between any two measurements you could record.

3. Matching Representations to Data Types

Now that you can classify data, the next step is knowing which diagram suits each type. Think of this as a decision framework — work through it every time an exam question asks you to choose or justify a representation.

Notice that histograms and frequency polygons are reserved for continuous (or continuously grouped) data. A bar chart uses gaps between bars to signal that the categories are separate; a histogram has no gaps because the data is continuous and the scale has no breaks.

4. Histograms and Frequency Density — The Key Calculation

Histograms are the representation that trips up the most students at GCSE. The trap is assuming you plot frequency on the vertical axis — you do not. You plot frequency density, which accounts for the width of each class interval.

Here is why this matters: if one class interval is twice as wide as another, a bar of the same height would represent twice as many values. Frequency density corrects for this by making the area of each bar proportional to the frequency, not the height.

Worked example: Constructing a Histogram from a Frequency Table

The table below shows the heights of 40 plants. Draw a histogram to represent this data. | Height $h$ (cm) | Frequency | |---|---| | $0 \leq h < 10$ | 4 | | $10 \leq h < 20$ | 10 | | $20 \leq h < 30$ | 14 | | $30 \leq h < 50$ | 12 |

1. Class $0 \leq h < 10$: width $= 10 - 0 = 10$ Class $10 \leq h < 20$: width $= 20 - 10 = 10$ Class $20 \leq h < 30$: width $= 30 - 20 = 10$ Class $30 \leq h < 50$: width $= 50 - 30 = 20$ — The last class is twice as wide as the others — this is why frequency density is essential here. [M1 for recognising unequal class widths]
2. $0 \leq h < 10$: $\frac{4}{10} = 0.4$ $10 \leq h < 20$: $\frac{10}{10} = 1.0$ $20 \leq h < 30$: $\frac{14}{10} = 1.4$ $30 \leq h < 50$: $\frac{12}{20} = 0.6$ — Divide each frequency by its class width. [M1 for correct frequency density calculation]
3. Horizontal axis: height $h$ (cm), continuous scale from 0 to 50. Vertical axis: frequency density, scale from 0 to at least 1.4. — No gaps between bars — the horizontal axis is a continuous scale, not separate categories.
4. Bar from 0 to 10, height 0.4 Bar from 10 to 20, height 1.0 Bar from 20 to 30, height 1.4 Bar from 30 to 50, height 0.6 — Check: area of last bar $= 20 \times 0.6 = 12$ ✓ This equals the frequency for that class. [A1 for all bars correct]
5. $(10 \times 0.4) + (10 \times 1.0) + (10 \times 1.4) + (20 \times 0.6)$ $= 4 + 10 + 14 + 12 = 40$ ✓ — Sum of all bar areas must equal the total frequency. This is your sanity check.

Answer: Frequency densities: 0.4, 1.0, 1.4, 0.6. Bars drawn with no gaps on a continuous scale; total area = 40.

5. Frequency Polygons, Pie Charts, and Scatter Diagrams

A frequency polygon is drawn by plotting the frequency (or frequency density) at the midpoint of each class interval and connecting the points with straight lines. It is particularly useful when you want to compare two distributions on the same diagram — something a histogram cannot do cleanly.

Worked example: Drawing a Frequency Polygon

Using the plant height data from the previous example (equal class widths only: $0 \leq h < 10$, $10 \leq h < 20$, $20 \leq h < 30$), plot a frequency polygon.

1. $0 \leq h < 10$: midpoint $= 5$ $10 \leq h < 20$: midpoint $= 15$ $20 \leq h < 30$: midpoint $= 25$ — Midpoint = (lower boundary + upper boundary) ÷ 2. [M1 for correct midpoints]
2. $(5, 4)$, $(15, 10)$, $(25, 14)$ — Each point is (midpoint, frequency). When class widths are equal, plotting frequency is acceptable.
3. Plot each coordinate on axes with height on the $x$-axis and frequency on the $y$-axis. Join consecutive points with straight lines. — The lines are straight — do not draw a curve. [A1 for correct points joined with straight lines]

Answer: Points at (5, 4), (15, 10), (25, 14) joined by straight lines.

Pie charts work differently — they show each category as a proportion of the whole. The angle for each sector is calculated using the formula: angle $= \frac{\text{frequency}}{\text{total}} \times 360^\circ$. They are best used when you want to compare parts of a whole, not when you need to read off exact frequencies.

Scatter diagrams are used exclusively when you have two continuous variables recorded for the same subject — for example, a student's revision hours and their exam score. Each data point represents one individual, and the pattern of points reveals whether a correlation exists between the two variables.

6. Choosing Between Representations — Decision Practice

You have now seen all the main representations. The skill the exam tests most often is justifying your choice — not just naming a diagram, but explaining why it suits the data type. Work through the practice set below before moving to the exam technique section.

7. Exam Technique and Common Mistakes

Exam questions on statistical representations test three things: choosing the right diagram, constructing it correctly, and reading values from it accurately. Each of these has its own set of traps.

Choosing and creating statistical representations comes down to one core skill: matching the structure of the diagram to the nature of the data. Discrete and categorical data use bar charts; continuous grouped data uses histograms (with frequency density) or frequency polygons; proportional comparisons use pie charts; and relationships between two continuous variables use scatter diagrams.