Histograms and scatter diagrams
Frequency density, and correlation between two variables.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 9.3.1Draw and interpret histograms with unequal class widths using frequency density.Supplement
- 9.3.2Draw scatter diagrams, describe correlation and use a line of best fit.
6 minute read
Histograms and scatter diagrams
Histograms
A histogram looks like a bar chart but differs in two crucial ways: the bars touch (the data is continuous), and when class widths are unequal the vertical axis is frequency density, not frequency. frequency density = frequency ÷ class width The rearrangement matters just as much: frequency = frequency density × class width which is to say frequency is represented by the AREA of each bar, not its height. That single fact answers most histogram questions.
Why area rather than height
With unequal widths, using height alone would make a wide class look far more common than it is. Using area corrects for the width, so the visual impression matches the data.
Scatter diagrams
Used to investigate a relationship between two variables. Correlation describes the pattern:
- Positive — as one increases, so does the other.
- Negative — as one increases, the other decreases.
- No correlation — no discernible pattern.
Strength is described as strong or weak by how closely the points cluster around a line.
Line of best fit
Draw a straight line following the trend with roughly equal numbers of points either side; it should pass through the mean point. Use it to estimate values.
- Interpolation — estimating within the data range, which is reasonably reliable.
- Extrapolation — estimating beyond the data, which is unreliable because the pattern may not continue.
Correlation is not causation
This is examined explicitly. Ice cream sales and drownings correlate, but neither causes the other — hot weather causes both. Always be prepared to name a possible third factor.
Think of it like this
In a histogram, area is the honest measure: a wide, low bar can hold just as many people as a narrow, tall one — exactly as a wide shallow puddle can hold as much water as a narrow deep one.
Worked examples
Method, step by step
A histogram bar covers the class 10 ≤ x < 30 and has frequency density 2.5. Find the frequency.
- 1Class width = 30 − 10 = 20.
- 2Frequency = frequency density × class width.
- 3= 2.5 × 20 = 50.
50
A scatter diagram of hours revised against exam mark shows points rising closely along a line. Describe the correlation and state what it suggests.
- 1The points rise from left to right, so the correlation is positive.
- 2They lie close to a straight line, so it is strong.
- 3This suggests students who revised longer tended to score higher — though it does not prove revision caused the higher marks.
Strong positive correlation: more revision is associated with higher marks, though correlation alone does not prove causation.
Common misconceptions
- Reading frequency directly from the height of a histogram bar. Frequency is the **area**.
- Leaving gaps between histogram bars. The data is continuous, so bars touch.
- Concluding that correlation proves causation, when a third variable may explain both.
In the exam
- For any histogram question, write `frequency = frequency density × class width` at the top of your working — it converts either way.
- When describing correlation, give **both** direction and strength, e.g. "strong positive correlation", then interpret it in context.