Cambridge O Level Mathematics: Scatter Diagrams, Correlation and Cumulative Frequency Diagrams (4024) – Revision Notes

These revision notes condense sections 9.5 (Scatter diagrams) and 9.6 (Cumulative frequency diagrams) of the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027. The content is the same for every candidate. For full explanations and longer worked examples, use the study guide, then test yourself with the practice questions. The course hub and printable checklist show the rest of the syllabus. To find your weak spots first, try the free 10-minute diagnostics.

Scatter diagrams (9.5)

• Each pair of values is one point. Mark points as small crosses (×).
• Label both axes, use even scales, and spread the points over most of the grid.

Correlation at a glance

Type	What the points do	Say in context
Positive	Rise left to right	“As x increases, y tends to increase”
Negative	Fall left to right	“As x increases, y tends to decrease”
Zero	No pattern	“No link between x and y”

Strong means the points sit close to a line. Weak means they are widely spread. Correlation shows a link in the data, not that one variable causes the other.

Line of best fit: method in steps

1. Work out the mean of each variable and plot the mean point.
2. Place a clear ruler so that it passes through or near the mean point.
3. Pivot it until the points are roughly evenly spread on both sides along the whole line.
4. Draw a single ruled line across the full data set.
5. To estimate, go from the known value to the line, then across to the other axis.

The line does not have to go through any plotted point.

Using the line

Task	How
Gradient	(change in y) ÷ (change in x), using two points far apart on the line
Meaning of gradient	y changes by that amount per 1 unit of x, on average
Equation	y = mx + c from the gradient and one point on the line
Reliability	Inside the data range is more reliable. Far outside it is unreliable.

Worked reminder. A line passes through (10, 25) and (30, 65). The gradient is (65 − 25) ÷ (30 − 10) = 2. Using (10, 25): c = 25 − 2 × 10 = 5, so y = 2x + 5. At x = 20, y = 45.

Worked reminder: describing and using a scatter diagram

A scatter diagram shows the hours of practice a player does each week (x) against the number of mistakes they make in a match (y). The points slope down from top left to bottom right and sit close to a straight line. Write: “Strong negative correlation: players who practise more tend to make fewer mistakes.” If the ruled line passes through (2, 14) and (8, 5), the gradient is (5 − 14) ÷ (8 − 2) = −1.5. This means each extra hour of practice goes with about 1.5 fewer mistakes, on average.

Choosing scales

Pick scales that are easy to read, such as 2 cm for 10 units, and use most of the grid. On a cumulative frequency diagram, make the vertical axis run to at least n so that every height you need is on the grid. Check that every point you plot is inside the grid before you start.

Cumulative frequency diagrams (9.6)

Building the table

Add up frequencies as a running total. Each total belongs to the upper class boundary. The final total must equal n.

Drawing the curve

1. Horizontal axis: the variable. Vertical axis: cumulative frequency. Label both.
2. Plot (upper boundary, cumulative frequency) for every class.
3. Plot the start point (lower boundary of the first class, 0).
4. Mark points as small crosses.
5. Join with a smooth curve. It never goes down.

Reading the curve

Measure	Height read	Then
Lower quartile	n/4	across to the curve, down to the axis
Median	n/2	across to the curve, down to the axis
Upper quartile	3n/4	across to the curve, down to the axis
p-th percentile	(p/100) × n	across to the curve, down to the axis
IQR	UQ − LQ	subtract the two readings

To find how many values are at or below x: go up from x to the curve, then across. For “more than x”, subtract that reading from n. For a percentage, divide by n and multiply by 100.

Worked reminder

For 100 journey times in minutes, the cumulative frequencies are 8, 30, 70, 88 and 100 at 10, 20, 30, 40 and 50 minutes, with 0 at 0 minutes.

Median: height 50, between (20, 30) and (30, 70) → about 25 minutes
LQ: height 25 → about 17.7 minutes
UQ: height 75 → about 32.8 minutes
IQR: 32.8 − 17.7 = about 15 minutes

Worked reminder: counts between two values

Using the same journey data, how many journeys took between 20 and 40 minutes? The reading at 40 minutes is 88 and the reading at 20 minutes is 30, so 88 − 30 = 58 journeys. Subtract the lower reading from the upper reading, because each reading counts everything up to that value.

Worked reminder: finding a missing value

A cumulative frequency table gives 9, 31 and 70 for three consecutive upper boundaries. The frequency of the second class is 31 − 9 = 22 and of the third class is 70 − 31 = 39. Subtract each cumulative frequency from the next one to recover the class frequencies.

Must-know distinctions

Idea	Do not confuse with
Plot at the upper boundary	Plotting at the midpoint, which is for other graphs
Smooth curve through crosses	Ruled line (that is for a scatter diagram’s line of best fit)
Median is at n/2	Median class (the class that contains the median)
Quartile heights are n/4 and 3n/4	Quartile values (the readings on the horizontal axis)
IQR is a spread	The median, which is an average

Comparing two distributions

Write one comparison of averages (medians) and one of spread (IQRs). Put each in context: “The median is lower, so journeys are typically shorter. The IQR is smaller, so the times are more consistent.”

Quick self-test

1. State the type of correlation you expect between the temperature outside and the amount of heating used in a house.
2. State the type of correlation you expect between a person’s height and the day of the month they were born.
3. A pupil draws a line of best fit that joins the first and last points of a scatter diagram, ignoring the rest. Why might this be a poor line?
4. A line of best fit passes through (10, 25) and (30, 65). Estimate y when x = 25.
5. A cumulative frequency diagram has n = 240. At what heights do you read the lower quartile, median and upper quartile?
6. Using the worked reminder above, how many journeys took more than 30 minutes?
7. Using the same data, what percentage of journeys took more than 40 minutes?
8. Using the same data, estimate the 90th percentile.
9. Explain why the cumulative frequency curve starts at (lower boundary, 0).
10. Explain why a cumulative frequency curve can never slope downwards.

Answers

1. Negative: as it gets warmer, less heating is used.
2. Zero correlation.
3. It ignores the other points, so there may not be a roughly even number of points on each side. A line of best fit must follow the whole data set.
4. From y = 2x + 5, y = 2 × 25 + 5 = 55.
5. 60 (lower quartile), 120 (median) and 180 (upper quartile).
6. The reading at 30 minutes is 70, so 100 − 70 = 30 journeys.
7. The reading at 40 minutes is 88, so 12 journeys, which is 12%.
8. Height = 0.9 × 100 = 90, between (40, 88) and (50, 100), which gives about 41.7 minutes.
9. No values lie below the lower boundary, so the running total there is 0.
10. It is a running total of frequencies, and frequencies cannot be negative, so the total can only stay level or rise.

Where marks are usually lost

• Plotting at class midpoints, or not plotting the starting point (lower boundary, 0).
• Joining the points of a cumulative frequency diagram with a ruler instead of a smooth curve.
• Drawing a line of best fit as a zigzag, a curve, or a line that stops partway along the data.
• Marking points with dots too large to read, or not marking them clearly.
• Reading the median at half of one frequency instead of at n/2 of the total.
• Giving the height read from the vertical axis when the question asks for the value on the horizontal axis.
• Forgetting to subtract from n for “more than” questions.
• Describing correlation as “going up” without saying what it means for the two variables.
• Leaving out units in a final answer.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027, published by Cambridge University Press & Assessment (Cambridge Assessment International Education), sections 9.5 and 9.6.