Cambridge O Level Mathematics: Statistics: Classifying Data, Averages, Spread and Statistical Charts (4024) – Revision Notes

These notes condense sections 9.1 to 9.4 of the Cambridge O Level Mathematics (Syllabus D) syllabus for exams in 2025, 2026 and 2027. For full explanations and longer worked examples, read the study guide. When you are ready to test yourself, use the practice questions, then check your coverage on the printable checklist and the course hub. The free diagnostics show which topics to revisit first.

Key definitions

Term	Meaning
Categorical data	Described in words (colour, transport type)
Discrete data	Numerical, separate values only (goals, shoe size)
Continuous data	Numerical, any value in a range (time, mass, length)
Frequency	How many times a value or class occurs
Mean	Sum of values ÷ number of values
Median	Middle value of ordered data
Mode	Most common value
Modal class	Class with the highest frequency (for classes of equal width)
Range	Largest − smallest
Outlier	A value far from the rest

9.1 Tables

• Tally table: strokes in fives, then count to get the frequency. The frequencies must add to the number of items.
• Two-way table: two features at once. Use the totals to find missing entries, then check that the rows and columns both add to the grand total.

Method for a missing entry

1. Find a row or column with only one blank.
2. Subtract the known entries from its total.
3. Repeat until the table is full.
4. Check the grand total from both directions.

9.3 Averages and spread

Measure	Method	Good for	Weakness
Mean	Σx ÷ n	Data with no extremes	Pulled by outliers
Median	Middle of ordered list	Skewed data, outliers	Ignores most values
Mode	Most frequent	Categorical data, “most popular”	May not exist or may not be unique
Range	Largest − smallest	Quick spread	Uses only two values

Median position: (n + 1) ÷ 2. For 20 values, the position is 10.5, so the median is halfway between the 10th and 11th values.

Frequency table: mean = Σfx ÷ Σf. The mode is the value (the x), not its frequency.

Grouped data: estimated mean = Σ(f × midpoint) ÷ Σf. Midpoint of a continuous class a < x ≤ b is (a + b) ÷ 2. For a discrete class such as 10 to 19, the midpoint is 14.5. The answer is always an estimate.

Method for an estimated mean

1. Add a midpoint row.
2. Multiply each frequency by its midpoint.
3. Add those products and add the frequencies.
4. Divide the first total by the second.

Worked reminder: frequency table

Number of pens in a pencil case, 20 cases:

Pens	10	11	12	13	14
Frequency	2	5	8	3	2

Σfx = 20 + 55 + 96 + 39 + 28 = 238
Mean = 238 ÷ 20 = 11.9

Worked reminder: grouped data

Ages of 20 visitors: 0–10 (6 people), 10–20 (9 people), 20–30 (5 people), each class meaning “from this age up to just below the next”.

Midpoints 5, 15, 25
Σ(f × midpoint) = 30 + 135 + 125 = 290
Estimated mean = 290 ÷ 20 = 14.5
Modal class = 10–20

Worked reminder: the effect of an outlier

Values: 3, 4, 4, 6, 8, 9. The sum is 34, so the mean is 34 ÷ 6 = 5.67 (3 s.f.) and the median is (4 + 6) ÷ 2 = 5.

Now add the value 60. The sum becomes 94 and the mean becomes 94 ÷ 7 = 13.4 (3 s.f.), more than double. The ordered list is 3, 4, 4, 6, 8, 9, 60, so the median is the fourth value, 6. The median moved by 1 and the mean by almost 8. That is why the median is the better “typical value” when an extreme value is present.

9.4 Charts

Chart	Key points
Bar chart	Gaps between bars, equal widths, scale from zero, labelled axes
Dual bar chart	Two bars side by side per category; compare the groups directly
Composite bar chart	Parts stacked on one bar; total is the bar height; read an upper section as the difference between its top and bottom
Pictogram	Use the key; half a symbol is half the key value
Pie chart	Angle = (frequency ÷ total) × 360°
Frequency distribution	Values along the bottom, frequency up the side

Worked reminder: reading a composite bar chart

A stacked bar shows items sold on one day: 30 bottles of water, 22 cartons of juice and 15 cans of cola, stacked in that order. The three sections run from 0 to 30, from 30 to 52 and from 52 to 67. The cola section is 67 − 52 = 15 units tall, so you read its value as the difference between its top and bottom. The whole bar shows 67 items.

Pie chart method

1. Add the frequencies to get the total.
2. Work out 360° ÷ total (the angle for one item).
3. Multiply each frequency by that angle.
4. Check that the angles add to 360°.

To read a pie chart: frequency = (angle ÷ 360°) × total.

Worked reminder: pie chart

Forty-eight people choose a drink: tea 18, juice 12, coffee 10, water 8. One person is 360° ÷ 48 = 7.5°.

Tea 135°, Juice 90°, Coffee 75°, Water 60°  (total 360°)

9.2 Interpreting data

• Quote figures from the chart or table, not just the shape.
• Compare: one comment on an average, one on spread, each in context.
• Limits on conclusions: small or biased sample, grouped data hides detail, averages hide spread, a chart does not prove cause, scales can mislead, predictions beyond the data are unreliable.

Must-know distinctions

• Mean is affected by outliers; median is not.
• Mode is a value; the frequency of the mode is how often it occurs.
• Range measures spread; the averages measure the centre.
• A grouped mean is an estimate; a mean from a list is exact.
• Composite bars show the total; dual bars compare the parts.

Quick self-test

1. Classify each as categorical, discrete or continuous: (a) number of goals in a match, (b) height of a plant, (c) nationality.
2. Find the mean, median, mode and range of 6, 9, 4, 9, 12.
3. Twenty values are listed in order. Which positions give the median?
4. A table shows x = 0, 1, 2, 3 with frequencies 5, 7, 6, 2. Find the mean.
5. Classes 10–19, 20–29, 30–39 have frequencies 4, 6, 10. Estimate the mean.
6. In a survey of 40 people, 14 prefer cycling. What angle is its pie sector?
7. A sector of 80° represents how many people if the pie chart shows 117 people?
8. In a pictogram, one symbol = 4 sandwiches. How many symbols show 18 sandwiches?
9. The mean of four numbers is 10. A fifth number, 20, is added. What is the new mean?
10. Which average best describes house prices in a town with a few very expensive mansions, and why?
11. Two classes have the same mean mark. Class A has range 8 and Class B has range 35. What does this tell you?
12. A survey of 15 members of one chess club finds that 12 like board games. Why should you not conclude that most pupils in the school like board games?

Answers

1. (a) discrete, (b) continuous, (c) categorical.
2. Sum 40, so mean 8. Ordered: 4, 6, 9, 9, 12, so median 9, mode 9, range 12 − 4 = 8.
3. The 10th and 11th values; the median is halfway between them.
4. Σf = 20, Σfx = 0 + 7 + 12 + 6 = 25, mean = 25 ÷ 20 = 1.25.
5. Midpoints 14.5, 24.5, 34.5. Σ(f × midpoint) = 58 + 147 + 345 = 550. Mean ≈ 550 ÷ 20 = 27.5.
6. (14 ÷ 40) × 360° = 126°.
7. (80 ÷ 360) × 117 = 26 people.
8. 18 ÷ 4 = 4.5, so four and a half symbols.
9. Total of the four = 40; new total 60; new mean 60 ÷ 5 = 12.
10. The median, because the mansions pull the mean up and the median stays near a typical price.
11. The averages are equal, but Class A’s marks are far more consistent. Class B’s marks are more spread out.
12. The sample is small and biased: chess club members are not typical of the whole school, so the result cannot be generalised.

Where marks are usually lost

• Not ordering the data before reading off the median.
• Using the frequency column total as the number of classes in a mean calculation.
• Using class limits instead of midpoints in a grouped mean.
• Stating the grouped mean without saying it is an estimate.
• Giving the highest frequency instead of the corresponding value as the mode.
• Pie angles that do not total 360°, or no check made.
• Reading a stacked bar’s upper section from the axis instead of from the top of the section below.
• Comparing two sets with an average only, and no comment on spread.
• Answering “which is better” with no figures from the data.
• Leaving out units or context in a written conclusion.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) syllabus for examination in 2025, 2026 and 2027, Cambridge University Press & Assessment, section 9: Statistics, parts 9.1 to 9.4.