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

This guide follows the Cambridge O Level Mathematics (Syllabus D) syllabus for exams in 2025, 2026 and 2027. It covers sections 9.1 to 9.4 of the statistics content: classifying and tabulating data, interpreting data, averages and spread, and statistical charts. The syllabus is not tiered, so every candidate meets all of this material.

Related pages: revision notes, practice questions, the Cambridge O Level Mathematics hub, the printable checklist and the free diagnostics.

What this unit covers

Section	What you must be able to do
9.1	Classify and tabulate data, for example with tally tables and two-way tables
9.2 (1)	Read, interpret and draw inferences from tables and statistical diagrams
9.2 (2)	Compare data sets using tables, graphs, averages and spread
9.2 (3)	Appreciate the restrictions on drawing conclusions from given data
9.3 (1)	Calculate mean, median, mode and range for individual data, and say when each is used
9.3 (2)	Estimate the mean of grouped discrete or grouped continuous data
9.3 (3)	Identify the modal class
9.4	Draw and interpret bar charts (including composite and dual), pie charts, pictograms and simple frequency distributions

9.1 Classifying and tabulating data

Data is either categorical (described in words, such as colour or transport type) or numerical. Numerical data is discrete if it can only take separate values (number of goals, shoe size) and continuous if it can take any value in a range (time, mass, length). The type of data decides which table and chart suit it.

A tally table records raw data as strokes in groups of five. You then count each row to get a frequency. A two-way table classifies each item by two features at once, and its row and column totals must agree.

Worked example 1: tally and frequency

Twenty pupils recorded the number of books they borrowed in a week:

2 0 1 3 2 2 4 1 0 2 3 1 2 5 2 1 3 2 0 4

Books	Tally	Frequency
0	III	3
1	IIII	4
2	IIII II	7
3	III	3
4	II	2
5	I	1
Total		20

The total of 20 matches the number of pupils, which is your check that nothing was missed or counted twice.

Worked example 2: a two-way table

Fifty pupils were asked how they travel to school. Some entries are missing.

	Walk	Bus	Car	Total
Boys	9	11	4	24
Girls	8		6	26
Total	17			50

Girls on the bus: 26 − 8 − 6 = 12. Bus total: 11 + 12 = 23. Car total: 4 + 6 = 10. Check along the bottom row: 17 + 23 + 10 = 50.

9.3 Averages and range

For a list of values:

• Mean = (sum of all values) ÷ (number of values)
• Median = the middle value when the data is in order. With an even number of values, it is halfway between the middle two.
• Mode = the most common value. There can be more than one, or none worth quoting.
• Range = largest value − smallest value. This measures spread.

Choosing an average. The mean uses every value, so it suits data without extreme values, but one outlier pulls it. The median is not affected by extreme values, so it suits skewed data such as house prices or salaries. The mode is the only average that works for categorical data (the most popular colour) and suits questions such as “which shoe size should a shop stock most of?”. The range is easy to find but depends on only two values.

Worked example 3: the four measures

Delivery times in minutes for eight parcels: 14, 9, 22, 11, 9, 35, 12, 9.

Order them: 9, 9, 9, 11, 12, 14, 22, 35.

Mean   = 121 ÷ 8 = 15.125 = 15.1 (3 s.f.)
Median = (11 + 12) ÷ 2 = 11.5
Mode   = 9
Range  = 35 − 9 = 26

The 35 is an outlier. It drags the mean above six of the eight values, so 11.5 describes a typical delivery better.

Frequency tables

When data is in a frequency table, multiply each value by its frequency.

Mean = Σfx ÷ Σf

For the median, find the position (n + 1) ÷ 2 using the cumulative frequency. The mode is the value with the highest frequency (not the highest frequency itself).

Worked example 4: goals per match

Goals x	0	1	2	3	4	5
Matches f	4	6	8	4	2	1

Σf  = 25
Σfx = 0 + 6 + 16 + 12 + 8 + 5 = 47
Mean = 47 ÷ 25 = 1.88
Median position = (25 + 1) ÷ 2 = 13th value
Cumulative frequencies: 4, 10, 18  → the 13th value is 2
Mode = 2

Grouped data: estimating the mean

When values are grouped you do not know the exact values, so you assume each lies at the midpoint of its class. The answer is an estimate.

Estimated mean = Σ(f × midpoint) ÷ Σf

For continuous classes such as 5 < t ≤ 10, the midpoint is (5 + 10) ÷ 2 = 7.5. For discrete classes such as 6 to 10, the midpoint is also (6 + 10) ÷ 2 = 8.

The modal class is the class that has the largest frequency. With classes of equal width, that is the tallest bar.

Worked example 5: grouped continuous data

Waiting times t minutes for 40 customers:

t (minutes)	0 < t ≤ 5	5 < t ≤ 10	10 < t ≤ 15	15 < t ≤ 20	20 < t ≤ 25
Frequency	4	11	13	8	4
Midpoint	2.5	7.5	12.5	17.5	22.5
f × midpoint	10	82.5	162.5	140	90

Σ(f × midpoint) = 485
Estimated mean  = 485 ÷ 40 = 12.125 ≈ 12.1 minutes
Modal class     = 10 < t ≤ 15

Worked example 6: grouped discrete data

Quiz scores out of 20 for 20 pupils:

Score	1–5	6–10	11–15	16–20
Frequency	3	9	6	2
Midpoint	3	8	13	18

Σ(f × midpoint) = 9 + 72 + 78 + 36 = 195
Estimated mean  = 195 ÷ 20 = 9.75
Modal class     = 6–10

9.4 Statistical charts and diagrams

Bar charts

A bar chart has gaps between bars, equal bar widths, and a vertical scale starting at zero. Label both axes and give a title or units. Bars show categories or discrete values.

• A dual (side-by-side) bar chart puts bars for two groups next to each other in each category. It is best for comparing the groups directly.
• A composite (stacked) bar chart stacks the parts of each total on one bar. It shows the total height clearly, but only the bottom section starts at zero, so the upper sections are harder to compare.

A simple frequency distribution drawn as a chart uses the values on the horizontal axis and frequency on the vertical axis. For worked example 4, the bar at 2 goals would be 8 units tall.

Worked example 7: composite and dual bars

A canteen sold hot and cold lunches:

Day	Mon	Tue	Wed	Thu	Fri
Hot	40	35	50	45	30
Cold	25	30	20	35	45
Total	65	65	70	80	75

For a composite chart, each bar reaches the total (Monday to 65). The hot section is drawn from 0 to 40 and the cold section from 40 to 65. For a dual chart, Monday has one bar of height 40 and one of height 25.

From the table, Thursday had the greatest total (80), and Friday is the only day on which cold lunches outsold hot ones.

Pie charts

Each sector angle is proportional to its frequency.

Angle = (frequency ÷ total) × 360°

Worked example 8: drawing a pie chart

Seventy-two pupils chose a favourite sport.

Sport	Football	Cricket	Swimming	Hockey	Other
Pupils	24	18	12	9	9

Each pupil is worth 360° ÷ 72 = 5°.

Football 24 × 5 = 120°
Cricket  18 × 5 =  90°
Swimming 12 × 5 =  60°
Hockey    9 × 5 =  45°
Other     9 × 5 =  45°
Total = 360°

Reading backwards works the same way. A 75° sector in this chart stands for 75 ÷ 5 = 15 pupils.

Pictograms

A pictogram uses a symbol, and the key says how many items one symbol stands for. A half symbol shows half the key value, a quarter symbol a quarter, and so on.

If one symbol = 8 cars, then 3½ symbols = 28 cars, 2½ symbols = 20 cars and 4½ symbols = 36 cars. A value such as 30 would need three-quarters of the last symbol, so choose a key that divides your data sensibly.

9.2 Interpreting and comparing data

To read a chart, use the scale and key, not the look of the picture. To draw an inference, state what the figures show and use words from the context.

To compare two data sets, give one comment on an average and one on spread, each with the numbers and a conclusion in context.

Worked example 9: comparing

Minutes late for seven buses on each of two routes:

Route 1:  2 3 3 4 5 5 6
Route 2:  0 1 2 4 5 8 8

	Mean	Median	Range
Route 1	4	4	4
Route 2	4	4	8

“The two routes have the same mean and median lateness (4 minutes), but Route 1 has the smaller range (4 compared with 8), so its buses are more consistent. Route 2 is sometimes on time but is also sometimes 8 minutes late.”

Restrictions on conclusions

• A small sample may not represent everyone.
• A biased sample (for example, asking only members of one club) can give a misleading picture.
• Grouped data hides exact values, so the mean is only an estimate and the range cannot be found exactly.
• Averages hide spread, so two data sets with equal means can behave very differently.
• A chart cannot prove cause. Two things rising together do not show that one causes the other.
• Changing the scale (a truncated axis, uneven symbols) can exaggerate a difference.
• Predicting beyond the data is unreliable.

Common errors

• Finding the median without putting the data in order first.
• Quoting the highest frequency as the mode, instead of the value that has it.
• Dividing Σfx by the number of classes instead of by Σf.
• Using class boundaries rather than midpoints when estimating a mean.
• Giving the mean as exact when it comes from grouped data.
• Drawing pie angles that add to something other than 360°.
• Forgetting to check that two-way table totals agree.
• Comparing data sets using only an average, with no comment on spread.
• Reading a composite chart’s upper section from zero instead of from the top of the section below.

Next steps

Work through the revision notes to fix the methods, then try the practice questions. The probability practice set uses frequency tables in a similar way, and the past papers page points you to official papers.

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.