OCR GCSE Mathematics: Sampling, charts, cumulative frequency and histograms (J560)

This guide follows the OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026). It covers sections 12.01 and 12.02: sampling, then drawing and reading charts, line graphs, cumulative frequency graphs and histograms. Sampling and the chart work in 12.02a are listed for both tiers. Cumulative frequency graphs and histograms (12.02b) are Higher tier only.

Links: revision notes, practice questions, OCR GCSE Mathematics hub, checklist and free 10-minute diagnostics.

What this unit covers

Reference	What you must be able to do	Tier
12.01a	Define the population; know the difference between population and sample; infer properties of a population from a sample; understand simple random sampling and bias	Both
12.02a	Interpret and construct frequency tables, bar charts, pie charts, pictograms and vertical line charts; interpret multiple and composite bar charts	Both
12.02a	Design tables to classify data; interpret and construct line graphs for time series; identify trends such as seasonal variation	Both
12.02b	Interpret and construct cumulative frequency graphs	Higher tier only
12.02b	Interpret and construct histograms with equal or unequal class intervals	Higher tier only

Populations, samples and bias (12.01a)

The population is everything you want to know about in a study: all the people, objects or events of interest. It is not always people. For a study of battery life, the population is every battery of that type.

A sample is a smaller group taken from the population. You use one because testing everything costs too much, takes too long, or destroys the item (a battery tested until flat is no use afterwards).

A sample lets you infer properties of the population. The usual method is to scale up the proportion found in the sample.

Worked example 1. A sports club has 840 members. A sample of 60 members is chosen and 21 say swimming is their main sport. Estimate how many of all members would say the same.

proportion in sample = 21/60 = 0.35
estimate for club     = 0.35 × 840 = 294

About 294 members. This is an estimate, not a fact. A different sample of 60 would probably give a slightly different answer. Larger samples usually give more reliable estimates.

Simple random sampling

In a simple random sample every member of the population has the same chance of being picked, and every possible sample of that size is equally likely. To draw one:

1. Make a list of the whole population (a sampling frame) and number it.
2. Generate random numbers, for example with a random number function or by drawing numbers from a bag.
3. Pick the members with those numbers. If a number repeats, ignore it and draw again.
4. Stop when you have the sample size you want.

Bias

A sample is biased if it favours some members of the population over others, so it does not represent the whole. Common causes:

• Sampling from a convenient place or time only, such as the first 20 people through the school gate.
• A sampling frame that leaves some people out, such as a phone survey that misses households without a phone.
• Letting people volunteer, so keen people are over-represented.
• Leading questions, such as “Don’t you agree the canteen is too expensive?”

A bigger sample does not cure bias. A large biased sample is still biased.

Charts for categorical and discrete data (12.02a)

Match the chart to the data type.

Chart	Best for	Key rule
Frequency table	Any data	Tally, then total; check the total matches the number surveyed
Bar chart	Categorical data	Equal-width bars with gaps; height shows frequency; label both axes
Pictogram	Categorical data	Use a key; part symbols show part values
Pie chart	Categorical data, shares of a whole	Angle = (frequency ÷ total) × 360°
Vertical line chart	Ungrouped discrete numbers	A line at each value, height shows frequency
Multiple bar chart	Comparing groups side by side	Bars for each group sit next to each other
Composite bar chart	Totals and their parts	Each bar is split into stacked sections

Worked example 2 (pie chart). 72 pupils say how they usually travel to school: walk 24, bus 18, car 12, cycle 6, train 12. Find the angles.

360° ÷ 72 = 5° per pupil
walk  24 × 5 = 120°
bus   18 × 5 = 90°
car   12 × 5 = 60°
cycle  6 × 5 = 30°
train 12 × 5 = 60°
check: 120 + 90 + 60 + 30 + 60 = 360°

Draw each sector from the centre using a protractor, label it, and give the total angle check in your working.

Reading a pictogram. If the key says one symbol stands for 8 books, then 2.5 symbols stand for 2.5 × 8 = 20 books. A half symbol is half the value.

Reading a composite bar chart. In a composite chart, the height of a section is the difference between its top and bottom edges, not the value at its top. A section running from 55 to 85 on the scale is worth 85 − 55 = 30. In a multiple bar chart you read each bar from the axis.

Designing tables to classify data

A good data collection table has:

• A heading for each variable (for example “Age group” and “Transport”).
• Non-overlapping categories that cover every possible answer. “0 to 2 hours” and “2 to 4 hours” overlap at exactly 2. Write “0 up to 2”, “2 up to 4” instead, or use inequalities.
• An “Other” row where answers cannot all be predicted.
• A tally column and a frequency column.

A two-way table classifies the same people by two features at once, for example year group by transport, with row and column totals. Missing entries come from subtracting known values from the totals.

Line graphs for time series (12.02a)

A time series is data recorded at regular intervals, such as monthly sales. Plot time on the horizontal axis, plot the values, and join the points with straight lines. The lines help you see the pattern, but values between the points are not measured.

• Trend: the overall direction over a long period (rising, falling, steady).
• Seasonal variation: a pattern that repeats over a fixed period, such as a peak every summer.

Worked example 3. Quarterly sales of garden furniture (in hundreds of pounds):

Year	Q1	Q2	Q3	Q4
Year 1	12	30	24	6
Year 2	15	36	27	9

Each year, sales peak in Q2 and are lowest in Q4. That repeating pattern is the seasonal variation. Each quarter in year 2 is higher than the same quarter in year 1 (by 3, 6, 3 and 3), so the trend is upward.

To predict ahead, use the same quarter from earlier years and the trend. Say how confident you are: a prediction assumes the pattern continues.

Cumulative frequency graphs (12.02b, Higher tier only)

Cumulative frequency is a running total of frequencies. For grouped data you plot each cumulative frequency against the upper class boundary, because that is the point by which that many values have been reached. Join the points with a smooth curve (or straight segments if told to), beginning at the lowest class boundary, where the cumulative frequency is 0.

Worked example 4. Waiting times (minutes) of 60 patients at a clinic:

Wait t	0<t≤10	10<t≤20	20<t≤30	30<t≤40	40<t≤50	50<t≤60
Frequency	4	11	19	15	8	3
Cumulative	4	15	34	49	57	60

Plot (10, 4), (20, 15), (30, 34), (40, 49), (50, 57), (60, 60), starting at (0, 0).

• Median: the value at cumulative frequency 60 ÷ 2 = 30. The curve passes 30 between (20, 15) and (30, 34), at about 28 minutes.
• Lower quartile: at 60 ÷ 4 = 15, which is exactly the point (20, 15). So LQ = 20.
• Upper quartile: at 3 × 60 ÷ 4 = 45, between (30, 34) and (40, 49): 30 + (45 − 34)/(49 − 34) × 10 ≈ 37.3.
• Interquartile range: 37.3 − 20 ≈ 17.3 minutes.
• More than 40 minutes: cumulative frequency at 40 is 49, so 60 − 49 = 11 people.

Values read from a drawn curve are estimates. State “about” and show your reading lines on the graph. This guide reads quartiles at n/4 and 3n/4 of the total.

Histograms (12.02b, Higher tier only)

A histogram shows grouped continuous data. There are no gaps between bars. When class widths differ, the area of a bar represents the frequency, so the vertical axis is frequency density:

density = frequency ÷ width      (so a wide class has a lower density)
frequency = density × width

Worked example 5 (construct). Ages of 170 visitors to a museum:

Age a	0<a≤10	10<a≤30	30<a≤40	40<a≤60	60<a≤100
Frequency	20	60	36	30	24
Class width	10	20	10	20	40
Frequency density	2	3	3.6	1.5	0.6

Draw bars touching one another, with widths matching the class widths and heights equal to the frequency density. The tallest bar is 30 to 40 years, even though the 10 to 30 class has the most visitors. Always judge frequency by area, not height.

Estimating from part of a bar. Assume the values in a class are spread evenly. Estimate the number aged between 25 and 40:

25 to 30 is part of the 10<a≤30 bar: 5 × 3 = 15
30 to 40 is the whole bar:           36
estimate = 15 + 36 = 51

Worked example 6 (reverse). A histogram bar covers 5<x≤20 with height 2.4 on the frequency density axis. Its frequency is 15 × 2.4 = 36. If instead the vertical axis has no scale, find what one unit of area is worth from a bar whose frequency you know, then scale the other bars.

Common errors

• Calling a sample “random” because it was picked without thinking. Random means a method that gives everyone an equal chance.
• Plotting cumulative frequency at the class midpoint or the lower boundary instead of the upper boundary.
• Using the bar height as frequency in a histogram with unequal classes.
• Leaving gaps between histogram bars, or adding gaps in a cumulative frequency graph.
• Reading a composite bar section from the axis instead of subtracting its two edges.
• Overlapping classes in a table, such as “10 to 20” followed by “20 to 30”.
• Giving pie angles that do not total 360°.

Official syllabus

OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026), for first assessment in 2017, published by OCR. Sections covered: 12.01a, 12.02a and 12.02b.