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

Quick recall for sections 12.01 and 12.02 of the OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026). For full explanations and worked examples, read the study guide. Test yourself with the practice questions, tick off topics on the checklist, see the OCR GCSE Mathematics hub, or try the free 10-minute diagnostics.

Sampling (12.01a), the charts and tables in 12.02a, and time series line graphs are listed for both tiers. Cumulative frequency graphs and histograms (12.02b) are Higher tier only.

Key definitions

Term	Meaning
Population	Everything the study is about (all people, items or events of interest)
Sample	A smaller group taken from the population
Simple random sample	Every member has an equal chance of selection, and every sample of that size is equally likely
Bias	The sample favours some members, so it does not represent the population
Sampling frame	The numbered list of the whole population you pick from
Time series	Data recorded at regular intervals over time
Trend	Long-term direction of a time series
Seasonal variation	A pattern that repeats over a fixed period
Cumulative frequency	Running total of frequencies (Higher tier only)
Frequency density	Frequency ÷ class width (Higher tier only)

Sampling in steps

Estimating a population property:

1. Find the proportion in the sample: number with the feature ÷ sample size.
2. Multiply by the population size.
3. Say it is an estimate, and that a larger unbiased sample is more reliable.

Taking a simple random sample:

1. Number everyone in the population.
2. Generate random numbers within that range.
3. Select the matching members and skip repeats.
4. Continue until you have the sample size.

Sources of bias to quote: sampling only at one place or time; a sampling frame that misses people; volunteers only; leading questions. A very small sample is unreliable, but size is a separate issue from bias.

Choosing and drawing charts

Chart	Data type	Reminder
Frequency table	Any	Tally, then total
Bar chart	Categorical	Equal widths, gaps, labelled axes
Pictogram	Categorical	Always use the key; part symbols give part values
Pie chart	Categorical	Angle = frequency ÷ total × 360
Vertical line chart	Ungrouped discrete	Line height is frequency
Multiple bar chart	Compare groups	Bars side by side, read from the axis
Composite bar chart	Totals and parts	Section size = top edge − bottom edge

Pie chart shortcut: find degrees per item first (360 ÷ total), then multiply. Check that the angles add to 360°.

Designing tables

• One heading per variable, with a tally column and a frequency column.
• Categories must not overlap and must cover every answer. Use “0 up to 2” and “2 up to 4”, or inequalities.
• Add an “Other” option when answers cannot all be predicted.
• In a two-way table, find missing cells by subtracting from the row and column totals.

Time series

• Plot time horizontally, join points with straight lines.
• Describe the trend first (rising, falling, steady), then the seasonal variation (for example a peak in the same quarter each year).
• To predict, compare the same period in earlier years and extend the trend, and say the prediction assumes the pattern continues.

Cumulative frequency (Higher tier only)

Method in steps:

1. Add a running total column.
2. Plot each total against the upper class boundary; add (lowest boundary, 0).
3. Join with a smooth curve.
4. Median at n/2, lower quartile at n/4, upper quartile at 3n/4. Read across to the curve, then down to the axis.
5. Interquartile range = upper quartile − lower quartile.
6. “More than x”: total − cumulative frequency at x.

Worked reminder: 45 values, classes of width 10 from 0 to 40, frequencies 5, 12, 20, 8. Cumulative: 5, 17, 37, 45. The median is the 22.5th value, so it lies in the 20 to 30 class.

Histograms (Higher tier only)

density = frequency ÷ width
frequency = density × width

• Bars touch. Bar widths match class widths. Area represents frequency.
• Judge frequency by area, never by height when widths differ.
• For part of a bar, assume an even spread: part width × frequency density.
• If the vertical axis has no scale, use one bar with known frequency to find the frequency for one unit of area.

Worked reminder: class 20<x≤50 with frequency 45 has width 30 and frequency density 45 ÷ 30 = 1.5.

Worked reminders

Pie chart. 24 pupils, 5 chose walking. 360 ÷ 24 = 15° per pupil, so walking is 5 × 15 = 75°.

Histogram table (Higher tier only). Class 0<x≤4 has frequency 10: width 4, frequency density 10 ÷ 4 = 2.5. Class 4<x≤10 has frequency 27: width 6, frequency density 27 ÷ 6 = 4.5. Going backwards, a class of width 8 with frequency density 1.5 has frequency 8 × 1.5 = 12.

Reading a cumulative frequency graph (Higher tier only). For 80 values, the median is at 40, the lower quartile at 20 and the upper quartile at 60. Draw a horizontal line from each of those heights to the curve, then drop a vertical line to the horizontal axis and read the value. Write “about” with the answer.

Fixing a flawed survey. Name the flaw, then the repair:

Flaw	Repair
Only asking people at one event	Number the whole population and pick with random numbers
Volunteers only	Choose people, do not let them choose themselves
Leading question	Use neutral wording with balanced response options
Very small sample (unreliable rather than biased)	Take a larger sample from the same frame

Describing a time series. Use a short model answer: “Sales are highest in the third quarter of every year and lowest in the fourth, which is seasonal variation. Each year is higher than the one before, so the trend is upward.”

Must-know distinctions

• Bar chart vs histogram: a bar chart has gaps and shows categories or single values; a histogram has no gaps and shows continuous grouped data, with area for frequency.
• Trend vs seasonal variation: trend is the long-run direction; seasonal variation repeats within each period.
• Population vs sample: conclusions about the population come from the sample, so a biased sample gives unreliable conclusions.
• Multiple vs composite bars: multiple bars sit side by side; composite bars stack.

Quick self-test

1. State the difference between a population and a sample.
2. A survey of leisure habits stands outside a gym on Monday mornings. Give one reason the sample may be biased.
3. A town has 3,000 voters. In a random sample of 40, 12 support a new library. Estimate how many voters support it.
4. In a pie chart of 40 people, 9 chose “cycling”. Find its angle.
5. In a pictogram, one symbol stands for 6 pupils. How many pupils are shown by 3.5 symbols?
6. In a composite bar chart, a section runs from 40 to 72 on the scale. What frequency does it show?
7. (Higher tier only) 45 values are grouped in classes 0<x≤10, 10<x≤20, 20<x≤30 and 30<x≤40 with frequencies 5, 12, 20 and 8. Write the cumulative frequencies, and say which class contains the median.
8. (Higher tier only) For the same data, estimate how many values are more than 25. Assume the values in a class are spread evenly.
9. (Higher tier only) A class has width 25 and frequency density 0.8. Find its frequency.
10. (Higher tier only) A histogram bar for 10<a≤30 has height 3. Estimate the number with 24<a≤30.

Answers

1. The population is everything the study is about; a sample is a smaller group taken from it.
2. Gym users on Monday mornings are not typical of everyone, so some groups are left out.
3. 12/40 × 3000 = 900.
4. 9/40 × 360 = 81°.
5. 3.5 × 6 = 21 pupils.
6. 72 − 40 = 32.
7. 5, 17, 37, 45. The median is the 22.5th value, which lies in the 20<x≤30 class.
8. Cumulative frequency at 25 is 17 + 0.5 × 20 = 27, so 45 − 27 = 18.
9. 25 × 0.8 = 20.
10. 6 × 3 = 18.

Where marks are usually lost

• Writing “random” without saying how every member gets an equal chance.
• Giving a bias reason that does not link to the sample (say who is missed, and why).
• Pie angles rounded so that they no longer total 360°.
• Reading the top of a composite section as its value.
• Plotting cumulative frequency at midpoints instead of upper boundaries.
• Forgetting to start the cumulative frequency graph at zero.
• Using histogram bar height as frequency when class widths differ.
• Answers read from a graph with no reading lines drawn, or stated with false precision.
• Describing a time series with “it goes up and down” and no mention of trend or the repeating pattern.

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.