OCR A Level Mathematics A: Sampling and interpreting single-variable data (H240)

This guide teaches specification sections 2.01a to 2.01d (statistical sampling) and 2.02a, 2.02b and 2.02f to 2.02j (single-variable data) of the OCR A Level Mathematics A (H240) specification, Version 3.1 (August 2026). All of this content is in Stage 1, the column shared with AS Level, and there is no A Level only item here. The specification places Statistics in component 02 and says some items there are set in the context of the pre-release large data set, so practise reading summary statistics and diagrams from real data.

Scatter diagrams, correlation and regression (2.02c to 2.02e) are taught separately and are not repeated here. Use this guide with the revision notes and the practice questions. The OCR A Level Mathematics hub and the printable checklist show the whole course, the Statistics overview sets this unit in context, and the free 10-minute diagnostics show where to start.

What you must be able to do

Ref	Outcome
2.01a	Use the terms population and sample
2.01b	Use samples to make informal inferences about the population
2.01c	Use simple random sampling and opportunity sampling
2.01d	Select or critique sampling techniques in context; know that different samples can lead to different conclusions. Systematic, stratified, cluster and quota sampling are to be critiqued, not carried out
2.02a	Interpret tables and diagrams for single-variable data, including non-standard ones
2.02b	Know that histogram area represents frequency; link histograms to probability distributions; weigh advantages and disadvantages of diagrams
2.02f	Calculate and interpret mean, median, mode, percentiles, quartiles, IQR, standard deviation and variance
2.02g	Mean and standard deviation from a list, summary statistics or a frequency distribution
2.02h	Recognise and interpret possible outliers
2.02i	Select or critique data presentation
2.02j	Clean data: missing data, errors and outliers

Population, sample and inference (2.01a, 2.01b)

A population is every item or person you want information about. A sample is the part of it you actually observe. Observing the whole population is a census. A census is often too slow or costly, or it destroys the items tested, so you sample.

A sample gives an informal inference: an estimate that something is probably true of the population, never a certainty. Different samples give different estimates.

Worked example 1. A theatre has 900 season-ticket holders. A sample of 60 is asked whether they would attend a matinee, and 21 say yes.

estimate = 21/60 × 900 = 315 holders

A second sample of 60 finds 15 saying yes.

estimate = 15/60 × 900 = 225 holders

The two samples disagree by 90 holders, yet both may be fair. This is the point of 2.01d. Larger samples usually vary less, but no size repairs a biased method.

Sampling techniques (2.01c, 2.01d)

For random samples, the specification lets you assume the population is large enough to sample without replacement unless told otherwise.

Simple random sampling (carry out). Each member has an equal chance of selection, and each possible sample of the chosen size can occur with equal probability. You need a sampling frame, a list of the population.

1. Number the members 1 to N.
2. Generate random numbers in that range.
3. Skip repeats and numbers outside the range. Stop when you have n members.

Worked example 2. A register lists 480 students, numbered 001 to 480. Random three-digit numbers come out as 417, 962, 083, 417, 255, 391, 504, 138, 462. Reject 962 (too big), the second 417 (repeat) and 504 (too big). The sample of six is students 417, 083, 255, 391, 138, 462.

Opportunity sampling (carry out). You take whoever is available and willing, such as the first 30 people to walk past. It is quick and cheap and needs no frame, but it is not random, so it is likely to be biased towards people who are in that place at that time.

Methods to critique. You need to know how these work, but you are not required to carry them out.

Method	How it works	Strength	Weakness
Systematic	Every kth member from a random start	Simple, spreads across the list	Can clash with a pattern in the list; not every sample is possible
Stratified	Population split into groups; each sampled in proportion to its size	Every group represented	Needs a full frame with group labels
Cluster	Choose whole groups at random and survey all in them	Cheap when the population is spread out	Clusters may differ from the population
Quota	Interviewer fills set numbers per group	No frame needed, quick	Not random; the interviewer picks who to ask

For the theatre, 360 holders sit in the stalls, 330 in the circle and 210 in the balcony. A stratified sample of 60 takes 24, 22 and 14 respectively, because 360/900 × 60 = 24 and so on. A systematic sample of 60 uses every 15th holder (900/60 = 15) from a random start between 1 and 15.

How to critique in context. Name the method, then name what it does to this population (who could be missed, which pattern could interfere), then say what the conclusion should be. Writing only “it is biased” gives a reader little to credit. Say who is under-represented and why.

Interpreting diagrams (2.02a, 2.02b, 2.02i)

Diagram	Best for	Limitation
Vertical line chart, dot plot	Small discrete data sets	Hard to read with many values
Bar chart	Categories	Not for continuous data
Stem-and-leaf	Keeps every value; shows shape	Awkward with large sets
Box-and-whisker	Comparing median, spread and skew	Hides the individual values and modes
Cumulative frequency	Medians, quartiles, percentiles	Shape is harder to see
Histogram	Shape of continuous grouped data	Original values are lost

The specification includes non-standard representations. Read the axis labels, the units and the scale before anything else, and ask what a bar or area stands for.

Histograms

In a histogram area represents frequency. With unequal class widths the vertical axis shows frequency density.

density = frequency ÷ width of class
frequency = density × width of class

Worked example 3. Waiting times (minutes) at a clinic for 110 patients.

Time t	0 to <5	5 to <10	10 to <20	20 to <40	40 to <60
Frequency	12	20	36	30	12
Frequency density	2.4	4.0	3.6	1.5	0.6

Estimate how many waited between 15 and 30 minutes.

15 to 20: 5 × 3.6 = 18
20 to 30: 10 × 1.5 = 15
total ≈ 33 patients

This assumes the patients are spread evenly inside each class, so it is an estimate.

Link to probability. If each frequency is divided by the total (110) before dividing by width, the total area is 1, and area between two values is a probability. Here P(T < 10) = (12 + 20)/110 = 32/110 ≈ 0.291. This is how a histogram of real data suggests the shape of a probability distribution.

Measures of average and spread (2.02f, 2.02g)

Mean = Σx/n, or Σfx/Σf. Median is the middle value; the mode is the most frequent value (modal class for grouped data). The lower quartile Q1 is a quarter of the way through the ordered data, the upper quartile Q3 is three quarters of the way, and a percentile is the same idea at any percentage. The interquartile range is IQR = Q3 − Q1.

Different methods exist for finding quartiles in a short list, so state the method you use. In the list examples below, n is chosen so that the usual methods agree.

Variance is found by averaging the squared distances of the values from the mean. Standard deviation is its square root, which is why the specification calls it the root mean square deviation from the mean.

sd = √[ Σ(x − x̄)²/n ] = √[ Σx²/n − x̄² ]
sd = √[ Σf(x − x̄)²/Σf ] = √[ Σfx²/Σf − x̄² ]

You must use calculator statistical functions, and the specification asks you to know what the symbols on your own calculator represent. Check which of its standard deviation outputs divides by n, because that matches these formulae.

Worked example 4 (summary statistics). Twenty readings have Σx = 300 and Σx² = 4820.

mean = 300/20 = 15
variance = 4820/20 − 15² = 241 − 225 = 16
sd = 4

Worked example 5 (grouped data). Use the clinic data and class midpoints 2.5, 7.5, 15, 30 and 50.

Σfx  = 30 + 150 + 540 + 900 + 600 = 2220
Σfx² = 75 + 1125 + 8100 + 27000 + 30000 = 66300
mean = 2220/110 ≈ 20.2
variance = 66300/110 − 20.18² ≈ 195.4
sd ≈ 14.0

For grouped data both values are estimates, because one midpoint is used to represent all the values in a class. For the median, interpolate. Half of 110 is 55, and 32 patients are below 10, so the median lies in the 10 to <20 class.

median ≈ 10 + (55 − 32)/36 × 10 ≈ 16.4

The same method gives Q1 ≈ 8.9 (position 27.5) and Q3 ≈ 29.7 (position 82.5), so the IQR is about 20.8. The mean (20.2) exceeds the median (16.4), which fits the long right tail.

Comparing distributions. Comment on both a measure of average and a measure of spread, in context. If two classes both average 62 on a test but have standard deviations 8 and 14, the second class is more spread out and less consistent. The median and IQR are not pulled by extreme values, while the mean and standard deviation are.

Outliers and cleaning data (2.02h, 2.02j)

The specification asks you to know two definitions. A value is a possible outlier if it is

1. more than 1.5 × IQR beyond the nearer quartile, or
2. more than 2 standard deviations from the mean.

Use the definition a question gives. If none is given, state the one you use.

Worked example 6. A courier’s parcels delivered on 11 days, in order: 14, 17, 18, 21, 22, 23, 25, 26, 28, 31, 52.

median = 23 (6th value), Q1 = 18 (3rd), Q3 = 28 (9th), IQR = 10
upper fence: 28 + 1.5 × 10 = 43
lower fence: 18 − 15 = 3

52 exceeds 43, so it is an outlier by definition 1. Also Σx = 277, mean ≈ 25.18, sd ≈ 9.71, and mean + 2sd ≈ 44.6, so 52 is an outlier by definition 2 as well. Without 52 the mean falls from 25.2 to 22.5. The median barely moves (23 to 22.5).

Cleaning data. An outlier is only a flag, not a reason to delete.

• Errors: if you can show a value is impossible or mis-typed (for example 52 typed for 25), correct it from the source or remove it, and say so.
• Outliers that may be genuine: keep them and comment, or report results with and without them.
• Missing data: do not enter 0. Exclude those cases from the calculation, state the new n, and ask whether the gaps are random or hide a pattern.

If 52 was a typing error for 25, the mean becomes (277 − 52 + 25)/11 ≈ 22.7 and the standard deviation about 4.79.

Common errors

• Plotting frequency, not frequency density, on a histogram with unequal widths.
• Using class boundaries instead of midpoints for grouped mean and standard deviation.
• Forgetting to subtract the mean squared: Σx²/n is not the variance.
• Quoting a grouped mean or standard deviation as exact.
• Calling every outlier an error, or removing one without giving a reason.
• Writing “it is random” for an opportunity or quota sample.

Official syllabus

The OCR A Level in Mathematics A (H240) specification, Version 3.1 (August 2026), published by OCR, sections 2.01 and 2.02.