OCR GCSE Maths: Statistics & Probability Revision Guide (J560)

Statistics and Probability are, for many students, the most rewarding parts of GCSE Maths to revise. The methods are consistent, the question types are predictable, and a great deal of the content builds on ideas you already use in everyday life -- reading a chart, working out an average, judging how likely something is. For anyone trying to lift their grade reliably, this is fertile ground.

This guide covers the Statistics strand and the Probability strand of the OCR GCSE Mathematics (J560) specification. We will work through the key sub-topics in turn -- data types and sampling, averages and spread, statistical diagrams, the probability scale, combined events, tree and Venn diagrams, and the Higher-tier material -- with worked examples written out in full prose, the pitfalls examiners flag every year, and a clear sense of how OCR phrases its questions. Higher-tier-only content is marked with a [H] flag.

If you want the broad view across the whole subject first, our OCR GCSE Maths revision guide hub sets out the structure of the course; then return here to drill into Statistics and Probability.

How the OCR J560 Papers Work

A quick orientation on the exam, because it shapes good revision -- and OCR has one structural quirk worth knowing.

OCR GCSE Mathematics is assessed by three papers per tier, each worth 100 marks and lasting 1 hour 30 minutes, for a total of 300 marks. Foundation tier (Papers 1-3) covers grades 1 to 5; Higher tier (Papers 4-6) covers grades 4 to 9.

The detail that surprises students is the calculator rule: the non-calculator paper is the middle one of the three -- Paper 2 at Foundation and Paper 5 at Higher. The two outer papers in each tier permit a calculator. Statistics and Probability appear on all three papers, including that non-calculator middle paper, so you cannot leave them to the last minute and you cannot lean on a calculator for every average.

The assessment objectives are weighted by tier:

Assessment objective	What it tests	Foundation	Higher
AO1	Use and apply standard techniques	50%	40%
AO2	Reason, interpret and communicate mathematically	25%	30%
AO3	Solve problems within mathematics and in context	25%	30%

Statistics in particular is a magnet for AO2 marks, because so many questions ask you to interpret and communicate -- to compare two data sets, criticise a sample, or explain what a chart shows. That makes the OCR command words especially important here. "Describe" and "Compare" call for written interpretation; "Give a reason for your answer" tells you a bare value will not score full marks; "Work out" and "Calculate" want a number with working; "Estimate" expects an approximate result, which matters a great deal when you read values off a cumulative frequency curve.

For paper-by-paper tactics, see our OCR GCSE Maths exam technique guide.

The Statistics Strand

Data Types and Sampling

Before analysing data you must classify it and understand how it was gathered.

Data is either qualitative (descriptive, such as colour or favourite subject) or quantitative (numerical). Quantitative data is further split into discrete data, which takes specific separate values (usually counted, such as the number of pets a household owns), and continuous data, which can take any value in a range (usually measured, such as height or mass).

There is also the distinction between primary data, which you collect yourself through experiments, surveys or observation, and secondary data, which someone else has already collected -- from a database or published source. Primary data gives you control over its accuracy but takes time; secondary data is quick to obtain but may carry errors or bias you cannot check.

When a population is too large to survey in full, you take a sample:

• Random sampling gives every member an equal chance of selection.
• Systematic sampling picks every $n$n-th member from an ordered list.
• Stratified sampling [H] divides the population into groups (strata) and samples from each in proportion to its size.

A worked stratified example: a school has 600 students in Year 11 and 400 in Year 10, and you want a sample of 50. The total is 1000, so each stratum contributes in proportion -- Year 11 gives $\frac{600}{1000} \times 50 = 30$1000600​×50=30 students and Year 10 gives $\frac{400}{1000} \times 50 = 20$1000400​×50=20 students.

A biased sample does not represent the population fairly -- perhaps because of where or when it was taken, or because the questions themselves were leading. Good questionnaire design avoids leading questions, overlapping response boxes (such as "1-5" and "5-10", which leave 5 ambiguous), and vague time frames. If a question asks you to describe a problem with a sampling method, name the flaw and explain why it matters: "The sample is biased because it only surveys people leaving a gym, so it over-represents people who exercise."

Averages and Spread

Averages are among the most frequently tested ideas in the whole specification, and you must be able to find them from raw data, from a frequency table, and from a grouped frequency table.

• Mean -- add the values and divide by how many there are. From a frequency table, multiply each value by its frequency, sum the products, and divide by the total frequency.
• Median -- the middle value once the data is in order. For an even count, take the mean of the two middle values; the median position is found at $\frac{n+1}{2}$2n+1​.
• Mode -- the most common value.
• Range -- the largest value minus the smallest; a simple measure of spread that is sensitive to outliers.

Grouped data deserves special care. From a grouped frequency table you can only estimate the mean, because you do not know the exact values within each class. You use the midpoint of each class as a representative value. For example, with the table below:

Mass $m$m (kg)	Frequency $f$f	Midpoint $x$x	$f \times x$f×x
$0 \leq m < 10$0≤m<10	4	5	20
$10 \leq m < 20$10≤m<20	7	15	105
$20 \leq m < 30$20≤m<30	9	25	225

the estimated mean is $\frac{20 + 105 + 225}{4 + 7 + 9} = \frac{350}{20} = 17.5$4+7+920+105+225​=20350​=17.5 kg. From grouped data you can also identify the modal class (the interval with the highest frequency -- here $20 \leq m < 30$20≤m<30), but not an exact mode or median.

At Higher tier [H] you also work with quartiles and the interquartile range (IQR). The lower quartile $Q_1$Q1​ sits a quarter of the way through the ordered data, the upper quartile $Q_3$Q3​ three quarters of the way, and the $\text{IQR} = Q_3 - Q_1$IQR=Q3​−Q1​ measures the spread of the middle half of the data -- which makes it far more resistant to outliers than the range.

Comparing Distributions

A classic OCR question asks you to compare two data sets. The reliable structure is to compare one average and one measure of spread, and to phrase both in context. For instance: "The median mark for Class A is 56, compared with 43 for Class B, so Class A generally scored higher. The interquartile range for Class B is larger, so their marks were more spread out." Vague statements such as "Class A did better" earn nothing on their own -- you must quote values and interpret them.

Statistical Diagrams

OCR expects you to draw, read and interpret a range of charts. Foundation tier covers the standard types; Higher tier adds cumulative frequency diagrams, box plots and histograms with unequal class widths.

Bar charts, pie charts and pictograms are straightforward but still tested, often as interpretation. For a pie chart, the angles must total $360°$360°, and each angle is $\frac{\text{frequency}}{\text{total}} \times 360°$totalfrequency​×360°.

Frequency polygons plot frequency against the midpoint of each class and join the points with straight lines -- handy for comparing two distributions on one set of axes.

Scatter graphs and correlation. A scatter graph plots two variables against each other. Positive correlation means that as one increases, the other tends to increase; negative correlation means one rises as the other falls; no correlation means no clear relationship. A line of best fit should pass close to the points with roughly equal numbers either side and through the mean point. Use it for interpolation -- estimating within the range of the data, which is reliable -- but be wary of extrapolation beyond the data, where the trend may not hold. A useful exam phrase: correlation does not imply causation; two variables can move together without one causing the other.

Cumulative frequency diagrams [H]. Plot the running total of frequencies against the upper boundary of each class, and join with a smooth curve. The curve lets you estimate the median (read across from $\frac{n}{2}$2n​ on the vertical axis), the lower quartile (from $\frac{n}{4}$4n​) and the upper quartile (from $\frac{3n}{4}$43n​). Draw the horizontal line to the curve, then read down to the horizontal axis. Because these are estimates from a curve, the command word "estimate" is doing real work -- a value to the nearest small unit is expected, not a single exact figure.

Box plots [H]. A box plot shows five numbers: minimum, lower quartile, median, upper quartile and maximum. They are excellent for comparison -- line up the medians for central tendency and the box lengths (the IQR) for spread -- and you should always quote specific values when comparing two of them.

Histograms with unequal class widths [H]. This is the topic students most often get wrong. The vertical axis shows frequency density, not frequency, where

$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}.$frequency density=class widthfrequency​.

The area of each bar represents the frequency. So to recover a frequency from a histogram, multiply frequency density by class width. For example, a bar covering the interval $20 \leq x < 40$20≤x<40 (width 20) at a frequency density of 3.5 represents $3.5 \times 20 = 70$3.5×20=70 items. Reading the height of the bar as if it were the frequency is the single most common histogram error.

Everything in this strand -- data, sampling, averages, spread and every diagram type -- is covered step by step in our OCR GCSE Mathematics: Statistics course.

The Probability Strand

The Probability Scale and Single Events

Probability is measured on a scale from 0 (impossible) to 1 (certain), and can be written as a fraction, a decimal or a percentage; fractions and decimals are the most common in the exam. A useful identity is that the probabilities of all possible outcomes sum to 1, so $P(\text{not } A) = 1 - P(A)$P(not A)=1−P(A).

For equally likely outcomes, the theoretical probability of an event is

$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$P(event)=total number of outcomesnumber of favourable outcomes​.

For instance, the probability of rolling a 3 on a fair six-sided die is $P(3) = \frac{1}{6}$P(3)=61​.

When outcomes are not equally likely -- a biased spinner, say -- you estimate probability from data using relative frequency: the number of times the event happened divided by the number of trials. More trials give a more reliable estimate, a principle often called the law of large numbers.

Expected outcomes multiply probability by the number of trials. If $P(\text{red}) = 0.3$P(red)=0.3 and you spin 200 times, you expect $0.3 \times 200 = 60$0.3×200=60 reds -- though, being a prediction, the actual count may differ.

Sample Spaces and Frequency Trees

A sample space diagram lists all possible outcomes of two combined events. Rolling two dice, for example, gives a $6 \times 6$6×6 grid of 36 equally likely combinations; to find the probability of a total of 7, count the combinations that give 7 (there are six) and write $\frac{6}{36} = \frac{1}{6}$366​=61​.

A frequency tree records actual counts as they split through two stages -- for example, 100 patients split into those who tested positive or negative, and then within each branch those who actually had the condition or not. Frequency trees deal in numbers of people, whereas probability trees (below) deal in probabilities; OCR uses both, so read carefully which one a question wants.

Mutually Exclusive and Independent Events

Two events are mutually exclusive if they cannot both happen at once -- rolling a 2 and rolling a 5 on a single die. For mutually exclusive events you add: $P(A \text{ or } B) = P(A) + P(B)$P(A or B)=P(A)+P(B).

Two events are independent if the outcome of one does not affect the other -- flipping a coin and rolling a die. For independent events you multiply: $P(A \text{ and } B) = P(A) \times P(B)$P(A and B)=P(A)×P(B). So the probability of a head and a six is $\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$21​×61​=121​.

A short way to remember it: "and" tends to mean multiply; "or" (for mutually exclusive events) tends to mean add.

Tree Diagrams

Tree diagrams are the most important tool for combined probability at GCSE. Each branch carries the probability of an outcome. Draw the first set of branches for the first event, then a second set from each outcome for the second event. Multiply along the branches to find the probability of a combined outcome, and add the relevant combined outcomes to answer the question.

The key distinction is with or without replacement. If an item is replaced after the first pick, the probabilities on the second set of branches are unchanged. If it is not replaced, they change because the total has fallen by one. Suppose a bag holds 5 red and 3 blue counters and you draw two without replacement. On the first draw $P(\text{red}) = \frac{5}{8}$P(red)=85​. If a red came out, the second draw has 4 red among 7, so $P(\text{red then red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$P(red then red)=85​×74​=5620​=145​. Forgetting to reduce both the favourable count and the total is the classic without-replacement error.

A handy check on any tree: the probabilities on each set of branches must sum to 1, and the probabilities of all the final combined outcomes must also sum to 1.

Venn Diagrams and Set Notation

A Venn diagram shows events and their overlap. The intersection $A \cap B$A∩B ("A and B") is the region in both sets; the union $A \cup B$A∪B ("A or B") is the region in at least one set; the complement $A'$A′ is everything not in $A$A. To read a probability off a Venn diagram, divide the relevant count by the grand total in the rectangle. A frequent slip is forgetting the region outside both circles but inside the rectangle -- those members still count towards the total.

A two-way table does a similar job for two categorical variables, laying out frequencies in rows and columns with totals; to find a probability, locate the relevant cell or cells and divide by the appropriate total.

Conditional Probability [H]

Conditional probability is the probability of one event given that another has already happened, written $P(A \mid B)$P(A∣B) and read "the probability of A given B". It follows naturally from the without-replacement work above.

From a tree diagram with dependent events, the second set of branches already shows conditional probabilities, because they have been adjusted for what happened first. From a Venn diagram, you restrict your attention to the "$B$B" region and ask what fraction of it also lies in $A$A:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$P(A∣B)=P(B)P(A∩B)​.

A worked example: of 30 students, 18 study French, 12 study Spanish and 7 study both. Given that a randomly chosen student studies French, what is the probability they also study Spanish? Restrict to the 18 French students; 7 of them also study Spanish, so $P(\text{Spanish} \mid \text{French}) = \frac{7}{18}$P(Spanish∣French)=187​. Conditional-probability questions are typically worth several marks and sit near the end of a Higher paper, but the method is consistent and the marks are very achievable with careful working.

Everything from the probability scale through to conditional probability is covered, with plenty of tree- and Venn-diagram practice, in our OCR GCSE Mathematics: Probability course.

Common Mistakes Examiners See

The same pitfalls recur on Statistics and Probability questions year after year.

• Misreading a frequency table. Treating data values as frequencies, or adding values instead of multiplying value by frequency when finding the mean.
• Over-claiming from grouped data. From grouped data you can find an estimated mean and a modal class, not an exact mean, mode or median.
• Reading a histogram height as a frequency. The vertical axis is frequency density; frequency is the area of the bar.
• Probabilities outside the range 0 to 1. Any probability above 1 or below 0 is impossible -- a clear sign to recheck.
• Forgetting to simplify a probability when the question asks for simplest form.
• Tree-diagram errors without replacement. Failing to reduce the total on the second set of branches -- if you start with 5 red and 3 blue and remove a red, the second branches must read 4 red and 3 blue out of 7, not out of 8.
• Ignoring the region outside the circles in a Venn diagram, so the totals do not add up.
• Bald comparisons. Writing "Class A did better" without quoting an average and a measure of spread in context.

How These Topics Are Examined

A few habits turn knowledge into marks on OCR papers in particular.

Read the command word and answer it. "Describe" and "Compare" want sentences, not just numbers -- and a comparison needs both an average and a spread, each interpreted in context. "Give a reason for your answer" means you must justify any criticism or conclusion. "Estimate" -- which attaches to cumulative frequency and histogram readings -- expects an approximate value read from the graph, so do not agonise over a single exact figure.

Show working, even when it looks obvious. OCR mark schemes award method marks separately from accuracy marks. Show the multiplication for an "and" probability; draw the construction lines on a cumulative frequency curve; write out the value-times-frequency products for a mean. A small arithmetic slip then still leaves most of the marks intact.

Use the built-in checks. Branch probabilities must sum to 1; Venn-diagram regions (including the one outside the circles) must sum to the total; a probability must lie between 0 and 1. These checks catch errors before they cost you.

Interpretation is a skill, not a formality. Statistics rewards clear communication. Name the flaw in a sample and say why it matters; state which distribution is higher and by reference to the actual values; explain what a correlation does and does not tell you. This is where the AO2 marks live.

Prepare with LearningBro

LearningBro's OCR GCSE Mathematics courses are built around the J560 specification and designed to develop both your technique and your judgement. The Statistics course covers data and sampling, averages and spread, and every diagram on the specification, including the Higher-tier cumulative frequency diagrams, box plots and histograms. The Probability course takes you from the probability scale through sample spaces, mutually exclusive and independent events, tree diagrams, Venn diagrams, frequency trees and conditional probability -- with practice questions that mirror the demand of the real papers, including the non-calculator middle paper.

When the individual topics feel secure, the OCR GCSE Maths exam preparation course moves you onto mixed-topic practice under timed conditions, with a focus on the interpretation and reasoning that earn the AO2 marks. Statistics and Probability are among the most reliable areas to bank marks -- the methods are consistent, the question types are predictable, and the marks are there for the taking.

Related Reading

• OCR GCSE Maths revision guide -- the hub overview of the whole J560 course.
• OCR GCSE Maths: Number, Ratio & Proportion revision guide -- the companion topic guide to this one.
• OCR GCSE Maths: Geometry revision guide -- the other major content strand.
• OCR GCSE Maths exam technique guide -- paper-by-paper tactics, including the non-calculator middle paper.

Good luck with your revision.