What does the Statistics option in OCR Further Maths A cover?

The optional Statistics paper (Y542) covers discrete random variables (probability distributions, expectation, variance and linear transformations), continuous random variables (the probability density function, cumulative distribution function, and expectation and variance by integration), the Poisson distribution (conditions, mean and variance, sums of independent Poisson variables and the approximation to the binomial) and the geometric distribution, and chi-squared tests (goodness of fit and contingency tables) together with non-parametric tests such as the sign test and the Wilcoxon signed-rank test. It is one of the two optional papers a candidate may choose.

Which Statistics topics carry the most marks in OCR Further Maths A?

The chi-squared tests and the continuous random variable integration are the heaviest, because each is a multi-step structured question. The Poisson distribution is a reliable source of marks, often combined with the binomial approximation or a goodness-of-fit test. Fluency with the expectation and variance formulae (sums for discrete, integrals for continuous) and the chi-squared test statistic is the single biggest enabler in this option.

What techniques recur across the OCR Statistics content?

Imposing that probabilities sum to one (or a pdf integrates to one) to find a constant, computing expectation and variance by the sum or integral formulae and the E(X squared) minus mean squared rule, applying the aX plus b rules for mean and variance, using the Poisson and geometric probability formulae, combining independent Poisson variables, and computing the chi-squared statistic with the correct expected frequencies and degrees of freedom. These are automatic by exam day.

What is the most common Statistics mistake in OCR Further Maths A?

Forgetting to square the multiplier in Var(aX + b), and using the wrong degrees of freedom in a chi-squared test. Other frequent errors are confusing E(X squared) with the square of the mean, not normalising a pdf before integrating, mixing up the Poisson and binomial conditions, getting the geometric exponent wrong, and not combining small expected frequencies. Always check the distribution sums or integrates to one and match the degrees of freedom to the test.

How should I revise the OCR Statistics option?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: the expectation and variance formulae for discrete and continuous variables, the aX plus b rules, the Poisson and geometric formulae and means, and the chi-squared statistic with both degrees-of-freedom rules. Always normalise a distribution first, and choose the right test for the data.

EnglandFurther Maths

OCR A-Level Further Maths A Statistics option: random variables, distributions and hypothesis tests

A deep-dive OCR A-Level Further Mathematics A guide to the optional Statistics paper (Y542): discrete and continuous random variables, the Poisson and geometric distributions, and chi-squared and non-parametric tests, with the techniques OCR repeats in the Statistics option.

Generated by Claude Opus 4.818 min readH245/Y542Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What the Statistics option demands
Random variables
Distributions and tests
How the Statistics content is examined
Check your knowledge

What the Statistics option demands

The Statistics option of OCR A-Level Further Mathematics A (H245) is one of the four optional papers (Y542), of which a candidate chooses two. It develops probability and inference beyond A-Level Mathematics, with discrete and continuous random variables, named distributions and a range of hypothesis tests. The examiners reward correct set-up, accurate integration or summation, and clear hypotheses and conclusions.

This guide walks through the four Statistics topics in a logical order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Random variables

Discrete random variables describes a variable by its probability distribution, computes the expectation and variance (with the $\mathrm{E}(X^2) - [\mathrm{E}(X)]^2$ formula), and gives the $aX + b$ rules for the mean and variance. Continuous random variables uses a probability density function (integrating to one), finds probabilities by integration, relates the pdf to the cumulative distribution function, and computes the expectation and variance by integration, mirroring the discrete case.

Distributions and tests

The Poisson and geometric distributions models counts at a constant rate (Poisson, mean equals variance equals $\lambda$ , with sums of independent Poisson variables and the approximation to the binomial) and waiting times (geometric, mean $\tfrac{1}{p}$ ). Chi-squared and non-parametric tests carries out the goodness-of-fit and contingency table tests with the right degrees of freedom, and the sign and Wilcoxon signed-rank tests where distributional assumptions fail.

How the Statistics content is examined

A typical OCR profile for this option:

Random variable questions. Find a constant, then the mean and variance, for a discrete or continuous variable.
Distribution questions. Compute a Poisson or geometric probability, or use the Poisson approximation.
Chi-squared questions. Carry out a goodness-of-fit or contingency table test.
Non-parametric questions. Apply the sign test or Wilcoxon test to data without a known distribution.

The OCR Statistics option covers discrete and continuous random variables, the Poisson and geometric distributions, and chi-squared and non-parametric tests, as one of the optional papers. Learn the expectation and variance formulae (sums for discrete, integrals for continuous) with the $\mathrm{E}(X^2) - [\mathrm{E}(X)]^2$ rule, the $aX + b$ rules, the Poisson formula ( $\mathrm{E} = \mathrm{Var} = \lambda$ ) and geometric formula (mean $\tfrac{1}{p}$ ), and the chi-squared statistic $\sum\frac{(O-E)^2}{E}$ with degrees of freedom $n - 1 - m$ (goodness of fit) or $(r-1)(c-1)$ (contingency). Always normalise first, square the multiplier in the variance, and choose the right test and degrees of freedom.

Check your knowledge

A mix of recall and technique questions covering the Statistics content. Attempt them under timed conditions, then check against the solutions.

State the computational formula for the variance of a discrete random variable. (1 mark)
State $\mathrm{Var}(aX + b)$ . (1 mark)
State the Poisson probability formula. (1 mark)
State the mean of a geometric distribution with success probability $p$ . (1 mark)
State the chi-squared test statistic. (1 mark)
State the degrees of freedom for an $r \times c$ contingency table. (1 mark)

Solutions

Step 1: Recall the variance formula

Variance is defined as the expected value of $X^2$ minus the square of the expected value of $X$ . This is derived from the definition $\mathrm{Var}(X) = \mathrm{E}[(X - \mu)^2]$ expanded out.

\mathrm{Var}(X) = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2

Step 2: Apply the linear transformation rule for variance

Adding a constant $b$ shifts all values but does not affect spread, so it does not change variance. Multiplying by $a$ scales deviations by $a$ , so variance scales by $a^2$ .

\mathrm{Var}(aX + b) = a^2\mathrm{Var}(X)

Step 3: State the Poisson probability formula

The Poisson distribution models the probability of exactly $x$ events in a fixed interval, given an average rate $\lambda$ . The factor $e^{-\lambda}$ normalises the distribution so all probabilities sum to 1.

\mathrm{P}(X = x) = e^{-\lambda}\dfrac{\lambda^x}{x!}

Step 4: State the mean of a geometric distribution

The geometric distribution counts the number of trials until the first success. On average, you need $1/p$ trials, because each trial independently succeeds with probability $p$ .

\mathrm{E}(X) = \dfrac{1}{p}

Step 5: State the chi-squared test statistic

The chi-squared statistic measures the total discrepancy between observed and expected frequencies. Squaring the differences removes sign, and dividing by $E$ standardises each cell's contribution relative to its expected size.

\chi^2 = \displaystyle\sum \dfrac{(O - E)^2}{E}

Step 6: State the degrees of freedom for a contingency table

In an $r \times c$ table, each row and column total is fixed, leaving $(r-1)(c-1)$ cells free to vary. This determines the appropriate chi-squared distribution to compare the test statistic against.

(r - 1)(c - 1)

Sources & how we know this

OCR A Level Further Mathematics A (H245) specification — OCR (2017)