Statistical distributions — Edexcel A-Level Mathematics
Test yourself on Statistical distributions with PEARSON EDEXCEL A-Level practice questions.
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Statistical distributions explained
A discrete probability distribution lists each possible value of a random variable with its probability, and the probabilities sum to 1.
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Simple distributions include the uniform distribution on a finite set, such as a fair die with P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6. The binomial distribution models the number of successes in n independent trials, each with the same success probability p, written X ~ B(n, p). Its probabilities are P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ, where C(n, r) is the binomial coefficient. For example, X ~ B(10, 0.3) gives P(X = 4) = C(10, 4) × 0.3⁴ × 0.7⁶ ≈ 0.2001. You must recognise when the binomial model is appropriate and calculate single, cumulative and complementary probabilities. Calculation of the mean and variance of discrete random variables is excluded from this specification content.
4.2 Understand and use the Normal distribution as a model; find probabilities using the Normal distribution. Link to histograms, mean, standard deviation, points of inflection and the binomial distribution.
The Normal distribution models continuous data with a roughly symmetric, bell-shaped histogram. The curve is defined by its mean, μ, and standard deviation, σ. Points of inflection occur at μ ± σ; reading left to right, the curve changes from convex to concave at μ − σ, and from concave to convex at μ + σ. To find probabilities, use the cumulative Normal function on a calculator, or standardise using z = (x − μ)/σ. The Normal distribution can approximate a binomial distribution B(n, p) when n is large and p is close to 0.5, using μ = np and σ² = np(1 − p). Since the binomial is discrete and the Normal is continuous, a continuity correction must be applied; e.g. P(X ≥ x) becomes P(X > x − 0.5).
4.3 Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.
Choosing a model means matching the context's structure to a distribution's assumptions. Use the binomial when there is a fixed number of independent trials, two outcomes per trial, and a constant probability of success; use the Normal for continuous, roughly symmetric data, often summarised by a mean and standard deviation. Justify the choice by naming these features. Also recognise failures: the binomial is inappropriate if trials are not independent, the number of trials is not fixed, or p changes; the Normal is inappropriate for strongly skewed data, outliers, or discrete counts. A binomial with small n or p far from 0.5 is poorly approximated by a Normal. State the model, its parameters, and the reason in context.
Your focus
- Identify simple discrete probability distributions and verify that probabilities sum to 1.
- Recognise when a binomial distribution is an appropriate model and state its parameters.
- Calculate single, cumulative and complementary binomial probabilities and interpret them in context.
Show all 9 objectives
- Sketch a Normal curve and identify its mean, standard deviation and points of inflection.
- Calculate cumulative, upper-tail and interval probabilities using standardisation or calculator functions.
- Apply the Normal approximation to a binomial distribution, including the use of a continuity correction.
- Match a described context to a binomial or Normal model and state its parameters.
- Justify a distribution choice using the underlying assumptions in context.
- Identify and explain when the binomial or Normal model would not be appropriate.
Statistical distributions exam tips
Marking Points
- Defines a discrete probability distribution as a set of values with probabilities that sum to 1.
- Recognises the conditions for a binomial model: a fixed number of independent trials, two outcomes per trial, and a constant probability of success.
- States the distribution using notation such as X ~ B(n, p) and identifies n and p from the context.
- Calculates probabilities using P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ, including cumulative and complementary cases.
- Uses the binomial distribution as a model for a real situation and comments on whether its assumptions are reasonable.
- Interprets a calculated probability in the context of the problem rather than leaving it as a bare number.
- Identifies the Normal distribution as a continuous model for symmetric, bell-shaped data, linking μ to the centre and σ to the spread.
- Uses the points of inflection at μ ± σ to describe the shape and relates the histogram's symmetry to the fitted Normal curve.
- Calculates cumulative, upper-tail and interval probabilities correctly using standardisation z = (x − μ)/σ or calculator functions.
- Approximates a binomial distribution using a Normal model with μ = np and σ² = np(1 − p), stating the conditions of large n and p near 0.5.
- Applies a correct continuity correction when using the Normal approximation to the binomial, such as P(X > x) becoming P(X > x + 0.5).
- Identifies a fixed number of trials, two outcomes and constant probability as the conditions for a binomial model.
- Identifies continuous, roughly symmetric data with a mean and standard deviation as suitable for a Normal model.
- Gives a contextual reason for the chosen distribution, referring to the features of the situation rather than naming the distribution alone.
- Recognises and explains at least one situation where the binomial or Normal model is not appropriate, such as dependent trials or heavily skewed data.
- States the parameters of the chosen model, for example B(n, p) or N(μ, σ²), and checks they are consistent with the context.
Examiner Tips
- 💡Write down X ~ B(n, p) with the values of n and p before calculating, so the model is explicit.
- 💡For 'at least' or 'at most' questions, decide whether the complement gives fewer terms and state which values are included.
- 💡Check that the answer is a probability between 0 and 1 and that it is plausible in the context.
- 💡Always sketch the Normal curve, mark the mean μ and the boundary value, and shade the required region to check if your probability makes sense.
- 💡Write down the calculator input or standardisation formula explicitly so the method is visible for partial credit if the final answer is incorrect.
- 💡Underline the modelling clues in the question, such as fixed number of trials or continuous measurements, before choosing.
- 💡Write one sentence naming the distribution, its parameters, and the contextual reason.
- 💡If the question asks whether a model is appropriate, give a balanced judgement covering both supporting and conflicting evidence.
Common Mistakes
- Using the binomial formula when trials are not independent, such as sampling without replacement from a small population; the correction is to use a suitable conditional or hypergeometric approach.
- Confusing P(X ≤ r) with P(X = r); the correction is to add the individual probabilities for all values up to r, or to use the complement where easier.
- Swapping n and p in the formula; the correction is to identify n as the number of trials and p as the probability of success on one trial before substituting.
- Using the variance σ² instead of the standard deviation σ when standardising or entering parameters into a calculator; correct by taking the square root of the variance.
- Forgetting to apply a continuity correction when approximating a discrete binomial distribution with a continuous Normal distribution; correct by adjusting the boundary by 0.5.
- Applying the wrong continuity correction direction, such as using x − 0.5 instead of x + 0.5 for P(X > x); correct by sketching the discrete bars to see which area is required.
- Choosing the binomial for trials whose probability changes between trials; correct by checking that p is constant before selecting it.
- Using the Normal for discrete count data with a small mean; correct by considering a Poisson or binomial model instead.
- Naming a distribution without justification; correct by linking each assumption to a feature of the context.