N: Statistical distributions: 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 idea

The Normal distribution models continuous data with a symmetric bell-shaped curve, defined by mean μ and variance σ^2, written X ~ N(μ, σ^2). Probabilities are areas under the curve, found with a calculator. Standardisation Z = (X - μ)/σ transforms any Normal variable to the standard Normal; it is used generally, and is essential when μ or σ are unknown. For example, if X ~ N(50, 4^2), then P(X < 55) = P(Z < (55 - 50)/4) = P(Z < 1.25), which a calculator evaluates directly. Link the curve to histograms: as sample size increases, a histogram of Normal data approximates the smooth curve. The mean is the centre of symmetry, and points of inflection occur at μ ± σ. The Normal distribution can approximate the binomial when n is large and p is close to 0.5; for AQA A-Level Mathematics, no continuity correction is required.

What earns the marks

• State the Normal distribution using notation X ~ N(μ, σ^2) with correct mean and variance.
• Find probabilities directly using calculator statistical functions for given ranges and inequalities.
• Standardise a value correctly using Z = (X - μ)/σ, both to find probabilities and to find unknown parameters μ or σ.
• Interpret the mean as the centre of symmetry and the standard deviation as the distance from the mean to a point of inflection.
• Apply the Normal distribution to approximate a binomial distribution when n is large and p is close to 0.5, using the same mean and variance as the binomial (μ = np, σ^2 = np(1 - p)).

Common mistakes and how to put them right

• Confusing variance and standard deviation.

  Put it right: The second parameter is the variance, so square the standard deviation.

• Inputting the variance instead of the standard deviation into the calculator when finding probabilities.

  Put it right: Check if the calculator requires σ or σ^2 and input accordingly.

• Using the wrong mean or variance in the Normal approximation to the binomial.

  Put it right: Match the binomial mean np and variance np(1 - p) to the approximating Normal distribution.

• Using standardisation unnecessarily for basic probabilities.

  Put it right: Use the calculator's Normal cumulative distribution function directly unless μ or σ is unknown, but remember standardisation is a valid general method.

Exam tips

• Always sketch the Normal curve and shade the required region to avoid errors with inequality directions.
• When finding unknown parameters, write down the standardisation step clearly, showing the substitution into Z = (X - μ)/σ.
• State the binomial parameters n and p before writing down the Normal approximation to show your method clearly.