Ch 5 · 5.5 The Normal Distribution · A Level Mathematics 9709
Topic 5 of 5 in Probability & Statistics 1 — A Level Mathematics 9709 — 60 lessons.
Introduction
Heights of adults, masses of apples from one orchard, errors in measurements: plot enough of them and you get a symmetric bell-shaped curve, the normal distribution. It is the most-used model in statistics. In Paper 5 you find probabilities from it, work backwards from probabilities to unknown means or standard deviations, and use it as a shortcut for binomial calculations with large n. (Syllabus section 5.5, Paper 5.)
Key ideas — The normal model
X sim N(μ, σ^2) means X is normally distributed with mean μ and variance σ^2 (standard deviation σ). The curve is symmetric about μ, and about 68% of values lie within σ of the mean, 95% within 2σ. X is continuous, so P(X = a) = 0 and P(X < a) = P(X ≤ a). Probabilities are areas under the curve: a quick sketch with the area shaded avoids sign errors.
🧑🏫 Teacher's explanation
Imagine you are measuring the heights of every student in a massive school. Most students will be around an average height, with fewer and fewer people being extremely tall or extremely short. If you plot these heights on a graph, they naturally form a symmetric, bell-shaped curve. This is the essence of the Normal Distribution.
📖 Definitions
🔍 Worked example — Easy
Given X sim N(50, 10^2), find P(X > 65).
🔍 Worked example — Standard
A variable X follows a normal distribution with mean μ = 100 and variance σ^2 = 25. Find the value of k such that P(X < k) = 0.975.
🔍 Worked example — Exam-style
A factory produces bolts with diameters following a normal distribution with mean 20 mm and standard deviation 0.5 mm. A bolt is rejected if its diameter is less than 19.2 mm or greater than 20.8 mm. Find the probability that a bolt is rejected.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Standardising
Every normal variable converts to the standard normal Z sim N(0, 1):
🧑🏫 Teacher's explanation
Imagine you are comparing two different tests. In Test A, the average score is 50 with a standard deviation of 10. In Test B, the average is 70 with a standard deviation of 15. If you score 65 on Test A and 80 on Test B, which performance is actually 'better' relative to the group? You cannot compare them directly because they use different scales. Standardisation is the mathematical process of converting these different scales into a single, universal 'language' called the Standard Normal Distribution, where the mean is always 0 and the standard deviation is always 1.
📖 Definitions
🔍 Worked example — Easy
Given X sim N(50, 10^2), find P(X > 65).
🔍 Worked example — Standard
Given X sim N(μ, 4^2), find the value of μ such that P(X < μ + 5) = 0.95.
🔍 Worked example — Exam-style
A continuous random variable X follows a normal distribution with mean 100 and variance 25. Find P(98 < X < 105).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Working backwards
Given a probability, find z from the tables (or the critical values table: e.g. z = 1.282 for 10% in one tail, 1.645 for 5%, 1.96 for 2.5%), then solve x - μ/σ = z. Two pieces of information give two equations, which find both μ and σ. A value below the mean has a negative z.
🧑🏫 Teacher's explanation
Think of the Normal Distribution as a bell-shaped curve that describes how data is spread out. Usually, we know the center (the mean, μ) and the spread (the standard deviation, σ), and we want to find the probability of an event. 'Working backwards' is like knowing how much area is under the bell curve and trying to figure out where the curve is positioned or how wide it is.
📖 Definitions
🔍 Worked example — Easy
A continuous random variable X follows a normal distribution with mean μ = 50 and standard deviation σ = 10. Find the value of x such that P(X > x) = 0.15.
🔍 Worked example — Standard
The weights of bags of flour follow a normal distribution. The mean weight is 1 ext kg and the standard deviation is 0.05 ext kg. Find the value of σ if P(X > 1.1) = 0.025.
🔍 Worked example — Exam-style
A variable X is normally distributed. When μ = 20, P(X < 15) = 0.0228. Find the value of μ when σ = 5 and P(X < 15) = 0.1587.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Normal approximation to the binomial
If X sim B(n, p) with n large enough that np > 5 and nq > 5 (where q = 1 - p), then approximately
🧑🏫 Teacher's explanation
Imagine you are trying to count the number of people who walk into a shop. The Binomial distribution is like looking at individual people—it's 'chunky' and discrete. You can have 10 people or 11 people, but never 10.5. However, if the number of people is very large, those 'chunks' become so small relative to the total that the distribution starts to look like a smooth, continuous wave. This smooth wave is the Normal distribution.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled 100 times. Let X be the number of times a '6' is rolled. Use a normal approximation to find P(X ≤ 55).
🔍 Worked example — Standard
In a large population, 10% of people are left-handed. A sample of 200 people is taken. Use a normal approximation to find the probability that more than 25 people are left-handed.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: probabilities
X sim N(50, 4^2). Find P(X > 56) and P(45 < X < 56).
🔍 Worked example — Easy
X sim N(50, 4^2). Find P(X > 56).
🔍 Worked example — Standard
X sim N(50, 4^2). Find P(45 < X < 56).
🔍 Worked example — Exam-style
Given X sim N(μ, σ^2), find the value of μ if P(X > 70) = 0.15 and σ = 5.
⚠️ Common errors
Worked examples — Example 2: a value from a probability
For the same X, find a such that P(X < a) = 0.9.
🔍 Worked example — Easy
A continuous random variable X follows a normal distribution with mean μ = 50 and standard deviation σ = 4. Find P(X > 58).
🔍 Worked example — Standard
For X sim N(50, 4^2), find the value of a such that P(X < a) = 0.9.
🔍 Worked example — Exam-style
A machine produces components with lengths X sim N(100, 2^2). A component is rejected if it is shorter than 97mm. Find the probability that a component is rejected. Give your answer to 3 decimal places.
⚠️ Common errors
Worked examples — Example 3: finding μ and σ
X sim N(μ, σ^2) with P(X < 20) = 0.1 and P(X > 40) = 0.2.
🔍 Worked example — Easy
Given X sim N(μ, σ^2), find μ and σ if P(X < 10) = 0.5 and P(X < 15) = 0.8413.
🔍 Worked example — Standard
X sim N(μ, σ^2). Given P(X < 20) = 0.1 and P(X > 40) = 0.2, find μ and σ.
🔍 Worked example — Exam-style
A machine produces bolts with lengths X sim N(μ, σ^2). It is known that P(X < 50) = 0.05 and P(X < 55) = 0.95. Find μ and σ.
⚠️ Common errors
Worked examples — Example 4: approximating a binomial
X sim B(100, 0.4). Estimate P(X ≤ 45).
🔍 Worked example — Standard
X sim B(100, 0.4). Estimate P(X ≤ 45).
⚠️ Common errors
Common mistakes
Practice
Answers: (1) P(Z < -1.333) = 0.0912; P(-0.667 < Z < 1.333) = 0.9088 - 0.2524 = 0.656. (2) L = 8.0 + 1.645 × 0.2 = 8.33 cm. (3) Phi(z) = 0.9772 ⇒ z = 2: 62 = μ + 2 × 5, μ = 52. (4) 35 - μ/σ = -0.674, 50 - μ/σ = 1.282: σ = 7.67, μ = 40.2. (5) N(100, 50): P(Y > 109.5) = P(Z > 1.344) = 0.0895.