About this chapter
The normal distribution closes Statistics 1. The routine probabilities are well answered; what separates the grades is working backwards from a probability to the mean or standard deviation, and conditional questions such as the chance of a value above 65 given that it is above 50.
This chapter covers standardising, reading the normal and percentage points tables, probabilities for ranges, finding the mean or the standard deviation or both, symmetry, conditional probability, and several independent values. The later questions combine the normal distribution with tree diagrams, outliers and expected counts.
The ten questions
- Q1 (5 marks): two probabilities for X ~ N(50, 4²), one of them for a range.
- Q2 (5 marks): a probability for X ~ N(120, 15²), then the value with 5% above it.
- Q3 (5 marks): find σ from a given probability, then use symmetry for a second one.
- Q4 (5 marks): find μ when 20% lies above 40, then another probability.
- Q5 (5 marks): the quartiles and interquartile range of a normal model, then the chance of an outlier.
- Q6 (12 marks): apple masses: proportions, exactly one large apple in three, and a conditional probability.
- Q7 (12 marks): puzzle times: two percentages give two equations in μ and σ, then a range and a conditional.
- Q8 (11 marks): plant heights: show σ = 3.90, a conditional probability, and at least one tall plant in four.
- Q9 (11 marks): bolts from two machines with different distributions, combined on a tree diagram and reversed.
- Q10 (12 marks): σ = μ/4 and one probability give μ = 60, then a conditional probability, a percentage point and an expected count.
Key skills tested
Standardising. If X ~ N(μ, σ²), then Z = (X − μ)/σ follows the standard normal distribution. The second number in the brackets is the variance, not the standard deviation.
Using the table. P(Z < z) = Φ(z). For negative z, Φ(−z) = 1 − Φ(z), and P(Z > z) = 1 − Φ(z).
Ranges. P(a < X < b) is the difference of the two Φ values. Draw a sketch and shade the region.
Percentage points. To go from a probability to a value, use the percentage points table in full: 5% in the upper tail gives z = 1.6449.
Finding μ or σ. Write (x − μ)/σ = z with the right sign (negative below the mean) and solve.
Finding μ and σ. Two probabilities give two equations, which you solve simultaneously.
Symmetry. The curve is symmetrical about μ, so P(X < μ) = 0.5, and a range centred on μ is found from one Φ value.
Conditional probability. For a above b, P(X > a given X > b) = P(X > a) ÷ P(X > b). Find the numerator from the overlap.
Samples and counts. Several independent items multiply, or use 1 minus the complement. The expected number is n times the probability.
Worked example
Question 1 from this chapter. X ~ N(50, 4²). Find (a) P(X < 56) and (b) P(44 < X < 58).
(a) Standardise: (56 − 50) ÷ 4 = 1.5, so P(X < 56) = Φ(1.5) = 0.9332.
(b) Standardise both ends: (44 − 50) ÷ 4 = −1.5 and (58 − 50) ÷ 4 = 2. Then P(44 < X < 58) = Φ(2) − Φ(−1.5) = 0.9772 − 0.0668 = 0.9104.
Here σ = 4, the square root of the 4² in the brackets. Using the variance to standardise is a classic error.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q5: standardising shown in full, z values from the tables, then finding a parameter from a probability, as in Q1 to Q4.
- June 2022, Q6: a normal probability, an expected number, then a conditional probability with a double inequality, as in Q6, Q8 and Q10.
Where marks are lost
- Not showing the standardisation. The June 2024 examiners stressed that when a question asks you to standardise, the standardising must be written out.
- Rounded z values. In June 2024 a common error was using a z value such as 0.25 instead of the 4 decimal place value from the percentage points table.
- The wrong tail. In June 2024 some subtracted from 1 when they did not need to, and in June 2022 others forgot to subtract from 1 for an upper tail.
- Variance instead of standard deviation. In June 2022 a number of students standardised using the variance.
- The conditional numerator. In June 2022 many set up the conditional probability correctly but used the wrong region on top of the fraction.
Common questions
Does an expected number have to be a whole number?
No. The June 2022 examiners pointed out that an expected number is a mean, so a value like 3.6 is fine.
Which table do I use?
The normal distribution table goes from z to a probability. The percentage points table goes from a probability back to z.
How do I find the mean and standard deviation together?
Turn each given probability into an equation of the form (x − μ)/σ = z, then solve the pair simultaneously.
Where this chapter leads
- Statistics 1 Chapter 4: Probability: the tree diagrams and conditionals that combine with normal probabilities.
- Statistics 1 Chapter 3: Representations of Data: quartiles and outlier rules, applied here to a normal model.