The normal distribution
Many measurements, such as heights or the masses of packets, follow a bell-shaped curve. The normal distribution models them, and your calculator does most of the arithmetic once you set the problem up correctly.
Part 1 of 3: Learn it
In short
- X ~ N(μ, σ²): mean μ, variance σ², so the standard deviation is σ.
- The curve is symmetrical about μ, and the total area under it is 1.
- Z = (X − μ) ÷ σ turns any normal variable into the standard normal Z ~ N(0, 1).
Where this is in your specification
Spec points: DfE N2 (AQA 7357 N2, Edexcel 9MA0 Statistics topic 4, OCR A H240 2.04)
| Board | Topic: The normal distribution |
|---|---|
| DfE content | N2, N3 |
| AQA 7357 | N2, N3 |
| Edexcel 9MA0 | Statistics topic 4 |
| OCR H240 | 2.04 |
Properties
- Bell shaped and symmetrical, so the mean, median and mode are equal.
- About 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
- The curve changes from bending one way to the other (points of inflection) one standard deviation either side of the mean.
- For a continuous variable, P(X = a) = 0, so P(X < a) and P(X ≤ a) are the same.
Finding probabilities
Use the normal cumulative distribution function on your calculator with the lower bound, the upper bound, σ and μ. For "more than a" use a large upper bound such as 10⁹⁹ (or 1 − P(X < a)). A quick sketch with the region shaded helps you check the answer is sensible: above or below 0.5?
Working backwards
The inverse normal function gives the value with a given area to its left. If you need an unknown μ or σ, standardise: find the z value for the given probability from the standard normal, then solve (a − μ) ÷ σ = z.
X ~ N(μ, σ²). What is P(X > μ)?
Show the answer
0.5, because the curve is symmetrical about the mean.
Part 2 of 3: See it worked
Worked examples
Example 1
Heights are modelled by X ~ N(170, 8²) in cm. Find P(X < 180) and P(160 < X < 180).
- P(X < 180): z = (180 − 170) ÷ 8 = 1.25, and P(Z < 1.25) = 0.8944
- By symmetry P(X < 160) = 1 − 0.8944 = 0.1056
- P(160 < X < 180) = 0.8944 − 0.1056 = 0.7887 (calculator)
Answer: 0.894 and 0.789 (3 s.f.).
Example 2
With X ~ N(170, 8²), find h such that P(X > h) = 0.1.
- P(X < h) = 0.9
- Inverse normal: h = 170 + 1.2816 × 8
- = 180.25
Answer: h = 180 cm (3 s.f.).
Example 3
X ~ N(50, σ²) and P(X > 56) = 0.2. Find σ.
- Area 0.8 to the left in the standard normal: z = 0.8416
- (56 − 50) ÷ σ = 0.8416
- σ = 6 ÷ 0.8416 = 7.129
Answer: σ = 7.13 (3 s.f.).
Common mistakes
- Entering the variance where the calculator asks for the standard deviation.
- Using the inverse normal with the area to the right when the calculator expects the area to the left.
- Getting a negative z value the wrong sign when the point is below the mean.
- Rounding z values early, which shifts the final answer.
Where are the points of inflection of a normal curve?
Show the answer
At μ − σ and μ + σ.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
X ~ N(μ, σ²). What is P(X > μ)?
0.5, because the curve is symmetrical about the mean.
Where are the points of inflection of a normal curve?
At μ − σ and μ + σ.
X ~ N(40, 25). What is the standard deviation?
5, the square root of the variance 25.
Jobs that use this
- Data scientist (opens a new tab)
- Actuary (opens a new tab)
- Data analyst-statistician (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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