X ~ N(40, σ2) and P(X < 46) = 0.8849, which corresponds to z = 1.2. Find σ.
- z = (x − μ)/σ, so 1.2 = (46 − 40)/σ
- 1.2 = 6/σ
- σ = 6 ÷ 1.2 = 5
Answer: σ = 5
The normal distribution
Maths: Analysis and Approaches · Statistics and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
The normal distribution models continuous data such as heights and masses. Its curve is bell-shaped and symmetric about the mean μ, so the mean, median and mode are equal. X ~ N(μ, σ2) gives the mean and the variance: the second number is σ squared. A larger σ gives a wider, lower curve.
Probability is area under the curve, and the total area is 1. Half the area lies on each side of the mean, so symmetry answers many questions without a calculator. Otherwise use the GDC.
An inverse normal calculation goes backwards: from a probability to the value of x. The z-value says how many standard deviations x is from the mean.
Rules to remember
- Total area under the curve = 1; P(X < μ) = P(X > μ) = 0.5
- Symmetry: P(X < μ − k) = P(X > μ + k)
- About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ
- Standardise: z = (x − μ)/σ, where Z ~ N(0, 1)
- Rearranged: x = μ + zσ
- A value below the mean has a negative z; a probability below 0.5 for P(X < x) means x is below the mean
- Unknown μ or σ: find z from the probability with inverse normal, then use z = (x − μ)/σ
Common mistake
Reading N(50, 16) as a standard deviation of 16. The second number is the variance, so here σ = 4.
Worked example
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