Many measurements, like heights, follow a normal distribution: a symmetric bell-shaped curve centred on the mean μ, with spread given by the standard deviation σ. We write X ~ N(μ, σ²).
About 68% of values lie within one σ of the mean and 95% within two. To find P(X > 23) for X ~ N(30, 5²), standardise: z = (23 − 30)/5 = −1.4, then P(Z > −1.4) = 0.919. A calculator's normal function can do this directly.
Worked example
X ~ N(30, 2²). Find P(X > 29) to 3 s.f.
- z = (29 − 30)/2 = −0.5.
- P(Z > −0.5) = 0.6915.
Answer:0.691
Key words
- normal distribution
- a symmetric bell-shaped distribution
- z-score
- (x − μ)/σ
- standardise
- convert to the standard normal Z ~ N(0, 1)
Practise this
New questions every time.
Common Core-referenced: HSS-ID.A.4