If you know a probability, use the inverse normal to find the z-value, then solve z = (x − μ)/σ. For X ~ N(20, σ²) with P(X < 34) = 0.9, z = 1.2816, so 1.2816 = 14/σ and σ = 10.9.
To find μ when σ is known: X ~ N(μ, 15²) with P(X < 67) = 0.75 gives z = 0.6745, so 67 − μ = 0.6745 × 15 = 10.1 and μ = 56.9. Be careful with the sign of z when the probability is below 0.5.
Worked example
X ~ N(30, σ²) and P(X < 26) = 0.2. Find σ to 3 s.f.
- Inverse normal: z = −0.8416.
- −0.8416 = (26 − 30)/σ, so σ = 4 ÷ 0.8416.
Answer:σ = 4.75
Key words
- inverse normal
- finds z from a probability
- μ
- the mean
- σ
- the standard deviation
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Common Core-referenced: HSS-ID.A.4