Inverse Normal and finding μ or σ

The idea

Inverse Normal works backwards from a probability to a value. To find an unknown mean or standard deviation, turn the probability into a $z$ -value and solve $z = \frac{x - \mu}{\sigma}$ .

Sketch: is the value above or below the mean? That sets the sign of $z$ .
Find $z$ from the probability (Inverse Normal with $\mu = 0$ , $\sigma = 1$ ).
Substitute into $z = \frac{x - \mu}{\sigma}$ and solve.

Watch out A value below the mean has a negative $z$ : check it on your sketch.

See it worked

Worked example

$X \sim N(30, 5^2)$ . Find $a$ such that $P(X < a) = 0.9$ .

$P(Z < z) = 0.9$ gives $z = 1.2816$ .
$a = 30 + 1.2816 \times 5$ .
$a = 36.4$ (3 s.f.).

Answer $a = 36.4$

Your turn

Question 1Calculator allowed3 marks

$X \sim N(200, 20^2)$ . Find $k$ such that $P(X > k) = 0.05$ . Give $k$ to 1 decimal place.

Question 2Calculator allowed3 marks

$X \sim N(\mu, 4^2)$ and $P(X < 30) = 0.1$ . Find $\mu$ to 2 decimal places.

Question 3Calculator allowed3 marks

$X \sim N(50, \sigma^2)$ and $P(X > 58) = 0.2$ . Find $\sigma$ to 3 significant figures.

In past papers

26 official questions in 22 papers since 2018. The latest:

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