Learn7357Paper 3 · Pure and StatisticsStatistical distributionsInverse normal and finding μ and σ

Statistical distributions · Subtopic 4 of 5

Inverse Normal: Finding Values, μ and σ

Run the standardising formula backwards. Turn the probability into a $z$ -value with the tables, then solve $\dfrac{x-\mu}{\sigma}=z$ for whichever quantity is missing.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find $z$ from a probability using the inverse of $\Phi$ .
Find the value $x$ with a given probability above or below it.
Find an unknown $\mu$ or $\sigma$ from one condition.
Find both $\mu$ and $\sigma$ from two conditions with simultaneous equations.

Pause and think

Exam marks are $N(55,144)$ . The top $5\%$ get an A*. Without tables, will the A* boundary be around $60$ , $75$ or $95$ ?

Section 2 of 14

Core idea

Start hereSetting the pass mark

An exam board wants the top 10% to get an A. Scores are $N(50,16)$ . What's the cut-off? This time we know the probability and want the value: the normal table, read backwards.

The big idea

Given a probability, find $z$ from the inverse table, then $x=\mu+z\sigma$ . Two such facts give two simultaneous equations for unknown $\mu$ and $\sigma$ .

1. Finding a value

The top 5%: z = 1.645

\Phi(1.645)=0.95

, so

95\%

of the area is to the left and

5\%

to the right.

Worked exampleTop 10%

$X\sim N(50,16)$ . Find $a$ with $P(X>a)=0.1$ .

Goal: $a$ .
Strategy: Left area $0.9$ .

$\Phi(z)=0.9\Rightarrow z=1.282$
Inverse table.
$a=50+1.282\times4=55.13$
$x=\mu+z\sigma$ .

Answer

55.1

✓ Check

a

is above the mean, as the top tail must be ✓.

Worked exampleLower tail

Find $b$ with $P(X>b)=0.85$ .

Goal: $b$ .
Strategy: Left area $0.15<0.5$ : negative $z$ .

$\Phi(z)=0.15\Rightarrow z=-1.036$
Symmetry: $\Phi(1.036)=0.85$ .
$b=50-4.144=45.86$
Formula.

Answer

45.9

✓ CheckBelow the mean ✓.

2. Unknown μ and σ

Worked example $P(X<35)=0.2$ , $P(X>60)=0.1$

Find $\mu$ and $\sigma$ .

Goal: Both.
Strategy: Two $z$ -equations.

$\frac{35-\mu}\sigma=-0.8416,\quad\frac{60-\mu}\sigma=1.2816$
Inverse table.
$35=\mu-0.8416\sigma,\quad60=\mu+1.2816\sigma$
Multiply out.
$25=2.1232\sigma\Rightarrow\sigma=11.77$
Subtract.
$\mu=35+0.8416(11.77)=44.9$
Back-substitute.

Answer

\mu=44.9

\sigma=11.8

✓ Check

\frac{60-44.9}{11.77}=1.28

✓.

Common misconceptionSign of z Left-tail probabilities below

0.5

give negative

z

Pause and think

$P(X<\mu+k\sigma)=0.975$ . $k$ ?

Summary

$x=\mu+z\sigma$ .
Two facts → simultaneous equations.

Inverse normal: z from a probability.

HistoryTables before calculators. The first detailed tables of the normal distribution were computed by hand in the late 1700s and 1800s, including by the French physicist Christian Kramp in 1799 (the same Kramp who invented

n!

). For two centuries, statisticians did every calculation in this lesson with printed tables, as Cambridge candidates still do.

Fun factWhy 1.96?

\Phi(1.96)=0.975

, so the middle

95\%

of any normal distribution lies within

1.96

standard deviations of the mean. This one number appears in opinion polls, medical studies and quality control everywhere; you will meet it again in confidence intervals in Paper 6.

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