The Standard Normal Distribution — Core Maths Compendium

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Understanding the standard normal distribution

Click each step below to reveal it — work through them in order the first time round.

Step 1 · What the standard normal distribution is

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The standard normal distribution is a special normal distribution with mean 0 and standard deviation 1. Any normal variable can be converted to this standard form using a z-score — exactly why z-scores are so useful, letting you use one single table (or calculator function) for any normal distribution problem.

Step 2 · Reading the standard normal table

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The standard normal table (often written $\Phi(z)$) gives the probability that a standardised value is less than a given z-score, i.e. $P(Z

Step 3 · Finding probabilities: less than, greater than, between

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For $P(Zz)$, use $1-P(Z

Step 4 · Working with negative z-scores

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Because the normal distribution is symmetric, $P(Z<-z) = 1-P(Zz)$. Many tables only list positive z-values, so you use this symmetry property to handle negative z-scores rather than looking them up directly.

Worked example

Find $P(Z<1.5)$ using the standard normal distribution, and use it to find $P(Z>1.5)$.

From the standard normal table, $P(Z<1.5)\approx 0.9332$.
$P(Z>1.5) = 1-0.9332 = $ 0.0668.

Test yourself

Past-paper style question

A machine fills bottles with a volume of liquid that is normally distributed with mean 500 ml and standard deviation 4 ml.

(a) Find the z-score for a bottle containing 508 ml.
(b) Given that $P(Z<2)\approx 0.9772$, find the probability that a randomly chosen bottle contains more than 508 ml. [4 marks]

Show the answer

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(a) $z=\frac{508-500}{4}=\frac{8}{4}=2$.

(b) $P(Z>2) = 1-P(Z<2) = 1-0.9772 = $ 0.0228. So the probability a bottle contains more than 508 ml is about 2.3%.

Practice

The Standard Normal Distribution worksheet

Five short questions on the standard normal distribution. Work through them, then reveal the mark scheme to check.

Using $P(Z<1)\approx0.8413$, find $P(Z>1)$.
Using $P(Z<0.5)\approx0.6915$ and $P(Z<1.5)\approx0.9332$, find $P(0.5
Use the symmetry of the normal distribution and $P(Z<1)\approx0.8413$ to find $P(Z<-1)$.
A variable is normally distributed with mean 20 and standard deviation 2.5. Find the z-score for a value of 25.
Describe how you would use a bell-curve sketch to help you find $P(-1

Mark scheme

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$P(Z>1)=1-0.8413=$ 0.1587.
$P(0.50.2417.
$P(Z<-1)=1-P(Z<1)=1-0.8413=$ 0.1587.
$z=\frac{25-20}{2.5}=\frac{5}{2.5}=$ 2.
Sketch a bell curve and shade the region between z=−1 and z=2. This equals $P(Z<2)-P(Z<-1)$ — you would look up (or calculate using symmetry) each of these individual probabilities, then subtract the smaller from the larger to find the shaded area/probability.

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