State three properties of the Normal curve.

Symmetrical bell shape, maximum at x=\mu (mean = median = mode), total area 1.

How do you standardise X\sim N(\mu,\sigma^2)?

Z=\frac{X-\mu}{\sigma}, which has the distribution N(0,1).

Where are the points of inflection of a Normal curve?

At x=\mu-\sigma and x=\mu+\sigma.

What is P(X=a) for a Normal variable?

0, because the variable is continuous; probability is area under the curve.

What effect does increasing \sigma have on the curve?

The curve becomes wider and flatter; the area stays 1.

Given P(X<x)=p, how do you find x?

Find z with P(Z<z)=p (inverse Normal), then x=\mu+z\sigma.

How do you find unknown \mu and \sigma?

Standardise each given probability to get two equations in \mu and \sigma, then solve simultaneously.

By what Normal distribution can B(n,p) be approximated?

N\big(np,\,np(1-p)\big) when n is large and p is not near 0 or 1.

What is a continuity correction?

Adjusting a discrete value by 0.5 when approximating by a continuous variable, e.g. P(X\le45)\approx P(Y<45.5).

The Normal distributionAQA A-Level Maths: Flashcards

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What does $X\sim N(\mu,\sigma^2)$ mean?

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What does $X\sim N(\mu,\sigma^2)$ mean?: $X$ is Normally distributed with mean $\mu$ and variance $\sigma^2$ (standard deviation $\sigma$ ).
State three properties of the Normal curve.: Symmetrical bell shape, maximum at $x=\mu$ (mean = median = mode), total area $1$ .
How do you standardise $X\sim N(\mu,\sigma^2)$ ?: $Z=\frac{X-\mu}{\sigma}$ , which has the distribution $N(0,1)$ .
Where are the points of inflection of a Normal curve?: At $x=\mu-\sigma$ and $x=\mu+\sigma$ .
What is $P(X=a)$ for a Normal variable?: $0$ , because the variable is continuous; probability is area under the curve.
Roughly what percentage lies within $1\sigma$ , $2\sigma$ , $3\sigma$ of the mean?: About $68\%$ , $95\%$ and $99.7\%$ .
What effect does increasing $\sigma$ have on the curve?: The curve becomes wider and flatter; the area stays $1$ .
How is $P(Z<-a)$ written using $P(Z<a)$ ?: $P(Z<-a)=1-P(Z<a)$ , by symmetry.
Given $P(X<x)=p$ , how do you find $x$ ?: Find $z$ with $P(Z<z)=p$ (inverse Normal), then $x=\mu+z\sigma$ .
How do you find unknown $\mu$ and $\sigma$ ?: Standardise each given probability to get two equations in $\mu$ and $\sigma$ , then solve simultaneously.
By what Normal distribution can $B(n,p)$ be approximated?: $N\big(np,\,np(1-p)\big)$ when $n$ is large and $p$ is not near $0$ or $1$ .
What is a continuity correction?: Adjusting a discrete value by $0.5$ when approximating by a continuous variable, e.g. $P(X\le45)\approx P(Y<45.5)$ .

Exam questions on The Normal distribution

The mass of eggs from a farm is modelled by the random variable $X\sim N(60,4^2)$ , where $X$ is measured in grams.
Eggs with a mass between $56$ g and $63$ g are classed as medium. Find the probability that an egg chosen at random is medium.2 marks
The time $T$ minutes that students take to complete a puzzle is modelled by the Normal distribution with mean $25$ and standard deviation $6$ .
Find the interquartile range of $T$ .2 marks
A nursery models the height $Y$ cm of its two-year-old saplings as $Y\sim N(\mu,\sigma^2)$ . It is found that $20\%$ of saplings are shorter than $30$ cm and $10\%$ of saplings are taller than $45$ cm.
Find the values of $\mu$ and $\sigma$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).