Normal distributionNormal distribution

1.1. Learn about the properties of a normal Learn about the properties of a normal distributiondistribution

2.2. Solve problems using tables of the Solve problems using tables of the normal distributionnormal distribution

3.3. Meet some other examples of continuous Meet some other examples of continuous probability distributionsprobability distributions

Types of variables Types of variables Discrete & Continuous Discrete & Continuous

Describe what types of data could Describe what types of data could be described as continuous be described as continuous random variable X=x random variable X=x

– Arm lengths Arm lengths – Eye heightsEye heights

REMEMBER your box REMEMBER your box plot plot

LQ UQ

Middle 50%

range

The Normal The Normal Distribution Distribution (Bell Curve)(Bell Curve)

Average contents 50

Mean = μ = 50

Standard deviation = σ = 5

The normal distribution The normal distribution is a theoretical is a theoretical probabilityprobabilitythe area under the curve adds up to the area under the curve adds up to

oneone

The normal distribution The normal distribution is a theoretical is a theoretical probabilityprobabilitythe area under the curve adds up to the area under the curve adds up to

oneone

A Normal distribution is a theoretical model of the whole population. It is perfectly symmetrical about the central value; the mean μ represented by zero.

As well as the mean the As well as the mean the standard deviation (standard deviation (σσ) ) must must also be known.also be known.

0

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-5 -4 -3 -2 -1 0 1 2 3 4 5

The X axis is divided up into deviations from the mean. Below the shaded area is one deviation from the mean.

Two standard Two standard deviations from the deviations from the meanmean

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Three standard Three standard deviations from the deviations from the meanmean

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-5 -4 -3 -2 -1 0 1 2 3 4 5

A handy estimate – known as the A handy estimate – known as the Imperial RuleImperial Rule for a set of normal for a set of normal data:data:

68% of data will fall within 168% of data will fall within 1σσ of the of the μμ

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P( -1 < z < 1 ) = 0.683 = 68.3%

95% of data fits within 295% of data fits within 2σσ of the of the μμ

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P( -2 < z < 2 ) = 0.954 = 95.4%

99.7% of data fits within 399.7% of data fits within 3σσ of of the the μμ

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P( -3 < z < 3 ) = 0.997 = 99.7%

Simple problems solved using the Simple problems solved using the imperial rule - firstly, make a table imperial rule - firstly, make a table out of the ruleout of the rule

<-3<-3 -3 to -3 to -2-2

-2 to -2 to -1-1

-1 to -1 to 00

0 to 10 to 1 1 to 21 to 2 2 to 32 to 3 >3>3

0%0% 2%2% 14%14% 34%34% 34%34% 14%14% 2%2% 0%0%

The heights of students at a college were found to follow a bell-shaped distribution with μ of 165cm and σ of 8 cm.

What proportion of students are smaller than 157 cm

zx

estandardisfirst

thebelow 1or

18

165157 is 157cmfirst

16%

Simple problems solved using the Simple problems solved using the imperial rule - firstly, make a table imperial rule - firstly, make a table out of the ruleout of the rule

<-3<-3 -3 to -3 to -2-2

-2 to -2 to -1-1

-1 to -1 to 00

0 to 10 to 1 1 to 21 to 2 2 to 32 to 3 >3>3

0%0% 2%2% 14%14% 34%34% 34%34% 14%14% 2%2% 0%0%

The heights of students at a college were found to follow a bell-shaped distribution with μ of 165cm and σ of 8 cm.

Above roughly what height are the tallest 2% of the students?

of 2 beyond are students of 2% tallest The

165 + 2 x 8 = 181 cm

Task – class 10 Task – class 10 minutes minutes finish for homeworkfinish for homework Exercise A Exercise A Page 76Page 76

The Bell shape curve The Bell shape curve happens so when recording happens so when recording continuous random continuous random variables that an equation variables that an equation is used to model the shape is used to model the shape exactly.exactly.

Put it into your Put it into your calculator and use calculator and use the graph function.the graph function.

221

2

1 xey

221

2

1)( zez

Sometimes you will Sometimes you will see it using phi =.see it using phi =.

Luckily you don’t have to use the Luckily you don’t have to use the equation each time and you don’t have equation each time and you don’t have to integrate it every time you need to to integrate it every time you need to work out the area under the curve – work out the area under the curve – the normal distribution probabilitythe normal distribution probability

There are normal distribution tablesThere are normal distribution tables

How to read the Normal How to read the Normal distribution tabledistribution table

Φ(z) means the area under the curve on the left of z

How to read the Normal How to read the Normal distribution tabledistribution table

Φ(0.24) means the area under the curve on the left of 0.24 and is this value here:

Values of Values of ΦΦ(z)(z)

ΦΦ(-1.5)=1- (-1.5)=1- ΦΦ(1.5)(1.5)

Values of Values of ΦΦ(z)(z)

ΦΦ(0.8)=0.78814 (this is for the (0.8)=0.78814 (this is for the left)left)

Area = 1-0.78814 = 0.21186Area = 1-0.78814 = 0.21186

Values of Values of ΦΦ(z)(z)

ΦΦ(1.5)=0.93319(1.5)=0.93319 ΦΦ(-1.00)(-1.00)

=1- =1- ΦΦ(1.00)(1.00)=1-0.84134=1-0.84134=0.15866=0.15866

Shaded area = Shaded area = ΦΦ(1.5)- (1.5)- ΦΦ(-1.00) (-1.00) = 0.93319 - 0.15866 = 0.93319 - 0.15866 = 0.77453= 0.77453

TaskTask

Exercise B page 79Exercise B page 79

Solving Problems Solving Problems using the tablesusing the tables

NORMAL DISTRIBUTION NORMAL DISTRIBUTION The area under the curve is the probability of getting less The area under the curve is the probability of getting less

than the z score. The total area is 1. than the z score. The total area is 1. The tables give the probability for z-scores in the The tables give the probability for z-scores in the

distribution X~N(0,1), that is mean =0, s.d. = 1. distribution X~N(0,1), that is mean =0, s.d. = 1.

ALWAYS SKETCH A DIAGRAM ALWAYS SKETCH A DIAGRAM Read the question carefully and shade the area you want to Read the question carefully and shade the area you want to

find. If the shaded area is more than half then you can read find. If the shaded area is more than half then you can read the probability directly from the table, if it is less than half, the probability directly from the table, if it is less than half, then you need to subtract it from 1. then you need to subtract it from 1.

NB If your z-score is negative then you would look up the NB If your z-score is negative then you would look up the positive from the table. The rule for the shaded area is the positive from the table. The rule for the shaded area is the same as above: more than half – read from the table, less same as above: more than half – read from the table, less than half subtract the reading from 1. than half subtract the reading from 1.

You will have to standardise if You will have to standardise if the mean is not zero and the the mean is not zero and the standard deviation is not onestandard deviation is not one

Task Task

Exercise C page 168Exercise C page 168

Normal distribution Normal distribution problems in reverseproblems in reverse Percentage points table on page Percentage points table on page

155155

Work through examples on page Work through examples on page 84 and do questions Exercise D 84 and do questions Exercise D on page 85on page 85

Key chapter pointsKey chapter points

The probability distribution of a continuous The probability distribution of a continuous random variable is represented by a curve. random variable is represented by a curve. The area under the curve in a given interval The area under the curve in a given interval gives the probability of the value lying in that gives the probability of the value lying in that interval.interval.

If a variable X follows a normal probability If a variable X follows a normal probability distribution, with mean distribution, with mean μμ and standard and standard deviation deviation σσ, we write X , we write X ̴ N (̴ N (μμ, , σσ22))

The variable Z= is called the The variable Z= is called the standard normal variable corresponding to Xstandard normal variable corresponding to X

Key chapter points Key chapter points cont.cont. If Z is a continuous random If Z is a continuous random

variable such that Z variable such that Z ̴ N (̴ N (00, , 11) then ) then ΦΦ(z)=P(Z<z)(z)=P(Z<z)

The percentage points table The percentage points table shows, for probability p, the value shows, for probability p, the value of z such that P(Z<z)=pof z such that P(Z<z)=p