The Normal Distribution - University of Wisconsin– · PDF file• Chapter 4...

The Normal Distribution

Mary Lindstrom(Adapted from notes provided by Professor Bret Larget)

February 10, 2004

Statistics 371 Last modified: February 11, 2004

The Normal Distribution

• The Normal Distribution (AKA Gaussian Distribution) is our

first distribution for continuous variables and is the most

commonly used distribution.

• The normal density curve is the famous symmetric, bell-

shaped curve.

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Statistics 371 1

Central Limit Theorem

Why is it such an important distribution? Two related reasons:

1. The central limit theorem states that many statistics we

calculate from large random samples will have approximate

normal distributions (or distributions derived from normal

distributions), even if the distributions of the underlying

variables are not normally distributed.

2. Many measured variables nave approximately normal distri-

butions. This is true because most things we measure are

the sum of many smaller units and the central limit theorem

applies.

Statistics 371 2

Central Limit Theorem

These facts are the basis for most of the methods of statistical

inference we will study in the last half of the course.

• Chapter 4 introduces the normal distribution as a probability

distribution.

• Chapter 5 culminates in the central limit theorem, the

primary theoretical justification for most of the methods of

statistical inference in the remainder of the textbook.

Statistics 371 3

The Normal Density

Normal curves have the following bell-shaped, symmetric density.

f(y) =1

σ√

2πe−1

2

(y−µ

σ

)2

Parameters: The parameters of a normal curve are the mean µ

and the standard deviation σ.

If Y is a normally distributed Random Variable with mean µ and

standard deviation σ, then we write:

Y ∼ N(µ, σ2)

Statistics 371 4

Standard Shape

All normal curves have the same shape so that every normal

curve can be drawn in exactly the same manner, just by changing

labels on the axis (this is not true for the Binomial or Poisson

distributions).

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Normal Distributionmu = 100 , sigma = 3

90 95 100 105 110

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−40 −20 0 20 40

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Normal Distributionmu = −20 , sigma = 5

−40 −30 −20 −10 0

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−4 −2 0 2 4

Statistics 371 5

The 68–95–99.7 Rule

For every normal curve,

• 68% of the area is within one SD of the mean,

• 95% of the area is within two SDs of the mean,

• 99.7% of the area is within three SDs of the mean.

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−4 −2 0 2 4

−3 −2 −1 0 1 2 3P

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−4 −2 0 2 4

P( −1 < X < 1 ) = 0.6827P( X < −1 ) = 0.1587

P( X > 1 ) = 0.1587

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−4 −2 0 2 4

P( −2 < X < 2 ) = 0.9545P( X < −2 ) = 0.0228

P( X > 2 ) = 0.0228

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−4 −2 0 2 4

P( −3 < X < 3 ) = 0.9973P( X < −3 ) = 0.0013

P( X > 3 ) = 0.0013

Statistics 371 6

Standardization

If Y is a normally distributed Random Variable Y ∼ N(µ, σ2)

And we define

Z =Y − µ

σ

Then Z is a standard normal random variable

Z ∼ N(0,1)

Every problem that asks for an area under a normal curve is

solved by first finding an equivalent problem for the standard

normal curve.

Statistics 371 7

StandardizationP

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−4 −2 0 2 4

P( −1 < X < 1 ) = 0.6827

P( X < −1 ) = 0.1587P( X > 1 ) = 0.1587

−3 −2 −1 0 1 2 3

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Normal Distributionmu = −1 , sigma = 2

−5 0 5

P( −3 < X < 1 ) = 0.6827

P( X < −3 ) = 0.1587P( X > 1 ) = 0.1587

−3 −2 −1 0 1 2 3

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6 8 10 12 14

P( 9 < X < 11 ) = 0.6827

P( X < 9 ) = 0.1587P( X > 11 ) = 0.1587

−3 −2 −1 0 1 2 3

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−10 −5 0 5 10 15 20

P( 1 < X < 9 ) = 0.6827

P( X < 1 ) = 0.1587P( X > 9 ) = 0.1587

−3 −2 −1 0 1 2 3

Statistics 371 8

Example Calculation

Suppose that egg shell thickness is normally distributed with a

mean of 0.381 mm and a standard deviation of 0.031 mm.

Find the proportion of eggs with shell thickness less than 0.34

mm.

1. We define our random variable Y = egg shell thickness.

2. We know that

Y ∼ N(0.381,0.0312)

3. We wish to know

Pr{Y < 0.34}

Statistics 371 9

Example Calculation

Here is a way to do it in R.

> source("prob.R")> gnorm(0.381, 0.031, b = 0.34)

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Normal Distributionmu = 0.381 , sigma = 0.031

0.25 0.30 0.35 0.40 0.45 0.50

P( X < 0.34 ) = 0.093

P( X > 0.34 ) = 0.907

−3 −2 −1 0 1 2 3

Statistics 371 10

Example Calculation

If you don’t have R handy, the way to do this is to transform it

into a question about a standard normal distribution.

Pr{Y < 0.34} = Pr{Y − 0.381 < 0.34− 0.381}= Pr{[Y − 0.381]/0.031 < [0.34− 0.381]/0.031}= Pr{Z < −1.322}

Where Z is a standard normal random variable: Z ∼ N(0,1).

This is exactly the type probability tabulated in a standard normal

table

Statistics 371 11

Standard Normal table

• The standard normal table lists the area to the left of z under

the standard normal curve for each value from −3.49 to 3.49

by 0.01 increments.

• The normal table is on the inside front cover of your

textbook.

• Numbers in the margins represent z.

• Numbers in the middle of the table are areas to the left of z.

Statistics 371 12


Here is a portion of the table on the inside cover of your book:

0 .01 .02 .03-1.4 0.0808 0.0793 0.0778 0.0764-1.3 0.0968 0.0951 0.0934 0.0918-1.2 0.1151 0.1131 0.1112 0.1093-1.1 0.1357 0.1335 0.1314 0.1292-1.0 0.1587 0.1562 0.1539 0.1515

We want the area to the left of −1.322

• we round to −1.32

• look up −1.3 in the row labels on the left

• look up .02 in the column labels

• to find P (Z < −1.32) = 0.0934.

Statistics 371 13


• R can do this for general values of z

• and R can do the standardization for you.

• Recall that our original question was given Y ∼ N(0.381,0.0312)what is Pr{Y < 0.34}.

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P( X < −1.322 ) = 0.0931P( X > −1.322 ) = 0.9069

z = −1.322

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0.25 0.30 0.35 0.40 0.45 0.50

P( X < 0.34 ) = 0.093P( X > 0.34 ) = 0.907

x = 0.34

Statistics 371 14

Example Area Calculations: Area to the

leftFind the proportion of eggs with shell thickness less than 0.34

mm.

> pnorm(q = 0.34, mean = 0.381, sd = 0.031)[1] 0.09298744

> gnorm(0.381, 0.031, b = 0.34, sigma.axis = F)

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0.25 0.35 0.45

P( X < 0.34 ) = 0.093P( X > 0.34 ) = 0.907

Pr{Y < 0.34} = Pr{[Y − 0.381]/0.031 < [0.34− 0.381]/0.031}= Pr{Z < −1.322} = 0.0934

Statistics 371 15

Example Area Calculations: Area to the

rightFind the proportion of eggs with shell thickness more than 0.4

mm.

> 1 - pnorm(q = 0.4, mean = 0.381, sd = 0.031)[1] 0.2699702

> gnorm(0.381, 0.031, a = 0.4, sigma.axis = F)

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0.25 0.35 0.45

P( X < 0.4 ) = 0.73P( X > 0.4 ) = 0.27

Pr{Y > 0.4} = Pr{[Y − 0.381]/0.031 > [0.4− 0.381]/0.031}= Pr{Z > 0.613} = 1− Pr{Z < 0.613} = 1− 0.7291 = 0.2709

Statistics 371 16

Example Area Calculations: Area

between two values.Find the proportion of eggs with shell thickness between 0.34

and 0.4 mm.

> gnorm(0.381, 0.031, a = 0.34, b = 0.4)

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0.25 0.30 0.35 0.40 0.45 0.50

P( 0.34 < X < 0.4 ) = 0.637

P( X < 0.34 ) = 0.093

P( X > 0.4 ) = 0.27

−3 −2 −1 0 1 2 3

Statistics 371 17

Example Area Calculations: Area

outside two values.Find the proportion of eggs with shell thickness smaller than 0.32

mm or greater than 0.40 mm.

> gnorm(0.381, 0.031, a = 0.32, b = 0.4)

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0.25 0.30 0.35 0.40 0.45 0.50

P( 0.32 < X < 0.4 ) = 0.7055

P( X < 0.32 ) = 0.0245

P( X > 0.4 ) = 0.27

−3 −2 −1 0 1 2 3

Statistics 371 18

Example Area Calculations: Central

area.Find the proportion of eggs with shell thickness within 0.05 mm

of the mean.

> gnorm(0.381, 0.031, a = 0.381 - 0.05, b = 0.381 + 0.05)

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0.25 0.30 0.35 0.40 0.45 0.50

P( 0.331 < X < 0.431 ) = 0.8932

P( X < 0.331 ) = 0.0534

P( X > 0.431 ) = 0.0534

−3 −2 −1 0 1 2 3

Statistics 371 19

Example Area Calculations: Two-tail

area.Two-tail area. Find the proportion of eggs with shell thickness

more than 0.07 mm from the mean.

> gnorm(0.381, 0.031, a = 0.381 - 0.07, b = 0.381 + 0.07)

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0.25 0.30 0.35 0.40 0.45 0.50

P( 0.311 < X < 0.451 ) = 0.9761

P( X < 0.311 ) = 0.012

P( X > 0.451 ) = 0.012

−3 −2 −1 0 1 2 3

Statistics 371 20

Using R

We have seen how to use R and the new function gnorm

to graph normal distributions where the graphs include some

probability calculations. You can also make the same calculations

without graphs using the function pnorm (the “p” stands for

probability). The following lists the commands for all of the

previous computations.

Area to the left. Find the proportion of eggs with shell

thickness less than 0.34 mm.

> pnorm(0.34, 0.381, 0.031)[1] 0.09298744

Area to the right. Find the proportion of eggs with shell

thickness more than 0.36 mm.

> 1 - pnorm(0.36, 0.381, 0.031)[1] 0.75093

Statistics 371 21

Using R

Area between two values. Find the proportion of eggs withshell thickness between 0.34 and 0.36 mm.

> pnorm(0.36, 0.381, 0.031) - pnorm(0.34, 0.381, 0.031)[1] 0.1560825

Area outside two values. Find the proportion of eggs withshell thickness smaller than 0.32 mm or greater than 0.40 mm.

> pnorm(0.32, 0.381, 0.031) + 1 - pnorm(0.4, 0.381, 0.031)[1] 0.2945190

Central area. Find the proportion of eggs with shell thicknesswithin 0.05 mm of the mean.

> 1 - 2 * pnorm(0.381 - 0.05, 0.381, 0.031)[1] 0.8932345

Two-tail area. Find the proportion of eggs with shell thicknessmore than 0.07 mm from the mean.

> 2 * pnorm(0.381 - 0.07, 0.381, 0.031)[1] 0.02394164

Statistics 371 21

Quantiles

Quantile calculations ask you to use the normal table backwards.

You know the area but need to find the point or points on the

horizontal axis.

Statistics 371 22

Example Quantile Calculations

Percentile. What is the 90th percentile of the egg shell

thickness distribution?

> source("prob.R")> gnorm(0.381, 0.031, quantile = 0.9)

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0.25 0.30 0.35 0.40 0.45 0.50

z = 1.28

P( X < 0.4207 ) = 0.9

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Statistics 371 23


Upper cut-off point. What value cuts off the top 15% of egg

shell thicknesses?

> gnorm(0.381, 0.031, quantile = 0.85)

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0.25 0.30 0.35 0.40 0.45 0.50

z = 1.04

P( X < 0.4131 ) = 0.85

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Statistics 371 23


Central cut-off points. The middle 75% egg shells have

thicknesses between which two values?

> gnorm(0.381, 0.031, quantile = 0.875)

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0.25 0.30 0.35 0.40 0.45 0.50

z = 1.15

P( X < 0.4167 ) = 0.875

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Statistics 371 23

Using R

You can also make the same calculations without graphs usingthe function qnorm (the “q” stands for quantile). The followinglists the commands for all of the previous computations.

Percentile. What is the 90th percentile of the egg shellthickness distribution?

> qnorm(0.9, 0.381, 0.031)[1] 0.4207281

Upper cut-off point. What value cuts off the top 15% of eggshell thicknesses?

> qnorm(0.85, 0.381, 0.031)[1] 0.4131294

Central cut-off points. The middle 75% egg shells havethicknesses between which two values?

> qnorm(c(0.25/2, 1 - 0.25/2), 0.381, 0.031)[1] 0.3453392 0.4166608

Statistics 371 24

Approximation of Discrete distributions

using the Normal distribution• The normal distribution can also be used to compute

approximate probabilities of discrete distributions.

• Approximations to the binomial distribution use a normal

curve with µ = np and σ =√

np(1− p).

• Approximations to the Poisson distribution use a normal

curve with µ = µ and σ =√

µ.

• When approximating a binomial probability, the approxima-

tion is usually pretty good when np > 24 and n(1− p) > 24.

• We will not be using the continuity correction described in

your book.

Statistics 371 25

Binomial Distributions: increasing n

0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Binomial Distribution n = 2 , p = 0.2

Possible Values

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0.0

0.1

0.2

0.3

0.4


Possible Values

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0 2 4 6 8 10

0.00

0.10

0.20

0.30


Possible Values

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0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25


Possible Values

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0 5 10 15

0.00

0.05

0.10

0.15


Possible Values

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0.00

0.04

0.08

0.12


Possible Values

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Statistics 371 26

Poisson Distributions: increasing n

0 1 2 3 4 5

0.0

0.1

0.2

0.3

Poisson Distribution mu = 1

Count

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0.00

0.10

0.20


Count

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0 2 4 6 8 10 12 14

0.00

0.05

0.10

0.15


Count

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0 5 10 15 20

0.00

0.04

0.08

0.12


Count

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0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10


Count

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0 10 20 30 40

0.00

0.02

0.04

0.06

0.08


Count

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Statistics 371 27

Example: binomial

Assume Y has a binomial distribution with n = 150 and p = 0.4

and we want to compute Pr{Y ≤ 55}.

Exact:

> pbinom(55, 150, 0.4)[1] 0.2274186

Normal approximation:

> mu = 150 * 0.4> mu[1] 60> 150 * (1 - 0.4)[1] 90> sigma = sqrt(150 * 0.4 * 0.6)> sigma[1] 6> pnorm(55, mu, sigma)[1] 0.2023284

Statistics 371 28

Example: Poisson

If Y has a Poisson distribution with µ = 25 what is the probability

that Y is greater than or equal to 16?

Exact:

> 1 - ppois(15, 25)[1] 0.977707

Normal approximation:

> mu = 25> sigma = sqrt(25)> sigma[1] 5> 1 - pnorm(15, mu, sigma)[1] 0.9772499

With R, do the exact calculation. With a calculator or normal

table, use the normal approximation.

Statistics 371 29

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