Download The Normal Distribution

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.
0.4
0.2
0.0
Probability Density
• The normal density curve is the famous symmetric, bellshaped curve.
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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 distributions. This is true because most things we measure are
the sum of many smaller units and the central limit theorem
applies.
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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.
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The Normal Density
Normal curves have the following bell-shaped, symmetric density.
y−µ 2
1
−2 σ
1
f (y) = √ e
σ 2π
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, σ 2)
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Standard Shape
Probability Density
90
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Normal Distribution
mu = 100 , sigma = 3
mu = 2 , sigma = 10
95
100
105
Probability Density
Normal Distribution
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−40
−20
0
20
Normal Distribution
Normal Distribution
mu = −20 , sigma = 5
mu = 0 , sigma = 1
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−20
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Probability Density
Probability Density
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).
0
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−2
0
2
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4
5
The 68–95–99.7 Rule
For every normal curve,
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Normal Distribution
Normal Distribution
mu = 0 , sigma = 1
mu = 0 , sigma = 1
−2
0
Probability Density
Probability Density
• 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.
2
P( −1 < X < 1 ) = 0.6827
P( X < −1 ) = 0.1587
4
−4
−2
−1
0
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−2
0
2
Normal Distribution
mu = 0 , sigma = 1
mu = 0 , sigma = 1
−2
P( X > 2 ) = 0.0228
0
2
4
3
Normal Distribution
P( −2 < X < 2 ) = 0.9545
P( X < −2 ) = 0.0228
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1
Probability Density
Probability Density
−3
P( X > 1 ) = 0.1587
4
P( −3 < X < 3 ) = 0.9973
P( X < −3 ) = 0.0013
−4
−2
P( X > 3 ) = 0.0013
0
2
4
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Standardization
If Y is a normally distributed Random Variable
Y ∼ N(µ, σ 2)
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.
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Normal Distribution
mu = 0 , sigma = 1
mu = −1 , sigma = 2
P( −1 < X < 1 ) = 0.6827
P( X < −1 ) = 0.1587
P( X > 1 ) = 0.1587
−4
−2
Probability Density
−3
0
−2
−1
0
2
1
2
P( X > 1 ) = 0.1587
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−3
0
−2
−1
0
5
1
2
Normal Distribution
Normal Distribution
mu = 10 , sigma = 1
mu = 5 , sigma = 4
P( X > 11 ) = 0.1587
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P( X < −3 ) = 0.1587
3
P( X < 9 ) = 0.1587
−3
P( −3 < X < 1 ) = 0.6827
4
P( 9 < X < 11 ) = 0.6827
6
Probability Density
Normal Distribution
−2
10
−1
0
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1
2
Probability Density
Probability Density
Standardization
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3
3
P( 1 < X < 9 ) = 0.6827
P( X < 1 ) = 0.1587
P( X > 9 ) = 0.1587
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0
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5
0
10
1
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2
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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}
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Example Calculation
Here is a way to do it in R.
> source("prob.R")
> gnorm(0.381, 0.031, b = 0.34)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.34 ) = 0.093
P( X > 0.34 ) = 0.907
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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
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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.
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Standard Normal table
Here is a portion of the table on the inside cover of your book:
-1.4
-1.3
-1.2
-1.1
-1.0
0
0.0808
0.0968
0.1151
0.1357
0.1587
.01
0.0793
0.0951
0.1131
0.1335
0.1562
.02
0.0778
0.0934
0.1112
0.1314
0.1539
.03
0.0764
0.0918
0.1093
0.1292
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.
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Standard Normal table
• R can do this for general values of z
• and R can do the standardization for you.
Normal Distribution
Normal Distribution
mu = 0 , sigma = 1
mu = 0.381 , sigma = 0.031
P( X < −1.322 ) = 0.0931
P( X > −1.322 ) = 0.9069
−4
−2
z = −1.322
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Probability Density
Probability Density
• 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 < 0.34 ) = 0.093
P( X > 0.34 ) = 0.907
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0.30
0.35
0.40
0.45
0.50
x = 0.34
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Example Area Calculations: Area to the
left
Find 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)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.34 ) = 0.093
P( X > 0.34 ) = 0.907
0.25
0.35
0.45
Pr{Y < 0.34} = Pr{[Y − 0.381]/0.031 < [0.34 − 0.381]/0.031}
= Pr{Z < −1.322} = 0.0934
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Example Area Calculations: Area to the
right
Find 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)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.4 ) = 0.73
P( X > 0.4 ) = 0.27
0.25
0.35
0.45
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
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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)
Normal Distribution
mu = 0.381 , sigma = 0.031
Probability Density
P( 0.34 < X < 0.4 ) = 0.637
P( X < 0.34 ) = 0.093
P( X > 0.4 ) = 0.27
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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)
Normal Distribution
mu = 0.381 , sigma = 0.031
Probability Density
P( 0.32 < X < 0.4 ) = 0.7055
P( X < 0.32 ) = 0.0245
P( X > 0.4 ) = 0.27
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0.40
0
0.45
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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)
Normal Distribution
mu = 0.381 , sigma = 0.031
Probability Density
P( 0.331 < X < 0.431 ) = 0.8932
P( X < 0.331 ) = 0.0534
P( X > 0.431 ) = 0.0534
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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)
Normal Distribution
mu = 0.381 , sigma = 0.031
Probability Density
P( 0.311 < X < 0.451 ) = 0.9761
P( X < 0.311 ) = 0.012
P( X > 0.451 ) = 0.012
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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
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Using R
Area between two values. Find the proportion of eggs with
shell 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 with
shell 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 thickness
within 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 thickness
more than 0.07 mm from the mean.
> 2 * pnorm(0.381 - 0.07, 0.381, 0.031)
[1] 0.02394164
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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.
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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)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.4207 ) = 0.9
z = 1.28
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0
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Example Quantile Calculations
Upper cut-off point. What value cuts off the top 15% of egg
shell thicknesses?
> gnorm(0.381, 0.031, quantile = 0.85)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.4131 ) = 0.85
z = 1.04
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Example Quantile Calculations
Central cut-off points. The middle 75% egg shells have
thicknesses between which two values?
> gnorm(0.381, 0.031, quantile = 0.875)
Normal Distribution
Probability Density
mu = 0.381 , sigma = 0.031
P( X < 0.4167 ) = 0.875
z = 1.15
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Using R
You can also make the same calculations without graphs using
the function qnorm (the “q” stands for quantile). The following
lists the commands for all of the previous computations.
Percentile.
What is the 90th percentile of the egg shell
thickness distribution?
> qnorm(0.9, 0.381, 0.031)
[1] 0.4207281
Upper cut-off point. What value cuts off the top 15% of egg
shell thicknesses?
> qnorm(0.85, 0.381, 0.031)
[1] 0.4131294
Central cut-off points. The middle 75% egg shells have
thicknesses between which two values?
> qnorm(c(0.25/2, 1 - 0.25/2), 0.381, 0.031)
[1] 0.3453392 0.4166608
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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
q
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 approximation is usually pretty good when np > 24 and n(1 − p) > 24.
• We will not be using the continuity correction described in
your book.
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Binomial Distributions: increasing n
Binomial Distribution
n = 10 , p = 0.2
0.00
0.0
1.0
1.5
2.0
0.20
Probability
0.3
0.2
0.1
Probability
0.5
0
1
2
3
4
5
0
2
4
6
8
Possible Values
Possible Values
Possible Values
Binomial Distribution
n = 15 , p = 0.2
Binomial Distribution
n = 30 , p = 0.2
Binomial Distribution
n = 50 , p = 0.2
10
2
4
6
Possible Values
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0.08
0.00
0.00
0
0.04
Probability
0.15
0.10
Probability
0.05
0.20
0.15
0.10
0.00
0.05
Probability
0.12
0.25
0.0
0.10
0.4
0.30
Binomial Distribution
n = 5 , p = 0.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Probability
Binomial Distribution
n = 2 , p = 0.2
0
5
10
Possible Values
15
0
5
10
15
Possible Values
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Poisson Distributions: increasing n
1
2
3
4
5
0.10
0.00
0
2
4
6
8
0
2
4
6
8
10
Count
Poisson Distribution
mu = 10
Poisson Distribution
mu = 15
Poisson Distribution
mu = 25
10
Count
Statistics 371
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20
0.04
0.00
0.02
Probability
0.06
0.00 0.02 0.04 0.06 0.08 0.10
Probability
0.08
0.04
5
12
0.08
Count
0.12
Count
0.00
0
0.05
Probability
0.20
0.0
0.00
0.10
Probability
0.2
0.1
Probability
0
Probability
Poisson Distribution
mu = 5
0.15
Poisson Distribution
mu = 2
0.3
Poisson Distribution
mu = 1
0
5
10
15
Count
20
25
30
0
10
20
30
Count
27
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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
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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.
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