Unit 5. The Normal Distribution - UMass
嚜燕ubHlth 540
The Normal Distribution
Page 1 of 23
Unit 5.
The Normal Distribution
Topics
1. Introduction ########..########..#
2. Definition of the Normal Distribution ########
3. The Sample Average is Often Normally Distributed
Introduction to the Central Limit Theorem ####.
4. A Feel for the Normal Distribution #######.#..
5. The Relevance of the Normal Distribution #####.
6. Calculation of Probabilities for the Normal(0,1) ###
7. From Normal( 米 , 考 2 ) to Normal(0,1) 每 The Z-Score ##.
8. From Normal(0,1) to Normal( 米 , 考 2 ) ########
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PubHlth 540
The Normal Distribution
Page 2 of 23
1. Introduction
Much of statistical inference is based on the normal distribution.
? The pattern of occurrence of many phenomena in nature
happens to be described well using a normal distribution model.
? Even when the phenomena in a sample distribution are not described well
by the normal distribution, the sampling distribution of sample
averages obtained by repeated sampling from the parent distribution
is often described well by the normal distribution (Central limit theory).
You may have noticed in your professional work (especially in reading the literature for
your field) that, often, researchers choose to report the average when he/she wishes to
summarize the information in a sample of data.
The normal distribution is appropriate for continuous random variables only.
?
Recall that, in theory, a continuous random variable can
assume any of an infinite number of values.
Therefore, we*ll have to refine our definition of a probability model to accommodate the
continuous variable setting.
? Pr[ X = x ] , the calculation of a point probability, is meaningless in the
continuous variable setting. In its place, we calculate
Pr [ a < X < b], the probability of an interval of values of X.
﹢
? For the above reason,
﹉ Pr [ X = x] is also without meaning.
?﹢
PubHlth 540
The Normal Distribution
Page 3 of 23
Following is the extension of the ideas of a probability distribution for a discrete random
variable to the ideas underlying the meaning of a probability distribution for a continuous
random variable. The ideas of calculus (sorry!) helps us out.
1st: ※List§ of all
possible values that
exhaust all possibilities
Discrete Random
Variable
E.g. 每
1, 2, 3, 4, #, N
2nd: Accompanying
probabilities of
※each value§
Pr [ X = x ]
Continuous Random
Variable
※List§ ? range
E.g. -﹢ to +﹢
0 to +﹢
※Point probability§ ? probability
density
Probability density of X , written
fX(x)
※Unit total§ ? unit integral
Total must be 1
max
﹉ Pr[ X = x] = 1
x = min
z
﹢
?﹢
f X ( x )dx = 1
PubHlth 540
The Normal Distribution
Page 4 of 23
2. Definition of the Normal Distribution
Definition of the normal probability distribution density function.
? The concept ※probability of X=x§ is replaced by the ※probability density function
fx ( ) evaluated at X=x§
? A picture of this function with X=x plotted on the horizontal and fx ( ) evaluated at
X=x§ plotted on the vertical is the familiar bell shaped (※Gaussian§) curve
fX(x)
.4
.3
.2
.1
0
-1
-2
fX (X=x) =
1
2
L ?a x ? 米f OP
exp M
N 2考 Q
X=x
2
1
2羽考
0
x
2
where
2
x = Value of X
Range of possible values of X: -﹢ to +﹢
Exp = e = Euler*s constant = 2.71828 #
羽 = mathematical constant = 3.14
note: e =
1 1 1 1 1
+ + + + + ...
0! 1! 2! 3! 4!
note: 羽 = (circumference/diameter) for any circle.
米 = Expected value of X (※the long run average§)
考2 = Variance of X. Recall 每 this is the expected value of [ X-米 ]2
PubHlth 540
The Normal Distribution
Page 5 of 23
The Standard Normal Distribution is a particular normal distribution. It is the one for
which 米=0 and 考2=1. It is an especially important tool in analysis of epidemiological
data.
? It is the one for which 米=0 and 考2=1.
? Tabulations of probabilities for this distribution are available.
? A random variable whose pattern of values is distributed
standard normal has the special name: z-score, or normal deviate
? By convention, it is usually written as Z, rather than X.
fZ (Z=z) =
LM OP
N Q
1
?z2
exp
2
2羽
Introduction to the Z-Score: A tool to compute probabilities of intervals of values for X
distributed Normal(米,考2).
? Of interest is a probability calculation for a random variable X that is
distributed Normal(米, 考2)
? However, tabulated normal probability calculations are available only for the
Normal Distribution with 米 = 0 and 考2=1. We solve our problem by exploiting an
equivalence argument.
? ※Standardization§ expresses the desired calculation for X as an equivalent
calculation for Z where Z is distributed standard normal, Normal(0,1).
pr a ≒ X ≒ b = pr
LMF a ? 米 I ≒ Z ≒ F b ? 米 I OP
NH 考 K H 考 K Q
Z-score =
. Thus,
X ?米
考
? Note - The technique of standardization of X involves ※centering§ (by
subtraction of the mean of X which is 米) followed by ※rescaling§ (using the
multiplier 1/考)
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