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PubHlth 540 The Normal Distribution Page 1 of 23
Unit 5.
The Normal Distribution
Topics 1. Introduction ....................................................... 3
2. Definition of the Normal Distribution ........................ 4
3. The Sample Average is Often Normally Distributed
Introduction to the Central Limit Theorem ............. 7
4. A Feel for the Normal Distribution ........................... 10
5. The Relevance of the Normal Distribution ................ 12
6. Calculation of Probabilities for the Normal(0,1) ......... 13
7. From Normal( , 2 ) to Normal(0,1)  The ZScore ....... 19
8. From Normal(0,1) to Normal( , 2 ) ........................ 22
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.
Discrete Random Continuous Random
Variable Variable
1st: "List" of all E.g.  "List" ? range
possible values that 1, 2, 3, 4, ..., N E.g.  to +
exhaust all possibilities 0 to +
"Point probability" ? probability
2nd: Accompanying Pr [ X = x ] density
probabilities of
"each value" Probability density of X , written
fX(x)
Total must be 1 "Unit total" ? unit integral
max
f X (x)dx = 1
Pr[ X = x] = 1 z
x=min 
How do you explain normal distribution? Normal Distribution is a bellshaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ ...
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