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NORMAL DISTRIBUTION
ashokkr
September 2015
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Introduction
NORMAL DISTRIBUTION:-Normal distributions are a family of distributions
that have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes
described as bell shaped. The simplest case of normal distribution, known as
the Standard Normal Distribution, has expected value zero and variance one.
This is written as N (0, 1).
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Characteristics
• The area under the curve is 1
• It is a continuous distribution
• It is symmetrical about the mean. Each half of the distribution is a mirror
image of the other half.
• It is asymptotic to the horizontal axis.
Examples:• Many things are normally distributed, or very close to it. For example, height
and intelligence are approximately normally distributed; measurement errors
also often have a normal distribution.
• The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even
when the distribution is not normal.
• There is a very strong connection between the size of a sample N and the
extent to which a sampling distribution approaches the normal form. Many
sampling distributions based on large N can be approximated by the normal
distribution even though the population distribution itself is not normal.
• The normal distribution is the most used statistical distribution. The principal reasons are: normality arises naturally in many physical, biological, and
social measurement situations.
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Test strengths and weaknesses
The strengths of normal distribution are:
• probably the most widely known and used of all distributions.
• infinitely divisible probability distributions.
• strictly stable probability distributions.
The weakness of normal distributions is for reliability calculations. In this case,
using the normal distribution starts at negative infinity. This case is able to
result in negative values for some of the results
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Statistical formula
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P(x) =e−(x−µ) /2σ σ√12π
W here : −
µ = mean of x.
σ=varience of x.
π =3.14159.....
e=2.7182.....
• We usually work with the standardized normal distribution, where
µ=0 and σ=1, N (0, 1).
• We first convert the problem into an equivalent one dealing
with a normal variable measured in standardized deviation units,
called a standardized normal variable. To do this, if X ∼ N (µ,
σ), then Z = X- µ/ σ ∼ N (0, 1)
• A table of standardized normal values (from the table in the first
appendix of this paper) can then be used to obtain an answer in
terms of the converted problem.
• If necessary, we can then convert back to the original units of
measurement. To do this, simply note that, if we take the formula for Z, multiply both sides by , and then add to both sides,
we get X= Z σ + µ
• The interpretation of Z values is straightforward. Since σ= 1, if Z
= 2, the corresponding X value is exactly 2 standard deviations
above the mean. If Z = -1, the corresponding X value is one
standard deviation below the mean. If Z = 0, X = the mean,
i.e. µ.
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Distribution graph
The distribution is symmetric about y-axis.
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