Transcript The normal distribution

The normal
distribution
(Session 08)
SADC Course in Statistics
Learning Objectives
At the end of this session you will be able to:
• describe the normal probability distribution
• state and interpret parameters associated
with the normal distribution
• use a calculator and statistical tables to
calculate normal probabilities
• appreciate the value of the normal
distribution in practical situations
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The Normal Distribution
• In the previous two sessions, you were
introduced to two discrete distributions,
the Binomial and the Poisson.
• In this session, we introduce the Normal
Distribution – one of the commonest
distributions followed by a continuous
random variable
• For example, heights of persons, their
blood pressure, time taken for banana
plants to grow, weights of animals, are
likely to follow a normal distribution
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Example: Weights of maize cobs
Graph shows
histogram of
100 maize
cobs.
Data which
follows the bell
shape of this
histogram are
said to follow a
normal
distribution.
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Frequency definition of probability
In a histogram, the bar areas correspond to
frequencies.
For example, there are 3 maize cobs with
weight < 100 gms, and 19 maize cobs with
weight < 120 gms.
Hence, using the frequency approach to
probability, we can say that
Prob(X<120) = 19/100 = 0.019
The areas under the curve can be regarded as
representing probabilities since the curve and
edges of histogram would coincide for n=.
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Probability Distribution Function
f(x)
0.4
Two parameters associated
with the normal
distribution, its mean 
and variance 2.
0.2
0.0

 
x
The mathematical expression describing
the form of the normal distribution is
f(x) = exp(–(x–)2/22)/(22)
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Properties of the Normal Distribution
f(x)
0.4
0.2
0.0

 
x
• Total area under the curve is 1
• characterised by mean & variance: N(,2 )
• symmetric about mean ()
• 95% of observations lie within ± 2 of mean
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The Standard Normal Distribution
This is a distribution with =0 and =1, shown
below in comparison with N(0,2), =3.
0.4 f(x)


0.2


-6
-4
-2
0
+2
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+4
x
+6
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The Standard Normal Distribution
This is a distribution with =0 and =1.
Tables give probabilities associated with this
distribution, i.e. for every value of a random
variable Z which has a standard normal
distribution, values of Pr(Z<z) are tabulated.
In graph on right,
P=Pr(Z<z).
P
Symmetry means
any area (prob)
can be found.
0
z
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Calculating normal probabilities
Any random variable, say X, having a
normal distribution with mean  and
standard deviation , can be converted to a
value (say z) from the standard normal
distribution.
This is done using the formula
z = (X - ) / 
The z values are called z-scores. The z
scores can be used to compute probabilities
associated with X.
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An example
The pulse rate (say X) of healthy individuals is
expected to have a normal distribution with
mean of 75 beats per minute and a standard
deviation of 8. What is the chance that a
randomly selected individual will have a pulse
rate < 65?
We need to find Pr(X < 65)
i.e. Pr(X - 75 < 65 - 75)
= Pr[ (X – 75/8) < (-10/8) ]
= Pr(Z < -1.28)  Pr(Z<-1.3) = 0.0968
n)
(using
tables
of
the
standard
normal
dist
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A practical application
Malnutrition amongst children is generally
measured by comparing their weight-for-age
with that of a standard, age-specific reference
distribution for well-nourished children.
A child’s weight-for-age is converted to a
standardised normal score (an z-score),
standardised to  and  of the reference
distribution for the child’s gender and age.
Children whose z-score<-2 are regarded as
being underweight.
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A Class Exercise
Similarly to the above, height-for-age is used
as a measure of stunting again converted to a
standardised z score (stunted if z-score<-2).
Suppose for example, the reference
distribution for 32 months old girls has mean
91 cms with standard deviation 3.6 cms.
What is the probability that a randomly
selected girl of 32 months will have height
between 83.8 and 87.4 cms?
Graph below shows the area required. A
class discussion will follow to get the answer.
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Depicting required probability as an
area under the normal curve
83.8
87.4
91.0
94.6
98.2
Answer =
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Is a child stunted?
Suppose a 32 month old girl has height-forage value = 82.1
Would you consider this child to be stunted?
Discuss this question with your neighbour
and write down your answer below.
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Cumulative normal distribution
Cumulative
distribution is
given by the
function
F(x) = P(X ≤a)
a
x
In example above, the shaded area is 0.6,
the value of a from tables of the standard
normal distribution is 0.726.
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a
b
P(a<X<b) = F(b)-F(a) is the area under the
cumulative normal curve between points a
and b.
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Practical work follows …
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