7.1
Properties of the
Normal Distribution
BETWEEN NUMBERS
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Sometimes we want to model a random
variable that is equally likely between two
limits
Examples


Choose a random time … the number of
seconds past the minute is random number in
the interval from 0 to 60
Observe a tire rolling at a high rate of speed …
choose a random time … the angle of the tire
valve to the vertical is a random number in the
interval from 0 to 360
UNIFORM PROBABILITY DISTRIBUTION

When “every number” is equally likely in an
interval, this is a uniform probability distribution
Any specific number has a zero probability of
occurring
 The mathematically correct way to phrase this is that
any two intervals of equal length have the same
probability

EXAMPLE
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●
For the seconds after the minute example
Every interval of length 3 has probability
3/60



The chance that it will be between 14.4 and
17.3 seconds after the minute is 3/60
The chance that it will be between 31.2 and
34.2 seconds after the minute is 3/60
The chance that it will be between 47.9 and
50.9 seconds after the minute is 3/60
EXAMPLE CONTINUED
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●
For the seconds after the minute example
Every individual value has probability 0
The chance that it will be exactly 13.4 seconds
after the minute (i.e. between 13.4 and 13.4) is
0/60 or 0
 The chance that it will be exactly 31.2 seconds
after the minute (i.e. between 31.2 and 31.2) is
0/60 or 0
 The chance that it will be exactly 47.9 seconds
after the minute (i.e. between 47.9 and 47.9) is
0/60 or 0

●
We need to use a different way to specify the
probabilities
PROBABILITY DENSITY FUNCTION
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●
A probability density function is an
equation used to specify and compute
probabilities of a continuous random
variable
This equation must have two properties


The total area under the graph of the equation
is equal to 1 (the total probability is 1)
The equation is always greater than or equal
to zero (probabilities are always greater than
or equal to zero)
PROBABILITY
This function method is used to represent the
probabilities for a continuous random variable
 For the probability of X between two numbers

Compute the area under the curve between the two
numbers
 That is the probability

PROBABILITY

The probability of being between 4 and 8
The probability
From 4 (here)
To 8 (here)
PROBABILITY
●
An interpretation of the probability
density function is
The random variable is more likely to be in
those regions where the function is larger
 The random variable is less likely to be in
those regions where the function is smaller
 The random variable is never in those regions
where the function is zero

GRAPH OF PROBABILITY

A graph showing where the random variable has
more likely and less likely values
More likely values
Less likely values
TIME EXAMPLE

The time example … uniform between 0 and 60

All values between 0 and 60 are equally likely, thus
the equation must have the same value between 0
and 60
TIME EXAMPLE CONTINUED

The time example … uniform between 0 and 60

Values outside 0 and 60 are impossible, thus the
equation must be zero outside 0 to 60
TIME EXAMPLE CONTINUED

The time example … uniform between 0 and 60

Because the total area must be one, and the width of
the rectangle is 60, the height must be 1/60
1/60
TIME EXAMPLE CONTINUED

The time example … uniform between 0 and 60

The probability that the variable is between two
numbers is the area under the curve between them
1/60
NORMAL CURVE
The normal curve has a very specific bell shaped
distribution
 The normal curve looks like

NORMAL
●
●
A normally distributed random variable, or a
variable with a normal probability distribution, is
a random variable that has a relative frequency
histogram in the shape of a normal curve
This curve is also called the normal density curve
(a particular probability density function)
INFLECTION POINT

In drawing the normal curve, the mean μ and the
standard deviation σ have specific roles
The mean μ is the center of the curve
 The values (μ – σ) and (μ + σ) are the inflection points
of the curve

DIFFERENT NORMAL?
There are normal curves for each combination of
μ and σ
 The curves look different, but the same too
 Different values of μ shift the curve left and right
 Different values of σ shift the curve up and down

NORMAL

Two normal curves with different means (but the
same standard deviation)

The curves are shifted left and right
NORMAL

Two normal curves with different standard
deviations (but the same mean)

The curves are shifted up and down
“NORMAL” PROPERTIES
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Properties of the normal density curve
The curve is symmetric about the mean
 The mean = median = mode, and this is the
highest point of the curve
 The curve has inflection points at (μ – σ) and
(μ + σ)
 The total area under the curve is equal to 1
 The area under the curve to the left of the
mean is equal to the area under the curve to
the right of the mean

NORMAL PROPERTIES
●
Properties of the normal density curve

●
As x increases, the curve goes to zero … as x
decreases, the curve goes to zero
In addition, the empirical rule holds
The area within 1 standard deviation of the
mean is approximately 0.68
 The area within 2 standard deviations of the
mean is approximately 0.95
 The area within 3 standard deviations of the
mean is approximately 0.997

NORMAL PROPERTIES
The curve is symmetric about the mean
 Because the curve is symmetric

The mean = median = mode = μ
 The area under the curve to the left of the mean is
equal to the area under the curve to the right of the
mean

NORMAL PROPERTIES
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The highest point of the curve is at x = μ

●
This can be seen in a previous chart
The total area is equal to 1
This is a complex calculation
 But it is true

NORMAL PROPERTIES
●
It has inflection points (where the
concavity changes) at (μ – σ) and (μ + σ)

●
This can be seen in a previous chart
As x increases, the curve goes to zero … as
x decreases, the curve goes to zero

This is clear from the chart also
NORMAL PROPERTIES
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The empirical rule is true



Approximately 68% of the values lie between
(μ – σ) and (μ + σ)
Approximately 95% of the values lie between
(μ – 2σ) and (μ + 2σ)
Approximately 99.7% of the values lie between
(μ – 3σ) and (μ + 3σ)
REMINDER…

An illustration of the Empirical Rule
NORMAL CURVE


The equation of the normal curve with mean μ
and standard deviation σ is
This is a complicated formula, but we will never
need to calculate it (thankfully)
AREA UNDER THE CURVE

When we model a distribution with a normal
probability distribution, we use the area under
the normal curve to
Approximate the areas of the histogram being
modeled
 Approximate probabilities that are too detailed to be
computed from just the histogram

EXAMPLE

Assume that the distribution of giraffe weights
has μ = 2200 pounds and σ = 200 pounds
EXAMPLE

What is an interpretation of the area under the
curve to the left of 2100?
EXAMPLE

It is the proportion of giraffes that weigh 2100
pounds and less
HOW?
●
How do we calculate the areas under a
normal curve?
If we need a table for every combination of μ
and σ, this would rapidly become unmanageable
 We would like to be able to compute these
probabilities using just one table
 The solution is to use the standard normal
random variable

HOW?

The standard normal random variable is the
specific normal random variable that has
μ = 0 and
 σ=1


We can relate general normal random variables to
the standard normal random variable using a Zscore calculation
Z- SCORE

If X is a general normal random variable with mean
μ and standard deviation σ then
is a standard normal random variable
 This equation connects general normal random
variables with the standard normal random variable
 We only need a standard normal table
EXAMPLE CONTINUED

The area to the left of 2100 for a normal curve
with mean 2200 and standard deviation 200
Z-SCORE

To compute the corresponding value of Z, we use
the Z-score
Thus the value of X = 2100 corresponds to a value
of Z = – 0.5
 In the next section, we will learn how to use this
to find the probability!!!

SUMMARY
Normal probability distributions can be used to
model data that have bell shaped distributions
 Normal probability distributions are specified by
their means and standard deviations
 Areas under the curve of general normal
probability distributions can be related to areas
under the curve of the standard normal
probability distribution

THE BIRTH WEIGHTS OF FULL-TERM BABIES ARE
NORMALLY DISTRIBUTED WITH MEAN 3,400 GRAMS
AND STANDARD DEVIATION 505 GRAMS
a)
Draw a normal curve with the parameters labeled
b)
Shade the region that represents the proportions of full-term
babies who weigh more than 4,410 grams
Suppose that the area under the curve to the right of x = 4,410
is .0228. Provide two interpretations of this result.
c)