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Section 6 – 3A: The Standard Normal Distribution Introduction
The Normal Distribution
There are many different normal distributions. Each normal distribution has a different bell shaped
curve based on the values for the mean and standard deviation of the distribution. The larger the
mean the farther to the right of x = 0 the curve is located on the x axis. The larger the standard
deviation the more short and spread out the curve is.
Each of the different normal curves has a different mean and / or standard deviation.
The family of curves this general equation represents looks like either of the curves below.
σ1
σ2
µ1
µ2
x
One of the many Normal Distributions is chosen to be the Standard Normal Distribution
We will choose one of the many possible different Normal Distribution curves and select it as the
distribution we will use to solve all problems that involve normal distributions. This curve will be a
normal distribution that has a mean of 0 and a standard deviation of 1. We call this distribution the
Standard Normal Distribution or the Z Distribution. There is only one Standard Normal
Distribution or Z Distribution. The curve that represents the Z Distribution is called the Z
curve. We label the horizontal axis with the letter Z to show that the curve is the Standard
Normal Curve.
A Normal Curve with µ x = 3 and σ x = .6
µx = 3
σ x = .6
Stat 300 6 – 3A Lecture
The Standard Normal Curve with µ = 0 and σ = 1
µ=0
σ =1
x
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z
© 2012 Eitel
The Standard Normal Curve or z Curve
always has a mean of 0 and a Standard Deviation of 1
µ=0
σ =1
z
The horizontal z axis is a real number line
with negative z values to the left of 0
and positive z values to the right of 0
–3
–2 –1
negative
z values
0
1
2
3
positive
z values
z
The total area under the standard normal or z curve is 1
and
the z curve is symmetric about the mean of 0
This means that
the area under the curve to the left of z = 0 is .5000
and the area under the curve to the right of z = 0 is also .5000
area to the
right of 0
= .5000
area to the
left of 0
= .5000
–3
Stat 300 6 – 3A Lecture
–2 –1
negative
z values
0
1
2
3
positive
z values
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z
© 2012 Eitel
The Standard Normal Curve or z Curve
always has a mean of 0 and a Standard Deviation of 1
µ=0
σ =1
z
Basic Properties of the Standard Normal Distribution or z Curve.
1. There is only 1 Standard Normal Distribution or z curve.
2. The horizontal axis is labeled z to distinguish it from normal curves with an x axis.
3. The curve extends indefinitely in both directions, approaching but never touching the z axis.
4. The Mean is 0 and the Standard Deviation is 1
5. The curve is symmetric about the mean of 0.
6. The total area under the curve = 1
7. If a vertical line is drawn at an given value of z on the z axis then the area under the curve is divided
into two areas. One area will be to the left of the z value (the yellow area) and the other area will
be to the right of the z value(the white area).
µ=0
z= 1.5
z = – 2.1
8.
The areas of both the yellow and the white regions will be a decimal number less than 1.
9.
The total of the yellow and white areas will be 1.
10. The yellow areas shown above represents the probability of selecting one number from the z
distribution and having that number be less than the z value. P( x < z )
11. The white areas shown above represents the probability of selecting one number from the z
distribution and having that number be more than the z value. P( x > z )
Stat 300 6 – 3A Lecture
Page 3 of 4
© 2012 Eitel
The shaded area under a z curve represents a probability.
The probability of a number in the z distribution being less than a given z value
The area under the z curve that is to the left of a given z value represents the probability of
selecting one number from the z distribution and having that number be less than the z value.
The yellow area to
the left of z = 2.1
represents
P( z < 2.1)
The yellow area to
the left of z = –2.1
represents
P( z < –2.1)
z = – 2.1
z = 2.1
The probability of a number in the z distribution being more than a given z value
The area under the z curve that is to the right of a given z value represents the probability of
selecting one number from the z distribution and having that number be greater than the z value.
The yellow area to
the right of z = –2.1
represents
P( z > –2.1)
The yellow area to
the right of z = 2.1
represents
P( z > 2.1)
z = – 2.1
z= 2.1
The probability of a number in the z distribution being between two given z values
The area under the z curve between two given z values represents the probability of selecting
one number from the z distribution and having that number be between the two given z values.
The yellow area between
z = – 2.1 and z = – 1.3
represents
P( –2.1 < z < –1.3)
The yellow area between
z = – 2.3 and z = 1.8
represents
P( –2.3 < z < 1.8)
z= –2.1 z= –1.3
Stat 300 6 – 3A Lecture
z= –2.3
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z= 1.8
© 2012 Eitel