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The normal distribution
The normal curve:
Shape defined by the standard deviation and the mean
The normal distribution
FREQUENCY
Various weights of a group
of female students
20
30
40
50
WEIGHT (kg)
60
70
80
The normal distribution
The equation of the normal curve
Allows us to compute the height of the curve
y for a given value of x
1
e
Y 
 2
( X i  )2
2
2
The normal distribution
As the term (x-) becomes smaller, y becomes
larger and is at a maximum when x = .
s=1
s = 1.5
s=2
PROBABILITY
0.4
0.3
0.2
0.1
0.0
-5
-4
-3
-2
-1
0
X
1
2
3
4
5
The normal distribution
As the term (x-) becomes smaller, y becomes
larger and is at a maximum when x = .
0
1
2
PROBABILITY
0.4
0.3
0.2
0.1
0.0
-5
-4
-3
-2
-1
0
X
1
2
3
4
5
The normal distribution
A normal curve is symmetrical
Axis of symmetry passes through the baseline where x = 
(one of the parameters of the curve)
Theoretically, two tails never touch the horizontal axis.
The normal distribution
Vertical axis of the distribution re-scaled by dividing by
the number of observations - becomes a probability
distribution
The total probability encompassed by the density is 1.
The normal distribution
Total area under the curve is 100%:
The area bounded by one standard deviation on either
side of the central axis is approximately 68.26% of the
total area.
Normal distribution
Probability determination
example
• A normal distribution of values
• Mean = 50
• Standard deviation = 15
• What is the probability of finding a value
greater than 75?
Normal distribution
Z (standard) distribution
To convert a data point (Xi) to a Z score (Zi):
Zi
=
Xi - X
S
Normal distribution
Z (standard) distribution
•
•
•
•
Xi = 75, mean = 50, standard deviation = 15
Z score in example is 1.67
Table value = 0.0475
Probability = 0.0475
Normal distribution
Z (standard) distribution
Area under the two tails: (0.0475 x 2) = 0.095
1 – 0.095 = 0.905
0.0475
0.905
0.0475