2 Normal Distribution

McGill University
Advanced Business Statistics
MGSC-372

Review
Normal Distribution

The Normal Distribution aka The Gaussian Distribution

The Normal Distribution y 1 f ( x)  e 2



1  x  
 

2  

2

x

Areas under the Normal Distribution curve

-3

-2

-


68%
95%
99.7%

+

+2

+3

X = N( ,

2 )

Determining Normal Probabilities
Since each pair of values for  and  represents a different distribution, there are an infinite number of possible normal distributions. The number of statistical tables would be limitless if we wished to determine probabilities for all of them.
The standard normal distribution z has a mean of 0 and a standard deviation of 1, i.e. z = N(0,1), and provides a basis for computing probabilities for all normal distributions. We must therefore convert each normal random variable X into the standard normal random variable z using the standardization formula: x z 

Standardization Formula

x z 
The standardization formula converts a normal distribution x = N(,2) into the standard normal distribution z = N(0,1)

1  12 z 2 z e
2

“Destandardization” formula

Standard Normal Distribution Areas

-3

-2

-1

0
68%
95%
99.7%

1

2

3

z=
N(0,1)

Characteristics of the Normal Distribution

1.

The distribution has two parameters: the mean () and the standard deviation ().

2. The mean will determine where the curve is centered on the x-axis and the standard deviation will determine how spread out the curve is. 3. It is symmetrical about its mean and is bell-shaped, regardless of  and . Therefore 50% of the area falls to the left of the mean and
50% of the area falls to the right of the mean.
4. The mean, median and mode are all equal i.e. it is symmetrical.
5. The total area under the curve is equal to 1.

…more properties of the normal distribution
6. The probability of a value falling between point a and point b is equal to the area under the bell curve between point a and point b.

7. Each