Download Properties of Normal Distributions

Properties of
Normal
Distribution
The bell curve is known as the normal
distribution. This can be described in a single
mathematical equation. It can be used to
calculate probabilities in wide ranges of
contexts.
A population can be described by its mean ()
and the standard deviation (sigma ).
The smaller the value of Sigma, the more the
data clusters around the mean, so the narrower
the bell shape. Larger values of Sigma create a
larger bell shape.
Make predictions using the normal model
Karen is 168 cm tall. In her high school, boys’
heights are normally distributed with a mean of
174 cm and a standard deviation of 6 cm.
What is the probability that the first boy Karen
meets at school tomorrow will be taller than
she is?
From a normal distribution curve, we know that
68% of the boy’s heights lie with 1 lambda (6
cm) of the mean which means that 68% of the
boys are within 168 and 180 cm tall. Since the
curve is symmetrical, we know that the bottom
half is 34%
1 Sigma to either side is 68%
2 Sigma to either side is 95%
3 Sigma to either side is 99.7%
The curve of the equation is given by
1
 2
e
1  x 
 

2  
2
This equation is not easily found, which means
that computers are needed to make it simpler for
us to comprehend.
The distribution of the z-scores of a normally
distributed variable is a normal distribution with
a mean of 0 and a standard deviation of 1. This
is called the Standard Normal Distribution.
Areas are found on page 606 for easy reference.
Example
Mike owns a company that produces cereal. In
every box, there are 350 g of cereal. The actual
masses have a normal distribution with mean
352 g and a standard deviation of 4 g. Mike
does not want a cereal box to be sold if it is
under 347 g. What percentage of cereal boxes
does not contain this much cereal?
Homework
Pg 430 # 1,2,3,4,6