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Normal Distribution
Section 2.2
Introduce the Normal Distribution
Properties of the Standard Normal
Use Normal Distribution in an inferential
Theoretical Distribution
Empirical distributions
based on data
Theoretical distribution
based on mathematics
derived from model or estimated from data
Normal Distribution
Why are normal distributions so important?
 Many variables are commonly assumed to be
normally distributed in the population
Height, weight, IQ scores, ACT scores, etc.
If a variable is approximately normally
distributed we can make inferences about
values of that variable
Normal Distribution
Since we know the shape of the curve, we can
calculate the area under the curve
The percentage of that area can be used to determine
the probability that a given value could be pulled
from a given distribution
The area under the curve tells us about the probability- in
other words we can obtain a p-value for our result (data)
by treating it as a normally distributed data set.
Key Areas under the Curve
For normal
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.7%
Standard Normal Distributions
Standard Normal Distribution – N(, )
We agree to use the
standard normal
Roughly symmetric
Recall: Z-score
If we know the population mean and
population standard deviation, for any value
of X we can compute a z-score by subtracting
the population mean and dividing the result by
the population standard deviation
X 
Total area under the curve
is 1
The area in red is equal to
p(z > 1)
The area in blue is equal to
p(-1< z <0)
Since the properties of the
normal distribution are
known, areas can be looked
up on tables or found with
a calculator.
Suppose Z has standard normal distribution
Find 0 < Z < 1.23
Find -1.57 < Z < 0
Find Z > 0.78
Z is standard normal
Calculate -1.2 < Z < 0.78
Work time...
What is the area for scores less than z = -1.5?
What is the area between z =1 and 1.5?
What z-score cuts off the highest 30% of the
What two z-scores enclose the middle 50% of the