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STANDARD DEVIATION & THE NORMAL
MODEL
What is a normal distribution?
• The normal distribution is pattern for the distribution
of a set of data which follows a bell shaped curve.
• This distribution is sometimes called the Gaussian
distribution in honor of Carl Friedrich Gauss, a
famous mathematician.
• The bell shaped curve has several properties:
• The curve concentrated in the center and decreases
on either side. This means that the data has less of a
tendency to produce unusually extreme values,
compared to some other distributions.
• The bell shaped curve is symmetric. This tells you
that he probability of deviations from the mean are
comparable in either direction.
1
STANDARD DEVIATION & THE NORMAL
MODEL
• When you want to describe probability for a
continuous variable, you do so by describing
a certain area.
• A large area implies a large probability and a
small area implies a small probability. Some
people don't like this, because it forces them
to remember a bit of geometry (or in more
complex situations, calculus). But the
relationship between probability and area is
also useful, because it provides a visual
interpretation for probability.
• Here's an example of a bell shaped curve.
This represents a normal distribution with a
mean of 50 and a standard deviation of 10.
2
DESCRIBING DISTRIBUTION
NUMERICALLY
3
STANDARD DEVIATION & THE NORMAL
MODEL
4
Formula
• Standardizing normal
variables:
• Formula:
Y   
Z 

  
5
68-95-99.7 Rule
• 68% of the observations are within 1
standard deviation unit
• 95% of the observations are within 2
standard deviation unit
• 99.7% of the observations are within 3
standard deviation unit
• http://davidmlane.com/hyperstat/normal_dist
ribution.html
6
Example
Some IQ tests are standardized to a Normal model with a mean of 100
and a standard deviation of 16.
a) Describe the 68-95-99.7 rule for this problem
b) About what percent of people should have IQ scores above 116?
c) About what percent of people should have IQ scores between 68 and
84?
d) About what percent of people should have IQ scores above 132?
e) About what percent of people should have IQ scores above 120?
f) About what percent of people should have IQ scores below 90?
g) About what percent of people should have IQ scores between 95 and 130?
h) A person is a genius if his/her IQ belong to the top 10% of the all IQ
scores. What minimum IQ score qualifies you to be a genius?
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Answers to the Example
Some IQ tests are standardized to a Normal model with a mean of 100 and a standard deviation of 16.
b) 16%
c) 13.5%
d) 2.5%
e) About what percent of people should have IQ scores above 120?
Z = (120 -100)/16 = 1.25
Find P(Z > 1.25) from standard normal chart or your TI calculator.
Answer: 1-.8944 = .1056
f) About what percent of people should have IQ scores below 90?
Z = (90 -100)/16 = -0.625
Find P(Z < -0.625) from standard normal chart or your TI calculator.
Answer: .26
g) About what percent of people should have IQ scores between 95 and 130?
Z = (95 -100)/16 = -0.3125
Z = (130 -100)/16 = 1.875
Find P(-0.3125< Z < 1.875) = .9699-.3783 = .5916
h) A person is a genius if his/her IQ belong to the top 10% of the all IQ scores. What
minimum IQ score qualifies you to be a genius?
The top 10% corresponds to the 90th percentile. For the standard normal the 90th
percentile is 1.28. Hnece solve 1.28 = (Y-100)/16. The value of Y is 120.48.
8
Example
• In 2006 combined verbal and math SAT scores
followed a normal distribution with mean 1020
and standard deviation 240.
• Suppose you know that Peter scored in the top
3% of SAT scores. What was Peter’s
approximate SAT score?
• Answer: 1471.2
9
Using the TI-83 to Find a Normal Percentage
Always draw a
• The TI-83 provides a function named normalcdf picture!
– Press 2nd, DISTR (found above VARS)
– Scroll to normalcdf ( and press ENTER, or press 2.
• If z has a standard normal distribution:
– Percent(a < z < b) = normalcdf ( a , b )
– Example: to find P( -1.2 < z < .8 ),
press 2nd, DISTR, 2, then -1.2 , .8 )
– Note that the comma between -1.2 and .8 must be
entered
– Read .6731
?
-1.2
.8
• To find Percent( z < a ), enter normalcdf ( -5 , a )
– Example: normalcdf( -5 , 1.96 ) gives .9750
?
• To find Percent( z > a ), enter normalcdf ( a , 5 )
1.96
– Example: normalcdf( -1.645 , 5 ) gives .9500
?
-1.645
10
Using the TI-83/84 for Normal Percentages
Without Computing z-Scores
We can let the TI find its own z-scores:
– Find Percent(90 < x < 105) if x follows the normal model with mean 100
and standard deviation 15:
• Percent(90 < x < 105) = normalcdf( 90 , 105 , 100 , 15)
= .378
Notice that this is a time-saver for this type of problem, but that you may still
need to be able to compute z-scores for other types of problems!
x1
x2
11
Suppose We’re Given a normal
Percentage and Need A z-score?
• IQ scores are distributed normally with a
mean of 100 and a standard deviation of
15. What score do you need to capture
the bottom 2%?
– That is, we must find a so that Percent(x < a) = 2%
when x has a normal distribution with a mean of 100
and a standard deviation of 15.
– With the TI 83/84:
a = invNorm( .02, 100 , 15) = 69.2
x
12