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DO YOU THINK YOU ARE NORMAL?
1.
2.
3.
Yes
33%
33%
No
I’m not average, but I’m probably within 2
standard deviations.
33%
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CHAPTER 6
The Standard Deviation as a Ruler and the
Normal Model
NORMAL PROBABILITY PLOTS
When you actually have your own data, you must
check to see whether a Normal model is
reasonable.
 Looking at a histogram of the data is a good way
to check that the underlying distribution is
roughly unimodal and symmetric.

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NORMAL PROBABILITY PLOTS (CONT.)
A more specialized graphical display that can
help you decide whether a Normal model is
appropriate is the Normal probability plot.
 If the distribution of the data is roughly Normal,
the Normal probability plot approximates a
diagonal straight line. Deviations from a straight
line indicate that the distribution is not Normal.

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NORMAL PROBABILITY PLOTS (CONT.)

Nearly Normal data have a histogram and a
Normal probability plot that look somewhat like
this example:
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NORMAL PROBABILITY PLOTS (CONT.)

A skewed distribution might have a histogram
and Normal probability plot like this:
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*RE-EXPRESSING SKEWED DATA TO
IMPROVE SYMMETRY
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TRANSFORMING DATA
y=Log(x)
 To
get original data back
x=10^y =10y
y=Sqrt(x)
 To
get original data back
x=y^2 = y*y
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*RE-EXPRESSING SKEWED DATA TO
IMPROVE SYMMETRY (CONT.)


One way to make a skewed
distribution more symmetric
is to re-express or transform
the data by applying a simple
function (e.g., logarithmic
function).
Note the change in skewness
from the raw data (previous
slide) to the transformed data
(right):
THE 68-95-99.7 RULE (CONT.)

The following shows what the 68-95-99.7 Rule
tells us:
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THREE TYPES OF QUESTIONS

What’s the probability of getting X or greater?

What’s the probability of getting X or less?

What’s the probability of X falling within in the
range Y1 and Y2?
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FINDING NORMAL PERCENTILES BY
HAND


When a data value doesn’t fall exactly 1, 2, or 3
standard deviations from the mean, we can look
it up in a table of Normal percentiles.
Table Z in Appendix E provides us with normal
percentiles, but many calculators and statistics
computer packages provide these as well.
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FINDING NORMAL PERCENTILES BY
HAND (CONT.)
Table Z is the standard Normal table. We have to convert
our data to z-scores before using the table.
 Figure 6.7 shows us how to find the area to the left when
we have a z-score of 1.80:

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FINDING NORMAL PERCENTILES

Use the table in Appendix E

Excel


=NORMDIST(z-stat, mean, stdev, 1)
Online

http://davidmlane.com/hyperstat/z_table.html
CATEGORIES OF RETARDATION
Severity of mental retardation can be broken into 4
levels:
 50-70 - Mild mental retardation
 35-50 - Moderate mental retardation
 20-35 - Severe mental retardation
 IQ < 20 - Profound mental retardation

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WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 20 OR LESS?
1.
2.
3.
4.
0.0001%
0.0000%
0.0004%
0.04%
25%
1
25%
25%
2
3
25%
4
WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 50 OR LESS?
1.
2.
3.
4.
0.0001%
0.0000%
0.0004%
0.04%
25%
1
25%
25%
2
3
25%
4
IQ - CATEGORIES
115-124 - Above average (e.g., university
students)
 125-134 - Gifted (e.g., post-graduate students)
 135-144 - Highly gifted (e.g., intellectuals)
 145-154 - Genius (e.g., professors)
 155-164 - Genius (e.g., Nobel Prize winners)
 165-179 - High genius
 180-200 - Highest genius
 >200 - "Unmeasurable genius"

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WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 155 OR MORE?
1.
2.
3.
4.
99.99%
.01%
.9999
.0001
25%
25%
25%
25%
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3.
4.
WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 120 OR MORE?
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2.
3.
4.
5.
1.333
.9082
.0918
90.82%
9.18%
20%
20%
20%
20%
20%
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5
AVERAGE HEIGHT IN INCHES
WHAT FRACTION OF MEN ARE LESS THAN
5’9 FOOT TALL?
1.
2.
3.
4.
50%
.1027
54.09%
45.91%
25%
1
25%
25%
2.
3
25%
4
WHAT FRACTION OF WOMEN ARE LESS
THAN 5’9 FOOT TALL?
1.
2.
3.
4.
1.78
96.25%
3.75%
45.91%
25%
25%
25%
25%
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FROM PERCENTILES TO SCORES: Z IN
REVERSE
Sometimes we start with areas and need to find
the corresponding z-score or even the original
data value.
 Example: What z-score represents the first
quartile in a Normal model?

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HEIGHT PROBLEM


At what height does a quarter of men fall below?
At what height does a quarter of women fall
below?
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FROM PERCENTILES TO SCORES: Z IN
REVERSE (CONT.)
Look in Table Z for an area of 0.2500.
 The exact area is not there, but 0.2514 is pretty
close.


This figure is associated with z = -0.67, so the
first quartile is 0.67 standard deviations below
the mean.
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Z SCORE CALCULATORS

Excel
=NORMINV(prob, mean, stdev)
 =NORMINV(0.25, 0, 1)

Online
 http://stattrek.com/online-calculator/normal.aspx


Calculator
TI – 83/84
 TI-89

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TI- 83/84
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TI - 89
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RECOVERING THE MEAN AND STANDARD
DEV.
17.5% 18 and under
 7.6% 65 and over


What is the mean age and the standard deviation
of the population?
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FOR NEXT WEEK…



Sunday HW3 (2/3) by midnight
Thursday Quiz 2, covers HW3 and HW4 of the
material learned in class.
Data Project Step 2 – Due Tuesday (2/12).