DO YOU THINK YOU ARE NORMAL?
1.
2.
3.
Yes
17%
17%
17%
17%
No
I’m not average, but I’m probably within 2
standard deviations.
17%
17%
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2
3
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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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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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IQ – CATEGORIZES
Over 140 - Genius or near genius
 120 - 140 - Very superior intelligence
 110 - 119 - Superior intelligence
 90 - 109 - Normal or average intelligence
 80 - 89 - Dullness
 70 - 79 - Borderline deficiency
 Under 70 - Definite feeble-mindedness

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ASKING QUESTIONS OF A DATASET



What is the probability that someone has an IQ
over 100?
What is the probability that someone has an IQ
lower than 85?
What is the probability that someone has an IQ
between 85 and 130?
PROBLEM 8
A
B
C

ABOUT WHAT PERCENT OF PEOPLE
SHOULD HAVE IQ SCORES ABOVE 145?
.3%
 .15%
 3%
 1.5%
 5%
 2.5%

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WHAT PERCENT OF PEOPLE SHOULD HAVE
IQ SCORES BELOW 130?
95%
 5%
 2.5%
 97.5%

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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

Use the table in Appendix E

Excel


=NORMDIST(z-stat, mean, stdev, 1)
Online

http://davidmlane.com/hyperstat/z_table.html
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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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?
0.0001%
 0.0000%
 0.0004%
 0.04%

WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 50 OR LESS?
0.0001%
 0.0000%
 0.0004%
 0.04%

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?
99.99%
 .01%
 .9999
 .0001

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WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 120 OR MORE?
1.333
 .9082
 .0918
 90.82%
 9.18%

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AVERAGE HEIGHT IN INCHES
WHAT FRACTION OF MEN ARE LESS THAN
5’9 FOOT TALL?
50%
 .1027
 54.09%
 45.91%

WHAT FRACTION OF WOMEN ARE LESS
THAN 5’9 FOOT TALL?
1.78
 96.25%
 3.75%
 45.91%

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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

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 and the standard deviation of
the population?
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FOR NEXT WEEK…

Monday HW3

Data Project Step 2 – Due Tuesday.

Thursday Quiz 2, covers HW2, HW3 and HW4 of
the material learned in class.