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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 1
Chapter 6
The Standard Deviation
as a Ruler and the
Normal Model
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Standard Deviation as a Ruler



The trick in comparing very different-looking
values is to use standard deviations as our rulers.
The standard deviation tells us how the whole
collection of values varies, so it’s a natural ruler
for comparing an individual to a group.
As the most common measure of variation, the
standard deviation plays a crucial role in how we
look at data.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 3
Standardizing with z-scores

We compare individual data values to their mean,
relative to their standard deviation using the
following formula:
y  y

z
s

We call the resulting values standardized values,
denoted as z. They can also be called z-scores.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 4
Standardizing with z-scores (cont.)



Standardized values have no units.
z-scores measure the distance of each data
value from the mean in standard deviations.
A negative z-score tells us that the data value is
below the mean, while a positive z-score tells us
that the data value is above the mean.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 5
Benefits of Standardizing


Standardized values have been converted from
their original units to the standard statistical unit
of standard deviations from the mean.
Thus, we can compare values that are measured
on different scales, with different units, or from
different populations.
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Slide 6- 6
Standardizing Variables

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Height of women
Height of men
y  66, s y  2.5
x  70, sx  3
I am 67 inches tall.
My brother is 72 inches
tall.
Who is taller
(comparatively)?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 7
Example: Standardizing Variables

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Height of women
Height of men
y  66, s y  2.5
x  70, sx  3
I am 67 inches tall.
My brother is 72 inches tall.
Who is taller (comparatively)?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 8
Standardizing Variables
y  y 67  66
z

 0.4
sy
2.5
x  x 72  70
z

 0.67
sx
3
I am 0.4 standard deviations above mean height for
women.
My brother is 0.67 standard deviations above mean
height for men.
My brother is taller (comparatively).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 9
Example: SAT vs. ACT


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You took SAT and scored 550.
Your friend took ACT and scored 30.
Which score is better?
 SAT has mean 500 and standard deviation 100.
 ACT has mean 18 and standard deviation 6.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 10
Example: SAT vs. ACT

Your score
550  500
 0.5
100

Friend’s score
30  18
2
6

Your friend scored better on ACT than you did
on SAT.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 11
Shifting Data

Shifting data:
 Adding (or subtracting) a constant amount to
each value just adds (or subtracts) the same
constant to (from) the mean. This is true for the
median and other measures of position too.
 In general, adding a constant to every data
value adds the same constant to measures of
center and percentiles, but leaves measures of
spread unchanged.
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Slide 6- 12
Shifting Data (cont.)

The following histograms show a shift from men’s
actual weights to kilograms above recommended
weight:
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Slide 6- 13
Rescaling Data

Rescaling data:
 When we divide or multiply all the data values
by any constant value, both measures of
location (e.g., mean and median) and
measures of spread (e.g., range, IQR,
standard deviation) are divided and multiplied
by the same value.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 14
Rescaling Data (cont.)

The men’s weight data set measured weights in
kilograms. If we want to think about these weights in
pounds, we would rescale the data:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 15
Example: An appliance repair shop charges a $30
service call to go to a home for a repair. It also
charges $25 per hour for labor. From past history,
the average length of repairs is 1 hour 15 minutes
(1.25 hours) with standard deviation of 20 minutes
(1/3 hour). Including the charge for the service call,
what is the mean and standard deviation for the
charges for labor?
  30  25(1.25)  $61.25
1
  25   $8.33
3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 16
Example: Plumbing
A local plumber makes house calls. She charges $30 to come out to
the house and $40 per hour for her services. For example, a 4 hour
service call costs $30 + $40(4)=$190. The table shows summary
statistics for the past month. Fill in the table to find out the cost of
the service calls.
Statistic
Mean
Median
SD
IQR
Minimum
Hours of
Service Call
4.5
3.5
1.2
2.0
0.5
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Cost of
Service Call
Slide 6- 17
Example continued
Statistic
Hours of Service
Call
Cost of Service
Call
Mean
4.5
$210
Median
3.5
$170
SD
1.2
$48
IQR
2.0
$80
Minimum
0.5
$50
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 18
Rules for Combining two variables

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To find the mean for the sum (or difference), add (or
subtract) the two means
To find the standard deviation of the sum (or
differences), ALWAYS add the variances, then take
the square root.
Formulas:
 a  b   a  b
a b  a  b
2
a
 a b    
2
b
If variables are independent
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Slide 6- 19
Example: Bicycles arrive at a bike shop in boxes. Before
they can be sold, they must be unpacked, assembled, and
tuned (lubricated, adjusted, etc.). Based on past experience,
the times for each setup phase are independent with the
following means & standard deviations (in minutes). What
are the mean and standard deviation for the total bicycle
setup times?
Phase
Mean
SD
Unpacking
3.5
0.7
Assembly
21.8
2.4
Tuning
12.3
2.7
T  3.5  21.8  12.3  37.6 minutes
T  0.7 2  2.42  2.7 2  3.680 minutes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 20
Back to z-scores

Standardizing data into z-scores shifts the data
by subtracting the mean and rescales the values
by dividing by their standard deviation.
 Standardizing into z-scores does not change
the shape of the distribution.
 Standardizing into z-scores changes the center
by making the mean 0.
 Standardizing into z-scores changes the
spread by making the standard deviation 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 21
When Is a z-score Big?
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A z-score gives us an indication of how unusual a
value is because it tells us how far it is from the
mean.
Remember that a negative z-score tells us that
the data value is below the mean, while a positive
z-score tells us that the data value is above the
mean.
The larger a z-score is (negative or positive), the
more unusual it is.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 22
When Is a z-score Big? (cont.)
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There is no universal standard for z-scores, but
there is a model that shows up over and over in
Statistics.
This model is called the Normal model (You may
have heard of “bell-shaped curves.”).
Normal models are appropriate for distributions
whose shapes are unimodal and roughly
symmetric.
These distributions provide a measure of how
extreme a z-score is.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 23
When Is a z-score Big? (cont.)

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There is a Normal model for every possible
combination of mean and standard deviation.
 We write N(μ,σ) to represent a Normal model
with a mean of μ and a standard deviation of σ.
We use Greek letters because this mean and
standard deviation do not come from data—they
are numbers (called parameters) that specify the
model.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 24
When Is a z-score Big? (cont.)


Summaries of data, like the sample mean and
standard deviation, are written with Latin letters.
Such summaries of data are called statistics.
When we standardize Normal data, we still call
the standardized value a z-score, and we write
z
y

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 25
When Is a z-score Big? (cont.)


Once we have standardized, we need only one
model:
 The N(0,1) model is called the standard
Normal model (or the standard Normal
distribution).
Be careful—don’t use a Normal model for just any
data set, since standardizing does not change the
shape of the distribution.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 26
When Is a z-score Big? (cont.)


When we use the Normal model, we are
assuming the distribution is Normal.
We cannot check this assumption in practice, so
we check the following condition:
 Nearly Normal Condition: The shape of the
data’s distribution is unimodal and symmetric.
 This condition can be checked with a
histogram or a Normal probability plot (to be
explained later).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 27
The 68-95-99.7 Rule


Normal models give us an idea of how extreme a
value is by telling us how likely it is to find one
that far from the mean.
We can find these numbers precisely, but until
then we will use a simple rule that tells us a lot
about the Normal model…
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 28
The 68-95-99.7 Rule (cont.)

It turns out that in a Normal model:
 about 68% of the values fall within one
standard deviation of the mean;
 about 95% of the values fall within two
standard deviations of the mean; and,
 about 99.7% (almost all!) of the values fall
within three standard deviations of the mean.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 29
The 68-95-99.7 Rule (cont.)

The following shows what the 68-95-99.7 Rule
tells us:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 30
The First Three Rules for Working with
Normal Models
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Make a picture.
Make a picture.
Make a picture.
And, when we have data, make a histogram to
check the Nearly Normal Condition to make sure
we can use the Normal model to model the
distribution.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 31
Example: pg. 125 #18
Some IQ tests are standardized to a Normal model,
with a mean of 100 and a standard deviation of
16.
a.
b.
c.
d.
e.
Draw the model for these IQ scores. Clearly label it, showing what
the 68-95-99.7 Rule predicts about the scores.
In what interval would you expect the central 95% of the IQ scores
to be found?
About what percent of people should have IQ scores above 116?
About what percent of people should have IQ scores between 68
and 84?
About what percent of people should have IQ scores above 132?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 32
Example: pg. 125 #18
Answers:
a.
See Normal model.
b.
68 to 132 IQ points
c.
16%
d.
13.5%
e.
2.5%
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Slide 6- 33
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 34
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 6.5 shows us how to find the area to the left when
we have a z-score of 1.80:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 35
Finding Normal Percentiles Using Technology

Many calculators and statistics programs have the
ability to find normal percentiles for us.
Look under: 2nd DISTR
You will see THREE “norm” functions that know the
Normal model. They are:
normalpdf(
normalcdf(
invNorm(
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Slide 6- 36
normalpdf(

Calculates y values for graphing a Normal curve.
You won’t use this often, if at all. Try graphing:
Y1=normalpdf(x)
Use graphing window xmin=-4, xmax=4, ymin=-.1,
and ymax = .5
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Slide 6- 37
normalcdf(


Finds the area between two z-score cut points, by
specifying normalcdf(zleft,zright).
You will use this function OFTEN!!
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 38
Example: Using normalcdf
Find the area of the shaded region between
z = -0.5 and z = 1.0. Always draw a picture
first.
normalcdf(-.5,1.0)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 39
Example: SAT Scores Again
Question: SAT scores are nearly Normal with a
mean of 500 and a standard deviation of 100
points. What proportion of SAT scores fall
between 450 and 600?
1.
Make a picture of this normal model. Shade
desired region.
2.
Find z-scores for the cut points.
3.
Use your calculator to find this area.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 40
Example: SAT Scores Again cont.
Answer: The Normal model estimates that
about 53.3% of SAT scores fall between
450 and 600.
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Slide 6- 41
To infinity and beyond! (not really)
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Theoretically, the standard Normal model extends
right and left forever. Since you can’t tell the
calculator to use infinity as a right of left cut point,
the book suggests that you use 99 (or -99).
What area of the Normal model does
normalcdf(.67,99) represent?
What area of the Normal model does normalcdf(99.1.38) represent?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 42
From Percentiles to Scores: z in Reverse
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
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?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 43
From Percentiles to Scores: z in Reverse
(cont.)
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
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 44
invNorm(


Finds the z scores of a specific percentile, in
decimal form
Find the z score at the 25th percentile (first
quartile):
invNorm(.25)=-.6744897495
The calculator says that the cut point for the
leftmost 25% of a Normal model is approximately
z = -0.674.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 45
Example: SAT once again
Question: Suppose a college says it admits only
people with SAT verbal test scores among the
top 10%. How high a score does it take to be
eligible?
1.
2.
3.
4.
Draw a picture and shade appropriate region.
What percentile will we be using?
Use calculator to find z score.
Convert z score back to the original units.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 46
Example: SAT once again cont.
Answer: z = 1.28, so the score is 628
points on the SAT.
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Slide 6- 47
Are You Normal? How Can You Tell?


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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 48
Are You Normal? How Can You Tell? (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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 49
Are You Normal? How Can You Tell? (cont.)

Nearly Normal data have a histogram and a
Normal probability plot that look somewhat like
this example:
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Slide 6- 50
Are You Normal? How Can You Tell? (cont.)

A skewed distribution might have a histogram
and Normal probability plot like this:
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Slide 6- 51
What Can Go Wrong?

Don’t use a Normal model when the distribution is
not unimodal and symmetric.
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Slide 6- 52
What Can Go Wrong? (cont.)
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


Don’t use the mean and standard deviation when
outliers are present—the mean and standard
deviation can both be distorted by outliers.
Don’t round off too soon.
Don’t round your results in the middle of a
calculation.
Don’t worry about minor differences in results.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 53
What have we learned?

The story data can tell may be easier to
understand after shifting or rescaling the data.
 Shifting data by adding or subtracting the same
amount from each value affects measures of
center and position but not measures of
spread.
 Rescaling data by multiplying or dividing every
value by a constant changes all the summary
statistics—center, position, and spread.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 54
What have we learned? (cont.)

We’ve learned the power of standardizing data.
 Standardizing uses the SD as a ruler to
measure distance from the mean (z-scores).
 With z-scores, we can compare values from
different distributions or values based on
different units.
 z-scores can identify unusual or surprising
values among data.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 55
What have we learned? (cont.)

A Normal model sometimes provides a useful
way to understand data.
 We need to Think about whether a method will
work—check the Nearly Normal Condition with
a histogram or Normal probability plot.
 Normal models follow the 68-95-99.7 Rule, and
we can use technology or tables for a more
detailed analysis.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 56