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Chapter 6
The Standard Deviation as a
Ruler and the Normal Model
SAT
ACT
1620
28
1720
30
1500
25
1342
26
1289
27
1450
28
1760
28
1800
30
25
24
24
25
31
Which one scored
better?
27
24
26
30
21
24
21
23
23
27
Standardizing with z-scores
The trick in comparing very different-looking values is to
standardize the values.
Expressing the distances in standard deviations
standardize the values.
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.
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.
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.
Shifting Data
Shifting data:
Adding (or subtracting) a constant to every
data value adds (or subtracts) the same
constant to measures of position.
Adding (or subtracting) a constant to each
value will increase (or decrease) measures of
position: center, percentiles, max or min by the
same constant.
Its shape and spread - range, IQR, standard
deviation - remain unchanged.
Shifting Data (cont.)
The following histograms show a shift from men’s
actual weights to kilograms above recommended
To compare their weights with recommended
weight:
maximum weight of 74 kg, we subtract this value
Mean weight 82.36
kg
from each weight
Rescaling Data
Rescaling data:
When we multiply (or divide) all the data values
by any constant, all measures of position (such
as the mean, median, and percentiles) and
measures of spread (such as the range, the
IQR, and the standard deviation) are multiplied
(or divided) by that same constant.
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:
Shape really hasn’t changed: Both unimodal and skewed to the right.
Spread gets larger by the amount that we used to rescale the data.
Just checking
In 1995 the educational testing service (ETS) adjusted the scores of
SAT tests. Before ETS recentered the SAT verbal test, the mean of
all test scores was 450.
A) How would adding 50 points to each score affect the mean?
B) The standard deviation was 100 points. What would the standard
deviation be after adding 50 points?
C) Suppose we drew box-plots of test takers’ scores a year before and a
year after the recentering. How would the box-plots of the two years
differ?
Just checking
In 1995 the educational testing service (ETS) adjusted the scores of
SAT tests. Before ETS recentered the SAT verbal test, the mean of
all test scores was 450.
A) How would adding 50 points to each score affect the mean?
New mean= 450+50
B) The standard deviation was 100 points. What would the standard
deviation be after adding 50 points?
New std= 100
C) Suppose we drew box-plots of test takers’ scores a year before and a
year after the recentering. How would the box-plots of the two years
differ?
All measures in the box-plot would increase by 50 points after
recentering.
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.
When Is a z-score BIG?
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.
To say more about how big we expect a z-score
to be, we need to model the data’s distribution.
A model will let us say much more precisely how
often we’d be likely to see z-scores of different
sizes.
Of course, like all models of the real world, the
model will be wrong-wrong in the sense that it
can’t match reality exactly. But it can still be
useful.
When Is a z-score Big? (cont.)
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.
When Is a z-score Big? (cont.)
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.
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−µ
σ
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.
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 by making a
histogram or a Normal probability plot (to be
explained later).
The 68-95-99.7 Rule (Empirical 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 modelG
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.
The 68-95-99.7 Rule (cont.)
The following shows what the 68-95-99.7 Rule
tells us:
Just checking
As a group, the Dutch are among the tallest people in the
world. The average Dutch man is 184 cm tall-just over 6
feet. If a Normal model is appropriate and the standard
deviation for men is about 8 cm, what percentage of all
Dutch men will be over 2 meters?
Solution:
184-2*8=168 cm
184+2*8=200( 2 meters)
95% of the Dutch men have heights between 168 cm and 200 cm.
We expect 5% of the men to be more than 200 cm or less than 168
cm.
So 2.5% of the men are expected to be more than 2 meters.
Just Checking
Suppose it takes you 20 minutes, on average, to
drive to school, with a standard deviation of 2
minutes. Suppose a Normal model is appropriate
for the distributions of driving times.
A) How often will you drive at school less than 22
minutes?
84% of time
B) How often will it take you more than 24 minutes?
2.5% of time
The First Three Rules for Working with
Normal Models
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.
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 D provides us with normal
percentiles, but many calculators and statistics
computer packages provide these as well.
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.
The figure shows us how to find the area to the left when
we have a z-score of 1.80:
Finding Normal Percentiles Using Technology
Many calculators and statistics programs have the
ability to find normal percentiles for us.
The ActivStats Multimedia Assistant offers two methods
for finding normal percentiles:
The “Normal Model Tool” makes it easy to see how
areas under parts of the Normal model correspond to
particular cut points.
There is also a Normal table in which the picture of
the normal model is interactive.
Finding Normal Percentiles Using Technology
(cont.)
The following was produced with the “Normal
Model Tool” in ActivStats:
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?
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.
Are You Normal? 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.
Are You Normal? 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.
Are You Normal? Normal Probability Plots (cont)
Nearly Normal data have a histogram and a
Normal probability plot that look somewhat like
this example:
These two values are a bit lower than we’d
expect of the lowest two values in a Normal
model.
Are You Normal? Normal Probability Plots (cont)
A skewed distribution might have a histogram
and Normal probability plot like this for which 6895-99.7 rule would not be accurate.
What Can Go Wrong?
Don’t use a Normal model
when the distribution is not
unimodal and symmetric.
Ex. 6.3
Here are the summary statistics for the
weekly payroll of a small company: lowest
salary=$300, mean salary=$700,
median=$500, range=$1200, IQR=$600,
first quartile=$350, standard dev.=$400.
a)
Do you think the distribution of salaries is
symmetric, skewed to the left, or skewed
to the right?
It is skewed to the right since mean > median
Ex. 6.3 (cont.)
b) Between what two values are the middle 50% of
the salaries found?
$350, $250(IQR-350)
c) Suppose business has been good and the
company gives every employee a $50 raise. Tell the
new value of each summary statistics.
Except the standard deviation every statistics will
increase 50 points. Standard dev. Will remain
unchanged.
d) Instead, suppose the company gives each employee a 10%
raise. Tell the new value of each of the summary statistics.
Ex. 6.3 (cont.)
d) Instead, suppose the company gives each
employee a 10% raise. Tell the new value of each of
the summary statistics.
New mean= 700+700*0.10=770
New median=500+500*0.1=550
New min=300+300*.1=330
New range=1200+1200*.10=1320
New IQR=600+600*.10=660
New std=400
Ex. 6.10
Cars currently sold in the US have an
average of 135 horsepower, with a standard
deviation of 40 horsepower. What is the zscore for a car with 195 horse power?
Z=(195-135)/40=1.5
Ex. 6.12
People with z-scores greater than 2.5 on an
IQ test are sometimes classified as
geniuses. If IQ test scores have a mean of
100 and a std. dev. of 16 points, what IQ
score do you need to be considered a
genious?
2.5=(x-100)/16
x=140
Frequency table for quiz1 grades
Descriptive statistics for Grades by
sections
Box plots for Grades by sections
Assume that I picked a student with a 10 point
from each section. Will this mean that these
students are equivalent by means of their
success?
Section 10
Section 11
Mean=13.33
Std=3.241
Z-score= (10-13.33)/3.241=-1.027
Mean=13.300
Std=3.064
Z-score= (10-13.3)/3.064=-1.07
Section 12
Mean=12.567
Std=3.07
Z-score= (10-12.567)/3.07=-0.8367
Ex. 6.42
In a standard Normal model, what value(s) of z
cut(s) off the region described?
A) The lowest 12%
-1.175
B) The highest 30%
0.53
C) The highest 7%
1.47
D) The middle 50%
(-0.67, 0.67)
Ex. 6.43
Based on the Normal model N(100,16) describing IQ scores, what
percent of people’s IQS would you expect to be
A) Over 80?
Z=(80-100)/16=-1.25
1-0.1056=0.8944 ⇒89.4%
B) Under 90?
Z=(90-100)/16=-0.625
The mean for the values of -0.62 and -0.63=(0.2676+0.2643)/2=0.2659
⇒26.6%
C) Between 112 and 132?
Z1=(112-100)/16=0.75
Z2=(132-100)/16=2.00
The valu for 2.00-The value for 0.75=0.9772-0.7734=0.2038 ⇒20.4%
Ex. 6.27
A)
B)
C)
D)
E)
Environmental protection agency (EPA) fuel economy
estimates for automobile models tested recently predicted a
mean of 24.8 mpg and a standard deviation of 6.2 mpg for
highway driving. Assume that the distribution is moundshaped(i.e; Normal model applies)
Draw the model for auto fuel economy. Clearly label it showing
what the 68-95-99.7 rule predicts about miles per gallon.
In what interval would you expect the central 68% of autos to
be found?
About what percent of autos should get more than 31 mpg?
About what percent of autos should get between 31 and 37
mpg?
Describe the gas mileage of the worst 2.5% of all cars?
What Can Go Wrong? (cont.)
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 your results in the middle of a
calculation.
Don’t worry about minor differences in results.
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.
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.
What have we learned? (cont.)
We’ve learned that the 68-95-99.7 Rule can be a
useful rule of thumb for understanding
distributions:
For data that are unimodal and symmetric,
about 68% fall within 1 SD of the mean, 95%
fall within 2 SDs of the mean, and 99.7% fall
within 3 SDs of the mean.
What have we learned? (cont.)
We see the importance of Thinking about
whether a method will work:
Normality Assumption: We sometimes work
with Normal tables (Table Z). These tables are
based on the Normal model.
Data can’t be exactly Normal, so we check the
Nearly Normal Condition by making a
histogram (is it unimodal, symmetric and free
of outliers?) or a normal probability plot (is it
straight enough?).