HAPPY TUESDAY
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UPCOMING IN CLASS

Homework #3 due Monday at 5pm

Part 2 of the Data Project due February 7th

Quiz #2 in class February 9th (open book)
CHAPTER 6
The Standard Deviation as a Ruler and the
Normal Model
WHAT ABOUT SPREAD? THE STANDARD
DEVIATION (CONT.)

The variance, notated by s2, is found by summing the
squared deviations and (almost) averaging them:
y  y



2
s

2
n 1
The variance will play a role later in our study, but it is
problematic as a measure of spread—it is measured in
squared units!
WHAT ABOUT SPREAD? THE STANDARD
DEVIATION (CONT.)

The standard deviation, s, is just the square root of the
variance and is measured in the same units as the
original data.
 y  y 
2
s
n 1
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.
NORMAL MODEL

The following shows what the 68-95-99.7 Rule
tells us:
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.
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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.
STANDARDIZING DATA

By calculating z-scores for each observation, we
change the distribution of the data by
Shifting the data
 Rescaling the data

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

The following histograms show a shift from men’s
actual weights to kilograms above recommended
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:
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1- 15
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.

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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.
The larger a z-score is (negative or positive), the
more unusual it is.
We use the theory of the Normal Model to see.
NORMAL MODEL

The following shows what the 68-95-99.7 Rule
tells us:
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NORMAL MODEL



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.
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NORMAL MODEL

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.
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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

y  y

z
s
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.
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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 Thursday).

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.
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TWO STANDARDIZED TESTS, A AND B USE VERY
DIFFERENT SCALES OF SCORES. THE FORMULA
A=50*B+200 APPROXIMATES THE
RELATIONSHIP BETWEEN SCORES ON THE TWO
TWO TEST. USE THE SUMMARY STATISTICS
WHO TOOK TEST B TO DETERMINE THE
SUMMARY STATISTICS FOR EQUIVALENT SCORES
ON TEST A.
 Lowest = 18
Mean = 26
St. Dev=5
 Q3=28
Median=30
IQR = 6
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WHAT IS THE LOWEST SCORE FOR TEST A?
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WHAT IS THE MEAN FOR TEST A?
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WHAT IS THE STANDARD DEVIATION?
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WHAT IS THE Q3 FOR TEST A?
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WHAT IS THE MEDIAN FOR TEST A?
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WHAT IS THE IQR FOR TEST A?
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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.

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TWO STUDENTS TAKE LANGUAGE EXAMS
Anna score 87 on both
 Megan scores 76 on first, and 91 on the second


Overall student scores on the first exam



Mean=83
St. Dev. 5
Second exam
Mean = 70
 St. Dev. 14

TO QUALIFY FOR LANGUAGE HONORS, A
STUDENT MUST AVERAGE AT LEAST 85 ACROSS
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ALL COURSE. DO ANNA AND MEGAN QUALIFY?
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Only Anna qualifies
Both qualify
Neither qualify
Only Megan
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WHO PERFORMED BETTER OVERALL?
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2.
Anna
Megan
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50%
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COW PROBLEM



The mean weight of a group of yearling steers is
1153 pounds. Suppose that weights of all such
animals can be described with a Normal model
with a standard deviation of 86 pounds.
Cattle buyers hope that yearling steers will
weight AT LEAST 1000 pounds. To see how
much over or under that goal the cattle are, we
could subtract 1000 pounds from all the weights.
What would the new mean and standard
deviation be?
COW PROBLEM (CONT.)

Suppose such cattle sell at auction for $0.35 a
pound. Find the mean and standard deviation of
their sale price.
COW PROBLEM (CONT)

Assume the Normal Model applies to the sale
price for steers. Use the 68-95-99.7 rule to draw
the model for the price of steers.
NEXT TIME…

The benefits of symmetry in the Normal Model