Transcript No Slide Title

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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1
2
3
UPCOMING IN CLASS

Part 1 of the Data Project due (1/31)

Homework #3 due 2/3 at 11:59pm

Quiz #2 in class 2/7 (open book)
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CHAPTER 6
The Standard Deviation as a Ruler and the
Normal Model
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.
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 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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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
 Median=28
Q3=30
IQR = 6
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WHAT IS THE LOWEST SCORE FOR TEST A?
1.
2.
3.
4.
5.
6.
7.
18
50
200
250
1100
1500
2000
14%
14%
14%
14%
14%
14%
14%
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1
2
3
4
5
6
7
WHAT IS THE MEAN FOR TEST A?
1.
2.
3.
4.
5.
6.
7.
26
50
200
250
1100
1500
2000
14%
14%
14%
14%
14%
14%
14%
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1
2
3
4
5
6
7
WHAT IS THE STANDARD DEVIATION?
1.
2.
3.
4.
5.
6.
200
250
450
500
1100
1500
17%
17%
17%
17%
17%
17%
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1
2
3
4
5
6
WHAT IS THE Q3 FOR TEST A?
1.
2.
3.
4.
5.
1000
1400
1500
1600
1700
20%
20%
20%
20%
20%
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1
2
3
4
5
WHAT IS THE MEDIAN FOR TEST A?
1.
2.
3.
4.
5.
1000
1400
1500
1600
1700
20%
20%
20%
20%
20%
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1
2
3
4
5
WHAT IS THE IQR FOR TEST A?
1.
2.
3.
4.
5.
200
250
300
500
1000
20%
20%
20%
20%
20%
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1
2
3
4
5
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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STANDARDIZING WITH Z-SCORES
Standardized values have no units.
 z-scores measure the distance of each data value
from the mean in standard deviations.
Standardized values have been converted from
their original units to the standard statistical
unit of standard deviations from the mean.
 We can compare values that are measured on
different scales, with different units, or from
different populations.
 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.

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
25%
25%
25%
25%
ALL COURSE. DO ANNA AND MEGAN QUALIFY?
1.
2.
3.
4.
Only Anna qualifies
Both qualify
Neither qualify
Only Megan
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1
2
3
4
WHO PERFORMED BETTER OVERALL?
1.
2.
Anna
Megan
50%
50%
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1
2
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

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
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?
ABOUT WHAT PERCENT OF PEOPLE
SHOULD HAVE IQ SCORES ABOVE 145?
1.
2.
3.
4.
5.
6.
.3%
.15%
3%
1.5%
5%
2.5%
17%
17%
17%
17%
17%
17%
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2
3
4
5
6
WHAT PERCENT OF PEOPLE SHOULD HAVE
IQ SCORES BELOW 130?
1.
2.
3.
4.
95%
5%
2.5%
97.5%
25%
25%
25%
25%
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2
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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
UPCOMING IN CLASS

Part 1 of the Data Project due (1/31)

Homework #3 due 2/3 at 11:59pm

Quiz #2 in class 2/7 (open book)
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