Download standard deviations from the mean

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 (9/5)

Homework #3 due Sunday (9/9) at 11:59 pm

Quiz #2 next Wed. (9/12) in class (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 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.
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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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?
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
2000
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
1700
2000
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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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
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
ASSUMING NORMALITY

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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ASSUMING NORMALITY
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).

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.
 A more specialized graphical display that can
help you decide 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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WHAT DOES THE DISTRIBUTION LOOK
LIKE?
33%
1.
2.
3.
A
B
C
1
33%
2
33%
3
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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1
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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1
2
3
4
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
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?
1.
2.
3.
4.
0.0001%
0.0000%
0.0004%
0.04%
25%
1
25%
25%
2
3
25%
4
WHAT PERCENT OF THE POPULATION HAS
AN IQ OF 50 OR LESS?
1.
2.
3.
4.
0.0001%
0.0000%
0.0004%
0.04%
25%
1
25%
25%
2
3
25%
4
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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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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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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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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