Transcript Lecture 18 - Measuring Variation 3

Standard Deviation
Lecture 18
Sec. 5.3.4
Fri, Oct 8, 2004
Deviations from the Mean


Each unit of a sample or population deviates
from the mean by a certain amount.
Define the deviation of x to be (x –x).
0
1
2
3 x = 4
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
deviation = –4
0
1
2
3 x = 4
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
dev = 1
0
1
2
3 x = 4
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
deviation = 3
0
1
2
3 x = 4
5
6
7
8
Sum of Squared Deviations
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
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We want to add up all the deviations, but to keep
the negative ones from canceling the positive
ones, so we square them all first.
Then we compute the sum of the squared
deviations.
We call this quantity SSX.
Sum of Squared Deviations

SSX = sum of squared deviations
SSX  x  x 
2

For example, if the sample is {0, 5, 7}, then
SSX = (0 – 4)2 + (5 – 4)2 + (7 – 4)2
= (-4)2 + (1)2 + (3)2
= 16 + 1 + 9
= 26.
The Population Variance
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Variance of the population – The average
squared deviation for the population.
The population variance is denoted by 2.
 x   
 
N
2
2
The Sample Variance
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Variance of a sample – The average squared
deviation for the sample, except that we divide by
n – 1 instead of n.
The sample variance is denoted by s2.
 x  x 
s 
n 1
This formula for s2 makes a better estimator of
2 than if we had divided by n.
2
2
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Example
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In the example, SSX = 26.
Therefore,
s2 = 26/2 = 13.
The Standard Deviation
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Standard deviation – The square root of the
variance of the sample or population.
The standard deviation of the population is
denoted .
The standard deviation of a sample is denoted s.
Example
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In our example, we found that s2 = 13.
Therefore, s = 13 = 3.606.
Example
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Example 5.10, p. 293.
Use Excel to compute the mean and standard
deviation of {0, 5, 7}.
Use basic operations.
 Use special functions.

Alternate Formula for the
Standard Deviation

An alternate way to compute SSX is to compute

x 

2
SSX  x


2
n
Note that only the second term is divided by n.
Then, as before
SSX
s 
n 1
2
Example



Let the sample be {0, 5, 7}.
Then  x = 12 and
 x2 = 0 + 25 + 49 = 74.
So
SSX = 74 – (12)2/3
= 74 – 48
= 26,
as before.
TI-83 – Standard Deviations
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Follow the instructions for computing the mean.
The display shows Sx and x.
Sx is the sample standard deviation.
 x is the population standard deviation.
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Using the data of the previous example, we have
Sx = 3.605551275.
 x = 2.943920289.
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Interpreting the Standard
Deviation
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Both the standard deviation and the variance are
measures of variation in a sample or population.
The standard deviation is measured in the same units as
the measurements in the sample.
Therefore, the standard deviation is directly comparable
to actual deviations.
Interpreting the Standard
Deviation
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The variance is not comparable to deviations.
The most basic interpretation of the standard
deviation is that it is roughly the average
deviation.
Interpreting the Standard
Deviation
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Observations that deviate fromx by much
more than s are unusually far from the mean.
Observations that deviate fromx by much less
than s are unusually close to the mean.
Interpreting the Standard
Deviation
x
Interpreting the Standard
Deviation
s
s
x
Interpreting the Standard
Deviation
s
x – s
s
x
x + s
Interpreting the Standard
Deviation
Closer than normal tox
x – s
x
x + s
Interpreting the Standard
Deviation
Farther than normal fromx
x – s
x
x + s
Interpreting the Standard
Deviation
Extraordinarily far fromx
x – 2s
x – s
x
x + s
x + 2s
Let’s Do It!
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Let’s do it! 5.13, p. 295 – Increasing Spread.
Let’s do it! 5.14, p. 297 – Variation in Scores.