Download 3.2 - MAT143

3.2
Measures of Dispersion
DATA
●
●
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Comparing two sets of data
The measures of central tendency (mean,
median, mode) measure the differences
between the “average” or “typical” values
between two sets of data
The measures of dispersion in this section
measure the differences between how far
“spread out” the data values are
RANGE
The range of a variable is the largest
data value minus the smallest data
value
● Example…Compute the range of
6, 1, 2, 6, 11, 7, 3, 3
●
RANGE
The range only uses two values in the
data set – the largest value and the
smallest value
● The range is not resistant
● If we made a mistake and
6, 1, 2
was recorded as
6000, 1, 2
 Find the Range of both sets of Data
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VARIANCE
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The variance is based on the deviation
from the mean
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
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( xi – μ ) for populations
( xi – ) for samples
To treat positive differences and negative
differences, we square the deviations
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
( xi – μ )2 for populations
( xi – )2 for samples
POPULATION VARIANCE
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The population variance of a variable is the sum
of these squared deviations divided by the
number in the population
 x   
2
i
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●
N
The population variance is represented by σ2
Note: For accuracy, use as many decimal places
as allowed by your calculator
POPULATION VARIANCE
●
Compute the population variance of
6, 1, 2, 11
Compute the population mean first
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Now compute the squared deviations
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xi   
2
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Average the squared deviations
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The population variance σ2 is ?
SAMPLE VARIANCE
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The sample variance of a variable is the sum of
these squared deviations divided by one less than
the number in the sample
 x  x 
2
i
n 1
●
●
The sample variance is represented by s2
We say that this statistic has n – 1 degrees of
freedom
SAMPLE VARIANCE
●
Compute the sample variance of
6, 1, 2, 11
Compute the sample mean first
●
Now compute the squared deviations
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xi  x 2
Average the squared deviations
(Remember…n-1)
The sample variance s2 is ?
DIFFERENCE
 Why
are the population variance and the
sample variance different for the same set
of numbers?
 Why do we use different formulas?
 The reason is that using the sample mean
is not quite as accurate as using the
population mean
 If we used “n” in the denominator for the
sample variance calculation, we would get
a “biased” result
STANDARD DEVIATION
●
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The standard deviation is the square root
of the variance
The population standard deviation
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
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Is the square root of the population variance
(σ2)
Is represented by σ
The sample standard deviation


Is the square root of the sample variance (s2)
Is represented by s
STANDARD DEVIATION
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If the population is { 6, 1, 2, 11 }


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If the sample is { 6, 1, 2, 11 }
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The population variance σ2 = 15.5
The population standard deviation σ =
The sample variance s2 = 20.7
The sample standard deviation s =
The population standard deviation and
the sample standard deviation apply in
different situations
WHY STANDARD DEVIATION?

The standard deviation is very useful for
estimating probabilities
EMPIRICAL RULE
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The empirical rule
If the distribution is roughly bell shaped,
then

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Approximately 68% of the data will lie within
1 standard deviation of the mean
Approximately 95% of the data will lie within
2 standard deviations of the mean
Approximately 99.7% of the data (i.e. almost
all) will lie within 3 standard deviations of the
mean
EMPIRICAL RULE
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For a variable with mean 17 and standard
deviation 3.4
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Approximately 68% of the values will lie between
(17 – 3.4) and (17 + 3.4), i.e. 13.6 and 20.4
Approximately 95% of the values will lie between
(17 – 2  3.4) and (17 + 2  3.4), i.e. 10.2 and 23.8
Approximately 99.7% of the values will lie between
(17 – 3  3.4) and (17 + 3  3.4), i.e. 6.8 and 27.2
A value of 2.1 (less than 6.8) and a value of
33.2 (greater than 27.2) would both be very
unusual
DRAWING THE PICTURE IS HELPFUL!
CALCULATOR
 How
Can My Calculator Help Me???
1 Var-Stats
 Remember…
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Greek Letters are Population
EXAMPLE

3.2 Handout