Post on 23-Dec-2015
Chapter Five
Measures of Variability: Range,
Variance, and Standard Deviation
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• indicates the sum of squared Xs.
• indicates the squared sum of X.
2X2)( X
New Statistical Notation
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Measures of Variability
Measures of variability describe the extent
to which scores in a distribution differ from
each other.
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A Chart Showing the Distance Between the Locations of Scores in Three Distributions
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Three Variations of the Normal Curve
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The Range, Variance, and
Standard Deviation
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The Range
• The range indicates the distance between the two most extreme scores in a distribution
Range = highest score – lowest score
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Variance and Standard Deviation
• The variance and standard deviation are two measures of variability that indicate how much the scores are spread out around the mean
• We use the mean as our reference point since it is at the center of the distribution
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The Sample Variance and theSample Standard Deviation
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N
XXSX
22 )(
Sample Variance
• The sample variance is the average of the squared deviations of scores around the sample mean
• Definitional formula
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NNX
XSX
22
2
)(
Sample Variance
• Computational formula
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N
XXSX
2)(
Sample Standard Deviation
• The sample standard deviation is the square root of the sample variance
• Definitional formula
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Sample Standard Deviation
• Computational formula
NNX
XSX
22 )(
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The Standard Deviation
• The standard deviation indicates the “average deviation” from the mean, the consistency in the scores, and how far scores are spread out around the mean
• The larger the value of X, the more the scores are spread out around the mean, and the wider the distribution
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Normal Distribution and the Standard Deviation
[Insert new Figure 5.2 here.]
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Normal Distribution and the Standard Deviation
Approximately 34% of the scores in a
perfect normal distribution are between
the mean and the score that is one
standard deviation from the mean.
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Standard Deviation and Range
For any roughly normal distribution, the
standard deviation should equal about
one-sixth of the range.
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The Population Variance and the Population Standard Deviation
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N
XX
22 )(
Population Variance
• The population variance is the true or actual variance of the population of scores.
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N
XX
2)(
Population Standard Deviation
• The population standard deviation is the true or actual standard deviation of the population of scores.
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The Estimated Population Variance and the Estimated Population Standard
Deviation
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• The sample variance is a biased estimator of the population variance.
• The sample standard deviation is a biased estimator of the population standard deviation.
)( 2XS
)( XS
Estimating the Population Variance and Standard Deviation
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1
)( 22
N
XXsX
Estimated Population Variance
• By dividing the numerator of the sample variance by N - 1, we have an unbiased estimator of the population variance.
• Definitional formula
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Estimated Population Variance
• Computational formula
1
)( 22
2
NNX
XsX
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1
)( 2
N
XXsX
Estimated Population Standard Deviation
• By dividing the numerator of the sample standard deviation by N - 1, we have an unbiased estimator of the population standard deviation.
• Definitional formula
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Estimated Population Standard Deviation
• Computational formula
1
)( 22
NN
XX
sX
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• is an unbiased estimator of
• is an unbiased estimator of
• The quantity N - 1 is called the degrees of freedom
2Xs
2Xs
Unbiased Estimators
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• Use the sample variance and the sample standard deviation to describe the variability of a sample.
• Use the estimated population variance and the estimated population standard deviation for inferential purposes when you need to estimate the variability in the population.
Uses of , , , and2Xs
2XS XS Xs
2XSXS
2Xs
Xs
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Organizational Chart of Descriptive and Inferential Measures of Variability
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Proportion of Variance Accounted For
The proportion of variance accounted
for is the proportion of error in our
predictions when we use the overall mean
to predict scores that is eliminated when
we use the relationship with another
variable to predict scores
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14 14 13 15 11 15
13 10 12 13 14 13
14 15 17 14 14 15
Example
• Using the following data set, find – The range, – The sample variance and standard deviation,– The estimated population variance and standard
deviation
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71017
Example Range
• The range is the largest value minus the smallest value.
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NNX
XSX
22
2
)(
44.218
33623406
1818
)246(3406
2
2
XS
ExampleSample Variance
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NNX
XSX
22 )(
56.144.218
18246
34062
XS
ExampleSample Standard Deviation
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1
)( 22
2
NNX
XsX
59.217
33623406
1718
)246(3406
2
2
Xs
ExampleEstimated Population Variance
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1
)( 22
NNX
XsX
61.159.217
18246
34062
Xs
Example—Estimated Population Standard Deviation