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VARIABILITY:RRangeVarianceVariance Standard Deviation
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Measures of Variability
Describe the extent to which scores in a distribution differ from each other.
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Distance Between the Locations ofDistance Between the Locations of Scores in Three Distributions
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Three Variations of the Normal Curve
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Th RThe Range, Variance, andVariance, and
Standard Deviation
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The Range
• The range indicates the distance between the two most extreme scores in a distribution
R hi h lRange = 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 pmean
• 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 thepSample Standard Deviation
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Sample Variance
• The sample variance is the average of the squared deviations of scores around the sample meanp
• Definitional formula
NXXS X
22 )( −Σ=
N
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Sample Variance
• Computational formula
NXX
22 )(Σ−Σ
NNSX
2 =N
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Sample Standard Deviation
• The sample standard deviation is the square root of the sample variance
• Definitional formula
XX 2)(Σ
Definitional formula
NXXSX
)( −Σ=
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Sample Standard Deviation
• Computational formula
XX2
2 )(Σ−Σ
NN
XSX
Σ=
N
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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 , paround the mean
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Normal Distribution andNormal Distribution and the Standard Deviation
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Normal Distribution andNormal Distribution and the Standard Deviation
Approximately 34% of the scores in a perfect pp y pnormal distribution are between the mean and the score that is one standard deviation from the meanthe mean.
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Standard Deviation and Range
For any roughly normal distribution, the y g y ,standard deviation should equal about one-sixth of the range.
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The Population Variance and the Population Standardthe Population Standard
DeviationDeviation
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Population Variance
• The population variance is the true or actual variance of the population of scores.
XX
22 )( μσ −Σ=
NX
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Population Standard Deviation
• The population standard deviation is the true or actual standard deviation of the population of scores.
X 2)(Σ
p p
NX
X
2)( μσ −Σ=
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The Estimated Population V iVariance
andand The Estimated Population p
Standard Deviation
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Estimating the PopulationEstimating the Population Variance and Standard Deviation
)( 2XS• The sample variance is a biased )( Xp
estimator of the population variance.)( XS• The sample standard deviation is a
biased estimator of the populationbiased estimator of the population standard deviation.
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Estimated Population Variance
• By dividing the numerator of the sample variance by N - 1, we have an unbiased estimator of the population variance.p p
• Definitional formula
)( 22 −Σ=
XXsX 1−NsX
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Estimated Population Variance
• Computational formula• Computational formula
)( 2ΣX )(2
2
Σ−Σ
= NXX
s1−
=N
sX
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Estimated PopulationEstimated Population Standard Deviation
• By dividing the numerator of the sample standard deviation by N - 1, we have an unbiased estimator of the population p pstandard deviation.
• Definition l form l• Definitional formula
)( 2−Σ=
XXsX 1−NsX
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Estimated PopulationEstimated Population Standard Deviation
C i l f l• Computational formula
)( 2ΣX )(2 Σ−Σ
= NXX
sX 1−=
NsX
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Unbiased Estimators2 2b d2Xs 2σ• is an unbiased estimator of
s σ• is an unbiased estimator ofXs σ• is an unbiased estimator of
• The quantity N 1 is called the degrees of• The quantity N - 1 is called the degrees of freedom
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2Xs2
XS XS XsUses of , , , andXsXS XS XsUses of , , , and
22XS
XS• Use the sample variance and the
sample standard deviation toXSsample standard deviation to describe the variability of a sample.
2Xs
s• Use the estimated population variance
d h i d l i d dXsand the estimated population standard deviation for inferential purposes whendeviation for inferential purposes when you need to estimate the variability in the population.
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Organizational Chart of Descriptive and Inferential
M f V i biliMeasures of Variability
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Proportion of VarianceProportion of Variance Accounted For
The proportion of variance accounted for is the p p f fproportion of error in our predictions when we use the overall mean to predict scores that is eliminated when we use the relationship witheliminated when we use the relationship with another variable to predict scores
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Example 1
• Using the following data set, find – The range, – The sample variance and standard deviation,
14 14 13 15 11 15
– The estimated population variance and standard deviation
14 14 13 15 11 15
13 10 12 13 14 13
14 15 17 14 14 15
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Example 2
• For the following sample data, compute the range, i d d d d i ivariance and standard deviation
8 8 10 7 9 6 11 9 10 7 11 11 7 9 11 10 11 8 10 7
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Example 3
• For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table.y g
• 15 12 13 15 16 17 13 16 11 18
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Example 4
• For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table.y g
• 1 3 2 2 2 4 3 3 4 1
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Example 5
• For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table.y g
• 1 3 30 12 15 20 5 13 2 4
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