chapter 4 summarizing scores with measures of variability
TRANSCRIPT
Chapter 4
SUMMARIZING SCORES WITH MEASURES OF VARIABILITY
Going Forward
Your goals in this chapter are to learn:• What is meant by variability• What the range indicates• What the standard deviation and variance are
and how to interpret them• How to compute the standard deviation and
variance when describing a sample, when describing the population, and when estimating the population
Measures of Variability
Measures of variability describe the extent to
which scores in a distribution differ from each
other.
Three Samples
Three Variations of the Normal Curve
Measures of Variability
• Smaller variability indicates– Scores are consistent– Measures of central tendency describe the distribution
more accurately– Less distances between the scores
• Larger variability indicates– Scores are inconsistent– Measures of central tendency describe the distribution
less accurately– Greater distances between the scores
The Range
The Range
• The range indicates the distance between the two most extreme scores in a distribution
• Range = Highest score – Lowest score
The Sample Variance andStandard Deviation
Variance and Standard Deviation
The variance and standard deviation indicate how much the scores are spread out around the mean.
Sample Variance
N
XXSX
22 )(
The sample variance is the average of the squared deviations of scores around the sample mean.
Sample Standard Deviation
The sample standard deviation is the square root of the average squared deviation of scores around the sample mean.
N
XXSX
2)(
The Standard Deviation
• The standard deviation indicates something like 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 SX, the more the scores are spread out around the mean, and the wider the distribution
Normal Distribution andthe Standard Deviation
Normal Distribution and theStandard Deviation
Approximately 34% of the scores in any normal
distribution are between the mean and the
score located one standard deviation from the
mean.
The Population Variance and Standard Deviation
Population Variance
The population variance is the true or actual variance of the population of scores.
N
XX
22 )(
Population Standard Deviation
The population standard deviation is the true or actual standard deviation of the population of scores.
N
XX
2)(
Estimating the PopulationVariance and Standard Deviation
• 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
)( 2X
)( X
Estimated Population Variance
By dividing by N – 1 instead of N, we have an unbiased estimator of the population variance called the estimated population variance.
1
)( 22
N
XXsX
Estimated PopulationStandard Deviation
Taking the square root of the estimated population variance results in the estimated population standard deviation.
1
)( 2
N
XXsX
Unbiased Estimators
• The estimated population variance is an unbiased estimator of the population variance
• The estimated population standard deviation is an unbiased estimator of the
population standard deviation
)( 2X
)( 2Xs
)( Xs)( X
Uses of
• 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
)( 2XS)( XS
)( 2Xs
)( Xs
,,, 22XXX sSS Xsand
Organizational Chart of Descriptive and Inferential Measures of Variability
New Symbols2X
2)( X
• The Sum of Squared Xs
First square each raw score and then sum the squared Xs
• The Squared Sum of X
First sum the raw scores and then square that sum
Computing Formulas
NNX
XSX
22
2
)(
The computing formula for the sample variance is
Computing Formulas
The computing formula for the sample standard deviation is
NNX
XSX
22 )(
Computing Formulas
The computing formula for the estimated population variance is
1
)( 22
2
NNX
XsX
Computing Formulas
The computing formula for the estimated population standard deviation is
1
)( 22
NNX
XsX
Example
Using the following data set, find • The range• The sample variance and standard deviation• 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
Example—Range
The range is the largest value minus the smallest value.
71017
ExampleSample Variance
NNX
XSX
22
2
)(
44.218
33623406
1818)246(
34062
2
XS
ExampleSample Standard Deviation
NNX
XSX
22 )(
56.144.21818
)246(3406
2
XS
ExampleEstimated Population Variance
1
)( 22
2
NNX
XsX
59.217
33623406
1718)246(
34062
2
Xs
Example—Estimated PopulationStandard Deviation
1
)( 22
NNX
XsX
61.159.21718)246(
34062
Xs