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