Measures of DispersionMeasures of Dispersion

Learning Objectives:Learning Objectives:

1.1. Explain what is meant by variabilityExplain what is meant by variability

2.2. Describe, know when to use, interpret Describe, know when to use, interpret and calculate: range, variance, and and calculate: range, variance, and standard deviationstandard deviation

22

2X

2)( X

More Statistical NotationMore Statistical Notation

indicates the sum of squared indicates the sum of squared Xs.Xs.Square ea score (2Square ea score (222+ 2+ 222))

Find sum of squared Xs =4+4=8Find sum of squared Xs =4+4=8

indicates the squared sum of indicates the squared sum of X.X.(2+2)(2+2)22

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Measures of VariabilityMeasures of Variability

… … describe the describe the

extent to which extent to which

scores in a scores in a

distribution differ distribution differ

from each other.from each other.

AA BB CC

00 88 66

22 77 66

66 66 66

1010 55 66

1212 44 66

X=6X=6 X=6X=6 X=6X=6

44

A Chart Showing the Distance A Chart Showing the Distance Between the Locations of Scores Between the Locations of Scores

in Three Distributionsin Three Distributions

VariabilityVariabilityProvides a quantitative measure of Provides a quantitative measure of

the degree to which scores in a the degree to which scores in a distribution are spread out or distribution are spread out or clustered togetherclustered together

Figure 4.1Figure 4.1

KurtosisKurtosis

Kurtosis based on size of a Kurtosis based on size of a distribution’s tail.distribution’s tail.Leptokurtic: thin or skinny distLeptokurtic: thin or skinny distPlatykurtic: flatPlatykurtic: flatMesokurtic: same kurtosis (normal Mesokurtic: same kurtosis (normal

distribution) distribution)

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Three Variations of the Three Variations of the Normal CurveNormal Curve

The Range, The Range, Semi-Interquartile Range, Semi-Interquartile Range,

Variance, and Standard Variance, and Standard DeviationDeviation

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The RangeThe Range

… … indicates the distance between the indicates the distance between the two most extreme scores in a two most extreme scores in a distributiondistributionCrude measurementCrude measurement

Used w/ nominal or ordinal dataUsed w/ nominal or ordinal data

RangeRangedifference btwn upper real difference btwn upper real limit of max score and lower real limit limit of max score and lower real limit of min scoreof min score

Range = highest score – lowest scoreRange = highest score – lowest score

1010

The Interquartile RangeThe Interquartile Range

Covered by the middle 50% of the Covered by the middle 50% of the distributiondistribution

Interquartile range= Q3-Q1Interquartile range= Q3-Q1Semi-Interquartile RangeSemi-Interquartile Range

Half of the interquartile rangeHalf of the interquartile range

1111

Variance and Standard Variance and Standard DeviationDeviation

Variance & standard deviation Variance & standard deviation communicate how different the communicate how different the scores in a distribution are from each scores in a distribution are from each otherother

We use the mean as our reference We use the mean as our reference point since it is at the center of the point since it is at the center of the distribution and calculate how spread distribution and calculate how spread out the scores are around the meanout the scores are around the mean

The Population Variance The Population Variance and the Population and the Population Standard DeviationStandard Deviation

1313

NNX

X

X

22

2

)(

Population VariancePopulation Variance

The population variance is the true or The population variance is the true or actual variance of the population of actual variance of the population of scores.scores.

1414

NNX

X

X

22 )(

Population Standard Population Standard DeviationDeviation

The population standard deviation is The population standard deviation is the true or actual standard deviation the true or actual standard deviation of the population of scores.of the population of scores.

Describing the Sample Describing the Sample Variance and the Sample Variance and the Sample

Standard DeviationStandard Deviation

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NNX

XSX

22

2

)(

Sample VarianceSample Variance

The The samplesample variance is the average variance is the average of the squared deviations of scores of the squared deviations of scores around the sample meanaround the sample mean

Sample VarianceSample Variance

Variance is average of squared Variance is average of squared deviations (usually large) & squared deviations (usually large) & squared unitsunits

Difficult to interpretDifficult to interpretCommunicates relative variabilityCommunicates relative variability

Standard DeviationStandard Deviation

Measure of Var. that communicates Measure of Var. that communicates the average deviationthe average deviation

Square root of varianceSquare root of variance

1919

NNX

XSX

22 )(

Sample Standard DeviationSample Standard Deviation

The sample standard deviation is the The sample standard deviation is the square root of the average squared square root of the average squared deviation of scores around the deviation of scores around the sample mean.sample mean.

2020

The Standard DeviationThe Standard Deviation

… … indicates indicates “ “average deviation” from mean, average deviation” from mean,

consistency in scores, consistency in scores,

& how far scores are spread out around & how far scores are spread out around meanmean

larger the value of SD, the more the larger the value of SD, the more the scores are spread out around mean, scores are spread out around mean, and the wider the distributionand the wider the distribution

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Normal Distribution and the Normal Distribution and the Standard DeviationStandard Deviation

2222

Normal Distribution and the Normal Distribution and the Standard DeviationStandard Deviation

Approximately 34% of the scores in a Approximately 34% of the scores in a

perfect normal distribution are perfect normal distribution are

between the mean and the score that between the mean and the score that

is one standard deviation from the is one standard deviation from the

mean.mean.

The Estimated Population The Estimated Population Variance and the Estimated Variance and the Estimated

Population Standard Population Standard DeviationDeviation

2424

)( 2XS

)( XS

Estimating the Population Estimating the Population Variance and Standard Variance and Standard

DeviationDeviation

The sample variance is a The sample variance is a biased estimator of the population biased estimator of the population variance.variance.

The sample standard deviation The sample standard deviation is a biased estimator of the is a biased estimator of the population standard deviation.population standard deviation.

2525

1

)( 22

2

NNX

XsX

Estimated Population Estimated Population VarianceVariance

By dividing the numerator of the By dividing the numerator of the sample variance by N - 1, we have an sample variance by N - 1, we have an unbiased estimator of the population unbiased estimator of the population variance.variance.

2626

1

)( 22

NNX

XsX

Estimated Population Estimated Population Standard DeviationStandard Deviation

By dividing the numerator of the By dividing the numerator of the sample standard deviation by N - 1, sample standard deviation by N - 1, we have an unbiased estimator of we have an unbiased estimator of the population standard deviation.the population standard deviation.

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2Xs

2

Xs

Unbiased EstimatorsUnbiased Estimators

is an unbiased estimator ofis an unbiased estimator of

is an unbiased estimator ofis an unbiased estimator of

The quantity N - 1 is called the The quantity N - 1 is called the degrees of freedomdegrees of freedomNumber of scores in a sample that are Number of scores in a sample that are

free to vary so that they reflect free to vary so that they reflect variability in popvariability in pop

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Uses of ,Uses of , , and, and

Use the sample variance and the Use the sample variance and the sample standard deviation to sample standard deviation to describe the variability of a sample.describe the variability of a sample.

Use the estimated population variance Use the estimated population variance and the estimated population and the estimated population standard deviation for inferential standard deviation for inferential purposes when you need to estimate purposes when you need to estimate the variability in the population.the variability in the population.

2Xs

2XS XS Xs

2XS

XS

2Xs

Xs

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Organizational Chart of Organizational Chart of Descriptive and Inferential Descriptive and Inferential

Measures of VariabilityMeasures of Variability

Always..Always..

Determine level of measurementDetermine level of measurementExamine type of distributionExamine type of distributionCalculate meanCalculate meanCalculate variabilityCalculate variability

American Psychological American Psychological Association (5Association (5thth ed) ed)

MeanMeanMM

Standard DeviationStandard DeviationSDSD

3232

1414 1414 1313 1515 1111 1515

1313 1010 1212 1313 1414 1313

1414 1515 1717 1414 1414 1515

ExampleExample

Using the following data set, find Using the following data set, find The range, The range, The semi-interquartile range, The semi-interquartile range, The sample variance and standard deviation,The sample variance and standard deviation, The estimated population variance standard The estimated population variance standard

deviationdeviation

3333

71017

Example RangeExample Range

The range is the largest value minus The range is the largest value minus the smallest value.the smallest value.

3434

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XSX

22

2

)(

44.218

33623406

1818)246(

34062

2

XS

ExampleExampleSample VarianceSample Variance

3535

NNX

XSX

22 )(

56.144.21818246

34062

XS

ExampleExampleSample Standard DeviationSample Standard Deviation

3636

1

)( 22

2

NNX

XsX

59.217

33623406

1718)246(

34062

2

Xs

ExampleExampleEstimated Population Estimated Population

VarianceVariance

3737

1

)( 22

NNX

XsX

61.159.21718246

34062

Xs

Example—Estimated Example—Estimated Population Standard Population Standard

DeviationDeviation