CRIM 483CRIM 483

Measuring VariabilityMeasuring Variability

VariabilityVariability Variability refers to the spread or dispersion of scoresVariability refers to the spread or dispersion of scores Variability captures the degree to which scores within a dataset Variability captures the degree to which scores within a dataset

differ from one anotherdiffer from one another– High variability=large distance between scoresHigh variability=large distance between scores

Score set 7,6,3,3,1Score set 7,6,3,3,1– Low variability=small distance between scoresLow variability=small distance between scores

Score set 4,2,3,3,1Score set 4,2,3,3,1– No variability=no distance between scoresNo variability=no distance between scores

Score set 4,4,4,4,4 Score set 4,4,4,4,4 Variability & mean are used together to describe the Variability & mean are used together to describe the

characteristics of a distribution (sample) and show how characteristics of a distribution (sample) and show how distributions differ from one anotherdistributions differ from one another

There are 3 measures for variability: range, standard There are 3 measures for variability: range, standard deviation, and variancedeviation, and variance

RangeRange The range is the most general measure of variabilityThe range is the most general measure of variability The formula: r=h-lThe formula: r=h-l

– r=range r=range – h=highest scoreh=highest score– l=lowest scorel=lowest score

Calculation of range provides a Calculation of range provides a generalgeneral estimate of estimate of how wide or how much scores differ from one how wide or how much scores differ from one anotheranother

Examples:Examples:– Highest age=35, lowest age=21Highest age=35, lowest age=21

35-21=14 years difference between age scores in sample35-21=14 years difference between age scores in sample– Highest age=50, lowest age=15Highest age=50, lowest age=15

50-15=35 years difference between age scores in sample50-15=35 years difference between age scores in sample– Which sample has the greatest variability with regard to Which sample has the greatest variability with regard to

age?age?

Standard DeviationStandard Deviation Standard deviation (SD)=average amount of Standard deviation (SD)=average amount of

variability from the mean in the set of scores variability from the mean in the set of scores (average distance from the mean)(average distance from the mean)– Standard deviation is used most often to measure Standard deviation is used most often to measure

variabilityvariability– Reported (as a rule) in combination with meansReported (as a rule) in combination with means– The greater the SD, the larger the distance between the The greater the SD, the larger the distance between the

score and the meanscore and the mean Formula to calculate the SDFormula to calculate the SD

s=√(s=√(∑(x-mean)∑(x-mean)22)/)/(n-1)(n-1) s=standard deviations=standard deviation ∑∑=sigma (sum of)=sigma (sum of) x=individual scorex=individual score Mean=mean of all scoresMean=mean of all scores n=sample sizen=sample size

Clarification of FormulaClarification of Formula

Why not add up the deviations from the Why not add up the deviations from the mean?mean?– Sum of deviations from the mean is always Sum of deviations from the mean is always

equal to zero (good way to check your work)equal to zero (good way to check your work) Why square the deviations?Why square the deviations?

– To rid of the negative sign in order to avoid To rid of the negative sign in order to avoid summing to 0summing to 0

Why take the square root?Why take the square root?– To return to the same units that you started To return to the same units that you started

withwith

Unbiased v. Biased EstimatesUnbiased v. Biased Estimates UnbiasedUnbiased

– You produce an unbiased estimate by dividing by (n-1) in the You produce an unbiased estimate by dividing by (n-1) in the SD formulaSD formula

– Artificially forces the SD to be larger than it would be otherwiseArtificially forces the SD to be larger than it would be otherwise– Why? This produces a more conservative estimate that we can Why? This produces a more conservative estimate that we can

feel more comfortable with–it is safer to overestimate than feel more comfortable with–it is safer to overestimate than underestimateunderestimate

BiasedBiased– You produce a biased estimate by dividing by (n) in the SD You produce a biased estimate by dividing by (n) in the SD

formulaformula– Use biased estimate when you are merely describing your Use biased estimate when you are merely describing your

sample and you have no intention of comparing it to the sample and you have no intention of comparing it to the populationpopulation

Ultimately, the larger your sample size the less difference Ultimately, the larger your sample size the less difference there is between the unbiased and biased estimates (p. 40)there is between the unbiased and biased estimates (p. 40)

In Sum…In Sum…

Must always compute the mean firstMust always compute the mean first SD play a critical role later when SD play a critical role later when

comparing scores between groups comparing scores between groups (e.g., do male and female attitudes (e.g., do male and female attitudes differ)differ)

Like means, SD are sensitive to Like means, SD are sensitive to extreme scoresextreme scores

VarianceVariance

Final method of measuring variability is Final method of measuring variability is variancevariance

Very similar to SD formulaVery similar to SD formula– Formula to calculate the SDFormula to calculate the SD

ss22= (= (∑(x-mean)∑(x-mean)22)/)/(n-1)(n-1)– ss22=standard deviation=standard deviation– ∑∑=sigma (sum of)=sigma (sum of)– x=individual scorex=individual score– Mean=mean of all scoresMean=mean of all scores– n=sample sizen=sample size

Variance is difficult to interpret and apply by itselfVariance is difficult to interpret and apply by itself Variance has greater utility in the formulas of more Variance has greater utility in the formulas of more

advanced statisticsadvanced statistics

Standard Deviation v. VarianceStandard Deviation v. Variance

Both measure variability, dispersion, or Both measure variability, dispersion, or spreadspread

SD produces variability in original units and SD produces variability in original units and variance produces variability in units variance produces variability in units squaredsquared– Example from book re: circuit board assemblyExample from book re: circuit board assembly

8.6 boards assembled/hour on average8.6 boards assembled/hour on average1.59=SD: difference across workers on 1.59=SD: difference across workers on

average boards produced is 1.59 boardsaverage boards produced is 1.59 boards2.53=Variance: difference is 2.53 boards 2.53=Variance: difference is 2.53 boards

squared from the mean squared from the mean

Differences in VariabilityDifferences in Variability