11

ReviewReview

Mean—arithmetic average, sum of all Mean—arithmetic average, sum of all scores divided by the number of scoresscores divided by the number of scores

Median—balance point of the data, exact Median—balance point of the data, exact middle of the distribution, 50middle of the distribution, 50thth percentile percentile

Mode—highest frequency, can be more Mode—highest frequency, can be more than onethan one

22

Review Review

Find the mean, Find the mean, median, modemedian, mode

XX ff

55 22

44 55

33 33

22 22

11 22

33

Review Review

Find the mean, Find the mean, median, modemedian, mode

Mean=sum of all Mean=sum of all scores(scores(∑fX)∑fX) /number /number of scores(N)of scores(N)

XX ff fXfX

55 22

44 55

33 33

22 22

11 22

NN ∑∑fXfX

44

Review Review

Find the mean, Find the mean, median, modemedian, mode

Mean=sum of all Mean=sum of all scores(scores(∑fX)∑fX) /number /number of scores(N)of scores(N)

Median=middle point Median=middle point (N-1/2)(N-1/2)thth position position

XX ff fXfX

55 22

44 55

33 33

22 22

11 22

55

Review Review

Find the mean, Find the mean, median, mode median, mode

Mean=sum of all Mean=sum of all scores(scores(∑fX)∑fX) /number /number of scores(N)of scores(N)

Median=middle point Median=middle point (N-1/2)(N-1/2)thth position position

Mode=greatest fMode=greatest f

XX ff fXfX

55 22

44 55

33 33

22 22

11 22

Measures of Variability

77

Major PointsMajor Points

The general problemThe general problem

Range and related statisticsRange and related statistics

Deviation scoresDeviation scores

The variance and standard deviationThe variance and standard deviation

BoxplotsBoxplots

Review questionsReview questions

88

The General ProblemThe General Problem

Central tendency only deals with the Central tendency only deals with the centercenter

DispersionDispersion Variability of the data around somethingVariability of the data around something The spread of the pointsThe spread of the points

Example: Mice and MusicExample: Mice and Music

99

Mice and MusicMice and Music

Study by David MerrellStudy by David Merrell

Raised some mice in quiet environmentRaised some mice in quiet environment

Raised some mice listening to MozartRaised some mice listening to Mozart

Raised other mice listening to AnthraxRaised other mice listening to Anthrax

Dependent variable is the time to run a Dependent variable is the time to run a straight alley maze after 4 weeks.straight alley maze after 4 weeks.

Borrowed from David Howell, 2000

1010

ResultsResults

Anthrax mice took much longer to runAnthrax mice took much longer to run

Much greater variability in Anthrax groupMuch greater variability in Anthrax group See following graphs for Anthrax and MozartSee following graphs for Anthrax and Mozart

We often see greater variability with larger We often see greater variability with larger meanmean

WEEK4

472.2416.7361.1305.6250.0194.4138.983.327.8

Mozart Group12

10

8

6

4

2

0

Std. Dev = 36.10

Mean = 114.6

N = 24.00

WEEK4

2050.0

2000.0

1950.0

1900.0

1850.0

1800.0

1750.0

1700.0

1650.0

1600.0

Anthrax Group10

8

6

4

2

0

Std. Dev = 103.14

Mean = 1825.9

N = 24.00

1313

Range and Related StatisticsRange and Related Statistics

The rangeThe range Distance from lowest to highest scoreDistance from lowest to highest score Too heavily influenced by extremesToo heavily influenced by extremes

The interquartile range (IQR)The interquartile range (IQR) Delete lowest and highest 25% of scoresDelete lowest and highest 25% of scores IQR is range of what remainsIQR is range of what remains May be too May be too littlelittle influenced by extremes influenced by extremes

1414

Trimmed SamplesTrimmed Samples

Delete a fixed (usually small) percentage Delete a fixed (usually small) percentage of extreme scoresof extreme scores

Trimmed statistics are statistics computed Trimmed statistics are statistics computed on trimmed samples.on trimmed samples.

1515

Deviation ScoresDeviation Scores

DefinitionDefinition distance between a score and a measure of distance between a score and a measure of

central tendencycentral tendency

usually deviation around the meanusually deviation around the mean

ImportanceImportance

)( XX

1616

VarianceVariance

Definitional formulaDefinitional formula

ExampleExample

See next slideSee next slide

1)( 2

2

NXX

s

1717

Definitional formulaDefinitional formula

Find the meanFind the mean

N=6N=6

∑∑X=30X=30

30/6=530/6=5

1

)( 22

N

XXs

XX X - XX - X (X - X)(X - X)22

22

44

55

88

77

44

3030

¯̄¯̄

Computing the VarianceComputing the Variance

1818

Computing the VarianceComputing the Variance

XX X - XX - X (X - X)(X - X)22

22 -3-3

44 -1-1

55 00

88 33

77 22

44 -1-1

3030 00

¯̄¯̄

Calculate the Calculate the difference between difference between each score and the each score and the mean and summean and sum

1

)( 22

N

XXs

1919

80.4524

1)( 2

2 NXX

s

Computing the VarianceComputing the Variance

XX X - XX - X (X - X)(X - X)22

22 -3-3 99

44 -1-1 11

55 00 00

88 33 99

77 22 44

44 -1-1 11

3030 00 2424

¯̄¯̄

Calculate the square Calculate the square of the difference of the difference between each score between each score and the mean and and the mean and sumsum

Standard Deviation is Standard Deviation is the square rootthe square root

2020

Standard DeviationStandard Deviation

Definitional formulaDefinitional formula

The square root of the varianceThe square root of the variance

Computational formula based on algebraic Computational formula based on algebraic manipulationmanipulation Makes it easier to calculateMakes it easier to calculate

1)( 2

2

NXX

ss

2121

Computational FormulaComputational Formula

2222

Try one Try one

XX ff

55 22

44 55

33 33

22 22

11 22

NN

1

)( 22

NNX

Xs

2323

Try one Try one

XX ff fXfX

55 22

44 55

33 33

22 22

11 22

NN ∑∑fXfX

1

)( 22

NNX

Xs

2424

Try one Try one

XX ff fXfX

55 22 1010

44 55 2020

33 33 99

22 22 44

11 22 22

1414 4545

1

)( 22

NNX

Xs

2525

Try one Try one

XX ff fXfX XX22 fXfX22

55 22 1010

44 55 2020

33 33 99

22 22 44

11 22 22

1414 4545 ∑∑fXfX22

1

)( 22

NNX

Xs

2626

Try one Try one

XX ff fXfX XX22 fXfX22

55 22 1010 2525 5050

44 55 2020 1616 8080

33 33 99 99 2727

22 22 44 44 88

11 22 22 11 22

1414 4545 167167

1

)( 22

NNX

Xs

2727

XX ff fXfX XX22 fXfX22

55 22 1010 2525 5050

44 55 2020 1616 8080

33 33 99 99 2727

22 22 44 44 88

11 22 22 11 22

1414 4545 167167 11414)45(

1672

s

1

)( 22

NNX

Xs

31.172.113

36.22

13

64.144167

13142025

167

s

2828

EstimatorsEstimators

MeanMean Unbiased estimate of population mean (Unbiased estimate of population mean ())

Define unbiasedDefine unbiased Long range average of statistic is equal to the parameter Long range average of statistic is equal to the parameter

being estimated.being estimated.

VarianceVariance

Unbiased estimate of Unbiased estimate of 22

1)( 2

2

NXX

s

Cont.

2929

Estimators--cont.Estimators--cont.

UsingUsing

gives biased estimategives biased estimate Standard deviationStandard deviation

use square root of use square root of unbiasedunbiased estimate. estimate.

NXX

s2

2 )(

3030

Merrell’s Music Study Merrell’s Music Study SPSS SPSS PrintoutPrintout

WEEK4

Treatment Mean N Std. Deviation

Quiet 307.2319 23 71.8267

Mozart 114.5833 24 36.1017

Anthrax 1825.8889 24 103.1392

Total 755.4601 71 777.9646

3131

BoxplotsBoxplotsThe general problemThe general problem A display that shows dispersion for center and tails of A display that shows dispersion for center and tails of

distributiondistribution

Calculational steps (simple solution)Calculational steps (simple solution) Find medianFind median Find top and bottom 25% points (quartiles)Find top and bottom 25% points (quartiles) eliminate top and bottom 2.5% (fences)eliminate top and bottom 2.5% (fences) Draw boxes to quartiles and whiskers to fences, with Draw boxes to quartiles and whiskers to fences, with

remaining points as outliers remaining points as outliers

Boxplots for comparing groupsBoxplots for comparing groups

3232

Combined Merrell DataCombined Merrell Data

71N =

WEEK4

3000

2000

1000

0

-1000

3333

Merrell Data by GroupMerrell Data by Group

242423N =

Treatment Condition

AnthraxMozartQuiet

WE

EK

4

3000

2000

1000

0

-1000

3434

Review QuestionsReview Questions

What do we look for in a measure of What do we look for in a measure of dispersion?dispersion?

What role do outliers play?What role do outliers play?

Why do we say that the variance is a Why do we say that the variance is a measure of average variability around the measure of average variability around the mean?mean?

Why do we take the square root of the Why do we take the square root of the variance to get the standard deviation?variance to get the standard deviation?

Cont.

3535

Review QuestionsReview Questions--cont.--cont.

How does a boxplot reveal dispersion?How does a boxplot reveal dispersion?

What do David Merrell’s data tell us about What do David Merrell’s data tell us about the effect of music on mice?the effect of music on mice?