individual differences & correlations psy 425 tests & measurements furr & bacharach ch...

Individual Differences&

Correlations

Psy 425

Tests & Measurements

Furr & Bacharach

Ch 3, Part 1

Nature of Variability

• Assumption: – Differences exist among people– A diagnostic measure is

capable of detecting those differences

• Two kinds of differences1. Interindividual (between

people)

2. Intraindividual (within a single person)

Questions…

• Who will be admitted?

• Who will benefit?

• Who should be hired?

• Who meets criteria for diagnosis?

Crucial assumption

• Psychological differences exist

• AND

• the differences can be detected through well-designed measurement processes

Psychometric Conceptsof Reliability & Validity

• Are entirely dependent on differences among people

Individual Differences & Psychological Tests

• Research– Exposing people to different

experimental conditions (experiences) & measuring effects of these conditions on behavior

– Determine the extent to which differences are a function of experimental conditions

• Clinical settings– Diagnosis– Change over time

Variability

• Differences among the scores within a distribution of scores

Assessment of Test Scores

• For a single test:– Detect and describe individual

differences within the distribution of scores

• Central tendency• Variability• Shape of the distribution

(X – X)

S # X {TS 1} Deviation 1 9 -8 2 26 9 3 20 3 4 11 -6 5 21 4 6 14 -3 7 24 7 8 13 -4 9 24 7

10 16 -1 11 12 -5 12 10 -7 13 13 -4 14 14 -3 15 21 4 16 14 -3 17 17 0 18 14 -3 19 14 -3 20 22 5 21 22 5 22 21 4 23 13 -4 24 24 7 25 17 0 26 31 14 27 14 -3 28 19 2 29 11 -6 30 14 -3

Sum (X) = 515 Mean (X) = 17.16667

XTEST

SCORES

Central Tendency

• “typical” score in a distribution of scores

• Mean =

• Arithmetic Mean

N

XX

(X – X)

S # X {TS 1} Deviation 1 9 -8 2 26 9 3 20 3 4 11 -6 5 21 4 6 14 -3 7 24 7 8 13 -4 9 24 7

10 16 -1 11 12 -5 12 10 -7 13 13 -4 14 14 -3 15 21 4 16 14 -3 17 17 0 18 14 -3 19 14 -3 20 22 5 21 22 5 22 21 4 23 13 -4 24 24 7 25 17 0 26 31 14 27 14 -3 28 19 2 29 11 -6 30 14 -3

Sum (X) = 515 Mean (X) = 17.16667

30

= 17.2

X =

N

XX

XMEAN

(X – X)

S # X {TS 1} Deviation 1 9 -8 2 26 9 3 20 3 4 11 -6 5 21 4 6 14 -3 7 24 7 8 13 -4 9 24 7

10 16 -1 11 12 -5 12 10 -7 13 13 -4 14 14 -3 15 21 4 16 14 -3 17 17 0 18 14 -3 19 14 -3 20 22 5 21 22 5 22 21 4 23 13 -4 24 24 7 25 17 0 26 31 14 27 14 -3 28 19 2 29 11 -6 30 14 -3

Sum (X) = 515 Mean (X) = 17.16667

30

= 17.2

X =

XMEAN

Variability

• Variance

• Standard Deviation

Computing Variance

S # X {TS 1} 1 9 2 26 3 20 4 11 5 21 6 14 7 24 8 13 9 24

10 16 11 12 12 10 13 13 14 14 15 21 16 14 17 17 18 14 19 14 20 22 21 22 22 21 23 13 24 24 25 17 26 31 27 14 28 19 29 11 30 14

Sum (X) = 515 Mean (X) = 17.16667

Mean = 17.17

XMEAN VARIANCE XX

(X – X) (X – X)2

S # X {TS 1} Deviation Deviation 1 9 -8 67 2 26 9 78 3 20 3 8 4 11 -6 38 5 21 4 15 6 14 -3 10 7 24 7 47 8 13 -4 17 9 24 7 47

10 16 -1 1 11 12 -5 27 12 10 -7 51 13 13 -4 17 14 14 -3 10 15 21 4 15 16 14 -3 10 17 17 0 0 18 14 -3 10 19 14 -3 10 20 22 5 23 21 22 5 23 22 21 4 15 23 13 -4 17 24 24 7 47 25 17 0 0 26 31 14 191 27 14 -3 10 28 19 2 3 29 11 -6 38 30 14 -3 10

Sum (X) = 515 856 Mean (X) = 17.16667 29

5.34218

(X – X)

1 9 – 17.17 = -8.17

2 26 – 17.17 = 8.83

3

.

.

.

.

.

.

Mean = 17.17

X XX

XX

DEVIATION

(X – X) (X – X)2


10 16 -1 1 11 12 -5 27 12 10 -7 51 13 13 -4 17 14 14 -3 10 15 21 4 15 16 14 -3 10 17 17 0 0 18 14 -3 10 19 14 -3 10 20 22 5 23 21 22 5 23 22 21 4 15 23 13 -4 17 24 24 7 47 25 17 0 0 26 31 14 191 27 14 -3 10 28 19 2 3 29 11 -6 38 30 14 -3 10

Sum (X) = 515 856 Mean (X) = 17.16667 29

5.34218

= SS = (X - X)2 = Variance (s2) = Standard Deviation (s)

XX 2XX X

XX Squared Deviation

(X – X) (X – X)2


10 16 -1 1 11 12 -5 27 12 10 -7 51 13 13 -4 17 14 14 -3 10 15 21 4 15 16 14 -3 10 17 17 0 0 18 14 -3 10 19 14 -3 10 20 22 5 23 21 22 5 23 22 21 4 15 23 13 -4 17 24 24 7 47 25 17 0 0 26 31 14 191 27 14 -3 10 28 19 2 3 29 11 -6 38 30 14 -3 10

Sum (X) = 515 856 Mean (X) = 17.16667 29

5.34218


s2 =856

30

= 29

XX 2XX

2

2

N

XXs

2

XXSS

X XX VARIANCE

Computing Standard Deviation

N

XXss

2

2

(X – X) (X – X)2


10 16 -1 1 11 12 -5 27 12 10 -7 51 13 13 -4 17 14 14 -3 10 15 21 4 15 16 14 -3 10 17 17 0 0 18 14 -3 10 19 14 -3 10 20 22 5 23 21 22 5 23 22 21 4 15 23 13 -4 17 24 24 7 47 25 17 0 0 26 31 14 191 27 14 -3 10 28 19 2 3 29 11 -6 38 30 14 -3 10

Sum (X) = 515 856 Mean (X) = 17.16667 29

5.34218


N

XXss

2

2

XX 2XX

2

XXSS

X XX

= 856/30

STDEV

Assessing the Distribution of Scores

• Frequency count– For each score or band of

scores, count the number of individuals who received that score or who are within that band of scores

– Plot the frequency distribution of scores

• Ideal distribution?– Normal = theoretically ideal

• What do you usually get?

Types of Distributions

• Normal– Symmetric on either side of the

mean– For psychological tests,

• Often assume that scores are normally distributed

• Important assumption…

• Skewed

0

5

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Distribution (2 wide)

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

TS 1

Num

ber

of P

artic

ipan

ts

Distribution (5 wide)

5 10 15 20 25 30 35

TS 1

Num

ber

of P

artic

ipan

ts

Normal Distribution

5 10 15 20 25 30 35

Normal(17.1667,5.43351)

TS 1

Num

ber

of P

artic

ipan

ts

Other Examples

5 10 15 20 25 30 35 40

Normal(16.55,5.80721)

AQ_Total

0 10 20 30 40 50 60 70

Normal(20.0606,13.048)

SPQ_total

Distributions

Worksheet #1

• Enter scores

• Determine central tendency and variability

• Graph frequency distribution of scores

Association between Distributions

• Covariability– Degree to which two distributions

of scores vary in a corresponding manner

• What scores might co-vary?– Depression & anxiety– Schizotypy & autism– IQ & GPA

1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

S # X Y

TEST SCORES

What do you want to know about these scores?

1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

S # X Y

1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

S # X Y

Direction & Magnitude

Direction of Relationship

• Positive or direct association– High on one, high on the other

• Negative or inverse association– High on one, low on the other

Magnitude of Relationship

• Strong or weak association?

• Strong– Consistency between variables

• Weak– Inconsistency between variables

LOOK!

1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

S # X Y

1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

515 646

17.17 21.53

5.43 14.87

S # X Y

X

X

s

• For each test:– Central

tendency– Variability

Deviation Deviation Cross-Product1 9 32 26 673 20 174 11 195 21 436 14 237 24 278 13 139 24 11

10 16 1311 12 2012 10 1613 13 1014 14 015 21 2016 14 017 17 2118 14 919 14 520 22 3721 22 4422 21 2823 13 2124 24 2525 17 3726 31 3527 14 1628 19 3729 11 2230 14 7

515 646

17.17 21.53

5.43 14.87

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

Covariance & Correlation

Deviation Deviation Cross-Product1 9 3 -8.172 26 67 8.833 20 17 2.834 11 19 -6.175 21 43 3.836 14 23 -3.177 24 27 6.838 13 13 -4.179 24 11 6.83

10 16 13 -1.1711 12 20 -5.1712 10 16 -7.1713 13 10 -4.1714 14 0 -3.1715 21 20 3.8316 14 0 -3.1717 17 21 -0.1718 14 9 -3.1719 14 5 -3.1720 22 37 4.8321 22 44 4.8322 21 28 3.8323 13 21 -4.1724 24 25 6.8325 17 37 -0.1726 31 35 13.8327 14 16 -3.1728 19 37 1.8329 11 22 -6.1730 14 7 -3.17

515 646

17.17 21.53

5.43 14.87

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

XX

Deviation Deviation Cross-Product1 9 3 -8.17 -18.532 26 67 8.83 45.473 20 17 2.83 -4.534 11 19 -6.17 -2.535 21 43 3.83 21.476 14 23 -3.17 1.477 24 27 6.83 5.478 13 13 -4.17 -8.539 24 11 6.83 -10.53

10 16 13 -1.17 -8.5311 12 20 -5.17 -1.5312 10 16 -7.17 -5.5313 13 10 -4.17 -11.5314 14 0 -3.17 -21.5315 21 20 3.83 -1.5316 14 0 -3.17 -21.5317 17 21 -0.17 -0.5318 14 9 -3.17 -12.5319 14 5 -3.17 -16.5320 22 37 4.83 15.4721 22 44 4.83 22.4722 21 28 3.83 6.4723 13 21 -4.17 -0.5324 24 25 6.83 3.4725 17 37 -0.17 15.4726 31 35 13.83 13.4727 14 16 -3.17 -5.5328 19 37 1.83 15.4729 11 22 -6.17 0.4730 14 7 -3.17 -14.53

515 646

17.17 21.53

5.43 14.87

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

YY

Deviation Deviation Cross-Product1 9 3 -8.17 -18.53 1512 26 67 8.83 45.47 4023 20 17 2.83 -4.53 -134 11 19 -6.17 -2.53 165 21 43 3.83 21.47 826 14 23 -3.17 1.47 -57 24 27 6.83 5.47 378 13 13 -4.17 -8.53 369 24 11 6.83 -10.53 -72

10 16 13 -1.17 -8.53 1011 12 20 -5.17 -1.53 812 10 16 -7.17 -5.53 4013 13 10 -4.17 -11.53 4814 14 0 -3.17 -21.53 6815 21 20 3.83 -1.53 -616 14 0 -3.17 -21.53 6817 17 21 -0.17 -0.53 018 14 9 -3.17 -12.53 4019 14 5 -3.17 -16.53 5220 22 37 4.83 15.47 7521 22 44 4.83 22.47 10922 21 28 3.83 6.47 2523 13 21 -4.17 -0.53 224 24 25 6.83 3.47 2425 17 37 -0.17 15.47 -326 31 35 13.83 13.47 18627 14 16 -3.17 -5.53 1828 19 37 1.83 15.47 2829 11 22 -6.17 0.47 -330 14 7 -3.17 -14.53 46

515 646

17.17 21.53

5.43 14.87

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

XX YY


10 16 13 -1.17 -8.53 1011 12 20 -5.17 -1.53 812 10 16 -7.17 -5.53 4013 13 10 -4.17 -11.53 4814 14 0 -3.17 -21.53 6815 21 20 3.83 -1.53 -616 14 0 -3.17 -21.53 6817 17 21 -0.17 -0.53 018 14 9 -3.17 -12.53 4019 14 5 -3.17 -16.53 5220 22 37 4.83 15.47 7521 22 44 4.83 22.47 10922 21 28 3.83 6.47 2523 13 21 -4.17 -0.53 224 24 25 6.83 3.47 2425 17 37 -0.17 15.47 -326 31 35 13.83 13.47 18627 14 16 -3.17 -5.53 1828 19 37 1.83 15.47 2829 11 22 -6.17 0.47 -330 14 7 -3.17 -14.53 46

515 646 1469

17.16667 21.53333

5.433506 14.87125

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

YYXX

Covariance

• Useful– Direction of association

1. Positive

2. Negative

• Limited information– Magnitude?

• Size of covariance effected by size of scales…

– Covariance between two small scale variables different than that between two large scale variables

N

YYXXcxy


10 16 13 -1.17 -8.53 1011 12 20 -5.17 -1.53 812 10 16 -7.17 -5.53 4013 13 10 -4.17 -11.53 4814 14 0 -3.17 -21.53 6815 21 20 3.83 -1.53 -616 14 0 -3.17 -21.53 6817 17 21 -0.17 -0.53 018 14 9 -3.17 -12.53 4019 14 5 -3.17 -16.53 5220 22 37 4.83 15.47 7521 22 44 4.83 22.47 10922 21 28 3.83 6.47 2523 13 21 -4.17 -0.53 224 24 25 6.83 3.47 2425 17 37 -0.17 15.47 -326 31 35 13.83 13.47 18627 14 16 -3.17 -5.53 1828 19 37 1.83 15.47 2829 11 22 -6.17 0.47 -330 14 7 -3.17 -14.53 46

515 646 1469

17.16667 21.53333 49

5.433506 14.87125

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

Covariance

Correlation

• INDEX OF CONSISTENCY OF INDIVIDUAL DIFFERENCE SCORES

• Easy to interpret

• Range between -1.0 and +1.0

• Reflects direction and magnitude of association

• “Bounded” quality is obtained by dividing the covariance by the standard deviations of the two variables.

yx

xyxy ss

cr


10 16 13 -1.17 -8.53 1011 12 20 -5.17 -1.53 812 10 16 -7.17 -5.53 4013 13 10 -4.17 -11.53 4814 14 0 -3.17 -21.53 6815 21 20 3.83 -1.53 -616 14 0 -3.17 -21.53 6817 17 21 -0.17 -0.53 018 14 9 -3.17 -12.53 4019 14 5 -3.17 -16.53 5220 22 37 4.83 15.47 7521 22 44 4.83 22.47 10922 21 28 3.83 6.47 2523 13 21 -4.17 -0.53 224 24 25 6.83 3.47 2425 17 37 -0.17 15.47 -326 31 35 13.83 13.47 18627 14 16 -3.17 -5.53 1828 19 37 1.83 15.47 2829 11 22 -6.17 0.47 -330 14 7 -3.17 -14.53 46

515 646 1469

17.16667 21.53333 49

5.433506 14.87125 0.15

S # X XX

Y YY XX YY

X

X

YYXX xyc xyrs

Correlation

Worksheet #2

• Enter scores

• Determine central tendency and variability

• Determine cross-products

• Determine covariance

• Determine correlation

individual differences & correlations psy 425 tests & measurements furr & bacharach ch...

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