Copyright © Allyn & Bacon (2007) Manual Statistical Computation Procedures Graziano and Raulin...

Copyright © Allyn & Bacon (2007)Copyright © Allyn & Bacon (2007)

Manual Statistical Manual Statistical Computation Computation ProceduresProcedures

Graziano and RaulinGraziano and RaulinResearch MethodsResearch MethodsThis multimedia product and its contents are protected under copyright law. The following are This multimedia product and its contents are protected under copyright law. The following are prohibited by law: (1) Any public performance or display, including transmission of any image prohibited by law: (1) Any public performance or display, including transmission of any image over a network; (2) Preparation of any derivative work, including the extraction, in whole or in over a network; (2) Preparation of any derivative work, including the extraction, in whole or in part, of any images; (3) Any rental, lease, or lending of the program.part, of any images; (3) Any rental, lease, or lending of the program.


Measures of Central Measures of Central TendencyTendency MeanMean

MedianMedian– The middle score when scores are The middle score when scores are

arranged in order from highest to lowestarranged in order from highest to lowest ModeMode

– The most frequently occurring score The most frequently occurring score – Easily identified from a frequency Easily identified from a frequency

distributiondistribution

N

XX


Measures of VariabilityMeasures of Variability

VarianceVariance

ComputationalComputationalFormulaFormula

Standard DeviationStandard Deviation

sSS

df

X X

N2

2

1

( )

SS X

X

N 2

2

s s 2


Computational Computational ExampleExample Compute mean, Compute mean,

variance, and variance, and standard deviation standard deviation of the following: 2, of the following: 2, 5, 7, 4, 6, 5 5, 7, 4, 6, 5

Compute sum of Compute sum of XX and sum of and sum of XX22

83.46

29

N

XX

X 29 X 2 155

SS X

X

N 2

2

sSS

df2 14 83

52 97

..

s s 2 2 97 172. .

SS 155

29

614 83

2

.


CorrelationsCorrelations

Pearson product-moment correlationPearson product-moment correlation

Spearman rank-order correlationSpearman rank-order correlation

r

XYX Y

N

XX

NY

Y

N

xy

2

2

2

2

rd

N Ns

1

6

1

2

2


Computational Computational ExampleExample Compute the Compute the

product-moment product-moment correlation for the correlation for the data at the rightdata at the right

Compute the sums, Compute the sums, sums of squares, sums of squares, and cross products and cross products as shownas shown

Computations on Computations on next slidenext slide

X Y XX Y X22 Y Y22 XY XY

4 5 16 25 204 5 16 25 20

3 5 9 25 153 5 9 25 15

5 6 25 36 305 6 25 36 30

2 4 4 16 82 4 4 16 8

6 7 36 49 426 7 36 49 42

5 7 25 49 355 7 25 49 35

25 34 115 200 25 34 115 200 150150


Product-Moment Product-Moment CorrelationCorrelation

r

XYX Y

N

XX

NY

Y

N

xy

2

2

2

2

rxy

150

25 34

6

11525

6200

34

6

8 33

1083 7 3393

2 2

.

. ..


Computational Computational ExampleExample Compute a rank-Compute a rank-

order correlation for order correlation for the ordinal data at the ordinal data at the rightthe right

Compute the Compute the difference in ranks difference in ranks (d) and d(d) and d22 as shown as shown

Computations on Computations on next slidenext slide

X Y d dX Y d d22

1 3 -2 41 3 -2 4

2 2 0 02 2 0 0

3 1 2 43 1 2 4

4 5 -1 14 5 -1 1

5 4 1 15 4 1 1

1010


Rank-Order CorrelationRank-Order Correlation

rd

N Ns

1

6

1

2

2

rs

16 10

5 5 11

60

1201 50 50

2

( ). .


The The tt-test-test

For independent samplesFor independent samples

For correlated samplesFor correlated samples

t

X X

SS SSN N N N

1 2

1 2

1 2 1 221 1

t

X X

sN

sN

rs

N

s

N

1 2

12

1

22

2

1

1

2

2

2


Computational Computational ExampleExample Group 1 Group 2Group 1 Group 2

5 3 5 3 8 5 8 5 7 2 7 2 8 3 8 3 7 7Mean 7.00 3.25Mean 7.00 3.25SS 6.00 4.75SS 6.00 4.75N 5 4 N 5 4


Independent Samples Independent Samples tt-test-test

t

X X

SS SSN N N N

1 2

1 2

1 2 1 221 1

t

7 00 3 25

6 00 4 755 4 2

15

14

3 75

834 52

. .

. .

.

..

tcrit 2 365. Reject Null Hypothesis


Computational Computational ExampleExample


Correlated Correlated tt-test-test

t

X X

sN

sN

rs

N

s

N

1 2

12

1

22

2

1

1

2

2

2

t

317 2 96

89391

97591

2 84893

91

975

91

0 21

0 0563 75

2 2

. .

. .(. )

. .

.

..

tcrit 199. Reject Null Hypothesis


One-Way ANOVA One-Way ANOVA FormulasFormulasSource df SS MS F

Between k-1 SSb SSb/dfb MSb/MSw

Within N-k SSw SSw/dfw

Total N-1 SST


One-Way ANOVA One-Way ANOVA FormulasFormulas

SS

X

n

X

Nbi

ii

kT

2

1

2

SS SSw ii

k

1

SS X

X

NT T

T 2

2


Raw data (one-way Raw data (one-way ANOVA)ANOVA)

Group 1 Group 2 Group 33 4 91 3 73 5 82 5 114 4 93


Summarized DataSummarized Data

Summary StatisticsGroup 1 Group 2 Group 3 Totals

X 16 21 44 81

X2 48 91 396 535

n 6 5 5 16

Mean 2.67 4.20 8.80

SS 5.33 2.80 8.80


ComputationsComputations

SS X

X

NT T

T 2

22

53516

124 94(81)

.

SS SSw ii

k

1

5 33 2 80 8 80 16 93. . . .

SS

X

n

X

Nbi

ii

kT

2

1

22 2 2 216

6

21

5

44

5

81

16108 00.


Summary TableSummary Table

Source df SS MS F

Between 2 108.00 54.00 41.46

Within 13 16.93 1.30

Total 15 124.94

Fcrit(2,13) = 3.80 Reject null hypothesis


Planned Comparisons Planned Comparisons FormulasFormulas

WhereWhere

c w Xi ii

k

1

wii

k

0

1

tc

MSwnwi

ii

k

2

1



Compare Group 3 against the average Compare Group 3 against the average of Groups 1 and 2 from the previous of Groups 1 and 2 from the previous ANOVAANOVA

( )( . ) ( )( . ) ( )(8. ) .c w Xi ii

k

1

1 2 67 1 4 20 2 80 10 73

tc

MSwnwi

ii

k

.

.( ) ( ) ( )

.2

1

2 2 2

10 73

13016

15

25

7 07

tcrit 2 160. Reject null hypothesis


Repeated-Measures Repeated-Measures FormulasFormulas

Source df SS MS F

Subjects n-1 SSs SSs/dfs

Between k-1 SSb SSb/dfb MSb/MSE

Error (n-1)(k-1) SSE SSE/dfE


Repeated-Measures Repeated-Measures FormulasFormulas

SS

X

k

X

n kS

ijj

k

i

n

ijj

k

i

n

1

2

1 11

2

SS

X

n

X

n kb

iji

n

j

k

ijj

k

i

n

1

2

1 11

2

SS X

X

n kSS SSE ij

j

k

i

n ijj

k

i

n

S b

2

11

11

2


Raw Data (Repeated Raw Data (Repeated Measures)Measures)

Subject Cond. 1 Cond. 2 Cond. 3 Sum

A 10 8 8 26

B 12 9 8 29

C 11 10 9 30

D 11 10 8 29

E 10 8 9 27

F 11 9 7 27


Summarized DataSummarized Data

Cond. 1 Cond. 2 Cond. 3 Sums

X 65 54 49 168

X2 707 490 403 4716

Mean 10.83 9.00 8.17



SSS

26 29 30 29 27 27

3

168

6 34 00

2 2 2 2 2 2 2

.

SSb

65 54 30

6

168

6 322 33

2 2 2 2

.

SSE ( ) . . .707 490 403

168

6 34 00 22 33 5 67

2



Source df SS MS F

Subjects 5 4.00 0.80

Between 2 22.33 11.17 19.60

Error 10 5.67 0.570

Fcrit(2,10) = 4.10 Reject null hypothesis


Two-Way ANOVA Two-Way ANOVA FormulasFormulas

SS

X

n

X

NA

ijj

b

ijj

bi

aT

1

2

1

1

2 SS

X

n

X

NB

iji

a

iji

aj

bT

1

2

1

1

2

SS

X

n

X

n

X

n

X

NAB

ij

ijj

b

i

a ijj

b

ijj

bi

a iji

a

iji

aj

bT

2

11

1

2

1

1

1

2

1

1

2

SS SSw ijj

b

i

a

11


Two-Way ANOVA Two-Way ANOVA FormulasFormulasSource df SS MS F

A a-1 SSA SSA/dfA MSA/MSw

B b-1 SSB SSB/dfB MSB/MSw

AB (a-1)(b-1) SSAB SSAB/dfAB MSAB/MSw

Within N-ab SSw SSw/dfw

Total N-1 SST


Raw DataRaw Data

B1 B2 B3

A1 A2 A3 A1 A2 A3 A1 A2 A3

10 7 11 1 6 4 3 2 58 4 9 2 7 3 2 1 67 3 10 1 6 6 3 2 49 2 9 4 5 4 3 3 56 1 2 3 4 5


Summarized Data Summarized Data (1/4)(1/4)

A Level Terms B1 B2 B3 Totals

X 40 16 50 106

A1X2 330 78 504 912

n 5 4 5 14

SS 10 14 4

(X)2/n 320 64 500 802.57

Mean 8 4 10 7.57




X 10 24 20 54

A2X2 26 146 86 258

n 5 4 5 14

SS 6 2 6

(X)2/n 20 144 80 208.29

Mean 2 6 4 3.86




X 15 8 25 48

A3X2 47 18 127 192

n 5 4 5 14

SS 2 2 2

(X)2/n 45 16 125 164.57

Mean 3 2 5 3.43




X 65 48 95 208

Totals X2 403 242 717 1362

n 15 12 15 42

SS

(X)2/n 281.67 192.00 601.67 1030.10

Mean 4.33 4.00 6.33 4.95


Computations (two-Computations (two-way) 1/3way) 1/3

SS

X

n

X

NA

ijj

b

ijj

bi

aT

1

2

1

1

2

802 57 208 29 164 57 103010 45 24[ . . . ] . .

SS

X

n

X

NB

iji

a

iji

aj

bT

1

2

1

1

2

28167 192 00 60167 103010 14533[ . . . ] . .



SS

X

n

X

n

X

n

X

NAB

ij

ijj

b

i

a ijj

b

ijj

bi

a iji

a

iji

aj

bT

2

11

1

2

1

1

1

2

1

1

2

SSAB [ ]320 64 500 20 144 80 45 16 125

[ . . . ] [ . . . ] [ . ]28167 192 00 60167 802 57 208 29 164 57 103010

SSAB 9333.



SS SSw ijj

b

i

a

11

10 14 4 6 2 6 2 2 2 48 00.

SS X

X

NT TT

2

22

1362208

4233190.



Source df SS MS F

A 2 145.33 72.67 49.96

B 2 45.24 22.62 15.55

AB 4 93.33 23.33 16.04

Within 33 48.00 1.45

Total 41 331.90

Fcrit(2, 33) = 3.29 Fcrit(4, 33) = 2.66


Chi-Square FormulasChi-Square Formulas

Use same formula for bothUse same formula for both– Chi-square goodness-of-fit testChi-square goodness-of-fit test– Chi-square test-for-independenceChi-square test-for-independence

Only difference is how expected cell Only difference is how expected cell frequencies are computedfrequencies are computed

2

2

1

O E

Ei i

ii

k


Goodness-of-fit Data Goodness-of-fit Data SetSet

Heads Tails

Observed 31 19

Expected 25 25


Goodness-of-fit Goodness-of-fit ComputationsComputations

2

2

1

O E

Ei i

ii

k

22 231 25

25

19 25

252 88

( ) ( ).

crit2 3841 . Fail to reject null hypothesis


Data for Test-for-Data for Test-for-IndendenceIndendence

Republican Democrat IndependentRow

Totals

Males 20(13.33)

10(10.00)

20(26.67)

50

Females 20(26.67)

20(20.00)

60(53.33)

100

ColumnTotals

40 30 80 150

The expected cell frequencies are computed by multiplying the rowtotal by the column total and dividing by the grand total. Theexpected cell frequencies are shown in parentheses.


Test-for-Independence Test-for-Independence ComputationsComputations

2

2

1

O E

Ei i

ii

k

22 2 220 13 33

13 33

10 10 00

10 00

20 26 67

26 67

( . )

.

( . )

.

( . )

.

( . )

.

( )

.

( . )

..

20 26 67

26 67

20 20

20 00

60 53 33

53 337 51

2 2 2

crit2 5991 . Reject null hypothesis


SummarySummary

The Student Resource Website The Student Resource Website illustrates how to compute each of illustrates how to compute each of these statistical procedures and these statistical procedures and also includes the statistical tables also includes the statistical tables needed to interpret the resultsneeded to interpret the results

The website also shows how to do The website also shows how to do the same computations using the same computations using SPSS for WindowsSPSS for Windows

Copyright © Allyn & Bacon (2007) Manual Statistical Computation Procedures Graziano and Raulin...

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