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Review Sheet

Analysis of Variance (ANOVA)

ANOVA Steps and Notation

Given the following hypothetical data from an experiment examining learning performance under three

temperature conditions

Temperature Conditions! " #$% 70 90

0 4 1 &'"!%1 3 2 G = 30

3 6 2 N = 15

1 3 0 k = 3

0 4 0

*! $ *" "% *# $

SS! SS" SS# +

n! $ n" $ n# $,,,

'! !,,,

'" +,,,

'# !

*he letter - identifies the num.er of treatment conditions (the num.er of levels of the factor)

*he num.er of scores in each treatment is identified .y a lowercase letter n

*he total num.er of scores in the entire study is specified .y a capital letter N*he total (&') for each treatment condition is identified .y the capital letter *

*he sum of all the scores in the research study (the grand total) is identified .y G/ 0ou can compute G .y

adding up all of the N scores or .y adding up the treatment totals G &*

*he final calculation for ANOVA is the 12ratio which is composed of two variances

1 variance .etween treatmentsvariance within treatments

sample variance s" SSdf

Recall that

SS &'"3 (&')"

N

Total Sum of Squares, SStotal

SStotal &'"3 G"

N

4n our example5 we have

SStotal !% 3 #%"

!$ +

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Within-Treatments Sum of Squares, SSwithin

SSwithin &SSinside each treatmentSSwithin 6 6 +

!

Between-Treatments Sum of Squares, SSbetween

SS.etween &(*"7n) 3 G"7N

SS.etween $"7$ 6 "%"7$ 6 $"7$ 2 #%"7!$

$ 6 8% 6 $ 3 %

9% 3 %

#%

*o verify our analysis5 chec- to see that the two components add up to the total/

SStotal SS.etween6 SSwithin

+ #% 6 !

Total Derees of !reedom, dftotal

dftotal N 3 !dftotal !$ 3 ! !+

Within-Treatments Derees of !reedom, dfwithin

dfwithin &(n 3 !) &dfin each treatment

dfwithin N 3 -

dfwithin !$ 3 # !"

Between-Treatments Derees of !reedom, dfbetween

df.etween - 3 !

df.etween # 3 ! "

*he two parts we o.tained from this analysis of degrees of freedom add up to e:ual the total degrees of freedom

dftotal dfwithin6 df.etween!+ !" 6 "

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;S (variance) s" SS

df

;S.etween s.etween" SS.etween #%7" !$

df.etween

;Swithin swithin" SSwithin !7!" !/##

dfwithin

1 ;S.etween;Swithin

1 !$7!/## !!/"8

*hen it is useful to organiithin treatments ! !" !/##*otal + !+

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?ypothesis testing with ANOVA

Given the data

@lace.o rug A rug = rug B% ! # C N "%

" + # G %

% ! + $ &'" ""

% ! # # # + +

* $ * !% * "% * "$

SS 8 SS 8 SS SS !%

?% D! D" D# D+

?a At least one of the treatment means is different

>e will use E /%$

a/ Analy