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95206_Graveter_Front_Endpapers
4/12/06
2:17 PM
Page 2
SUMMARY OF STATISTICS FORMULAS
THE MEAN
Population:
t STATISTIC (RELATED SAMPLES)
X
N
X
M n
Sample:
SUM OF SQUARES
t Statistic (Single Sample)
M tsM
2
(X)
SS X 2 N
t Statistic (Independent Measures)
1 2 M1 M2 ts(M1 M2)
VARIANCE
Population:
s2
n
ESTIMATION
Definitional: SS (X )2
Computational:
MD D
t where sMD sMD
SS
2 N
Sample:
SS
s 2 n1
STANDARD DEVIATION
SNS
t Statistic (Related Samples)
D MD tsMD
INDEPENDENT-MEASURES ANOVA
SS
n1
G2
SStotal X 2 N
dftotal N 1
z-SCORE (FOR LOCATING AN X VALUE)
T 2 G2
SSbetween n
N
dfbetween k 1
X
z SSwithin SSinside each treatment
dfwithin N k
Population:
Sample:
s
BINOMIAL (NORMAL APPROXIMATION)
X pn
z or
npq
X/n p
z p
q/n
where M n
2
n
t STATISTIC (SINGLE SAMPLE)
M
t sM
where sM sn
(M1 M2) (1 2)
t s(M1M2)
s2p
s2p
and
n1 n2
T2 G2
SSbetween n
N
dfbetween k 1
SSerror SSwithin SSsubjects
dferror (N k) (n 1)
where SSwithin SSinside each treatment
P2 G2
and SSsubjects k
N
2
t STATISTIC (INDEPENDENT MEASURES)
where s(M1M2) SS
where each MS df
REPEATED-MEASURES ANOVA
z-SCORE (FOR LOCATING A SAMPLE MEAN)
M
z M
MSbetween
F
MSwithin
SS1 SS2
s2p df1 df2
MSbetween
F MSerror
SS
where each MS df
95206_Graveter_Front_Endpapers
4/12/06
2:17 PM
Page 3
TWO-FACTOR ANOVA
SSregression R2SSY
df 2
T2 G2
SSbetween treatments n
N
SSresidual (1 R2)SSY
SS
MSregression df
df n 3
SS
MSresidual df
dfbetween treatments number of cells 1
SSwithin treatments SSeach treatment
LINEAR AND MULTIPLE REGRESSION
dfwithin treatments dfeach treatment
Standard error of estimate M
Sresidual
T R2 OW G 2
SSA nROW
N
F MSregression
MSresidual
dfA (number of levels of A) 1
CHI-SQUARE STATISTIC
(fo fe)2
2 fe
2
T COL G 2
SSB nCOL
N
dfB (number of levels of B) 1
MANN-WHITNEY U
SSAB SSbetween treatments SSA SSB
dfAB dfbetween treatments dfA dfB
MSA
FA MSwithin
MSB
FB MSwithin
MSAB
FAB MSwithin
nA(nA 1)
UA nAnB RA
2
nB(nB 1)
UB nAnB RB
2
SS
where each MS df
KRUSKAL-WALLIS TEST
PEARSON CORRELATION
T2
12
H 3(N 1)
n
N(N 1)
SP
r S
SXSSY
FRIEDMAN TEST
(X)(Y)
where SP (X MX)(Y MY) XY n
12
2r R2 3n(k 1)
nk(k 1)
SPEARMAN CORRELATION
MEASURES OF EFFECT SIZE
6 D2
rs 1 n(n2 1)
mean difference
Cohen’s d standard deviation
r 2 and 2 (Percentage of Variance Accounted For)
t2
r2 (for Independent and Repeated t)
t 2 df
LINEAR REGRESSION
Ŷ bX a
SP
where b SSX
SSregression r2SSY df 1
and
SS
MSregression df
SSresidual (1 r )SSY df n 2
2
MULTIPLE REGRESSION
Ŷ b1X1 b2X2 a
b1SPX1Y b2SPX2Y
R2 SSY
a MY bMX
SS
MSresidual df
SStreatment
2 SStreatment SSerror term
phi 2
n
Cramér’s V 2
n(df)
(for Analysis of
Variance)
(for Chi-square
Test for
Independence)