Download Formula Sheet – Statistics section

Formula Sheet – Statistics section
Mean
X

X
N
Where:
X = the data set mean

∑ = the sum of
X = the scores in the distribution
N = the number of scores in the distribution
Range
range  X highest  X lowest
Where:
X highest
= largest score 
X lowest = smallest score

Variance

SD 2 
(X  X) 2
N
The simplified variance formula

SD2 
(X) 2
N
N
X 2 
Where:
SD2 = the variance

∑ = the sum of
X = the obtained score
X = the mean score of the data
N = the number of scores

Standard Deviation (N)
SD 
The simplified standard deviation formula

(X  X) 2
N
(X) 2
X 
N
SD 
N
2
Where:
SD = the standard deviation

∑ = the sum of
X = the obtained score
X = the mean score of the data
N = the number of scores

The Pearson product-moment correlation
r
zX zY
N
Where:
r = correlation coefficient

∑ = the sum of
zX = Z score for variable X
zY = Z score for variable Y
zXzY = the cross product of Z scores
N = the number of scores
Bivariate Regression
Predicted ZY  ()(Zx )
Where:
ZY = the predicted Z score on the criterion variable Y

 = the standardized regression
coefficient
ZX = the predicted Z score on the predictor variable X

Multiple Regression
Predicted ZY  (1 )(Z x1 )  ( 2 )(Z x2 )  ( 3 )(Z x3 )  ... ( last )(Z xlast )
Where:
ZY = the predicted Z score on the criterion variable Y
 = the standardized regression coefficient
ZX = the predicted Z score on the predictor variable X

T-test
Case I (single sample)
tobt 
X 
sX
Where:
tobt = obtained t

X = the sample mean
µ = the population mean
sX
= the estimated standard error of the mean


Case II (Dependent means)

tobt 
D
sD
Where:
tobt = obtained t

D = mean of the difference scores
sD
= standard error of the difference scores


Case II (Independent means)

tobt 
X1  X 2
s12 s22

n1 n 2
Where:
tobt = obtained t

X 1 and X 2 = means for the two groups
s12 and s22 = variances of the two groups
n1 and n2 = number of participants in each of the two groups




 ANOVA


df Between  NGroups 1
Where:
dfBetween = degrees of freedom for between subjects
NGroups = number of total
groups
GX 
X
NGroups
Where:
G X = grand mean
X = sum of all group means
NGroups = number of total groups





SX2 
(X  GX) 2
df Between
Where:
SX2 = the estimated variance of the distribution of means

 = sum of
(X  GX) 2 = the square of each group mean minus the grand mean
dfBetween = degrees of freedom
2
SBetween
 (SX2 )(n)
Where:
2
SBetween
= the estimated between-group
population variance

SX2 = the estimated variance of the distribution of means
n = the number of scores in each group




SW2 ithin 
2
S12  S22  ... SLast
NGroups
Where:
SW2 ithin = the estimated within-group population variance

S12 = the estimated population variance from the first group’s scores
S22 = the estimated population variance from the second group’s scores
2
SLast
= the estimated population variance from the last group’s scores
NGroups = number of total groups


F
2
SBetween
SW2 ithin
Where:
F = the F score

2
SBetween
= the estimated between-group population variance
SW2 ithin = the estimated within-group population variance


Chi – Square – Goodness of Fit and Test of Independence
2  

(O  E) 2
E
Where:
 2 = Chi Square obtained

∑ = the sum of
O = observed score
E = expected score
References
Aron, A., Aron, E.N., & Coups, E.J. (2008). Statistics for the behavioral and social
sciences. Upper Saddle River, NJ: Pearson Education, Inc.
Heiman, G.W. (2001). Understanding research methods and statistics: an integrated
introduction for psychology. Boston, MA: Houghton Mifflin Company.
Jackson, S.L. (2006). Research Methods and statistics: A critical thinking approach.
Belmont, CA: Thomson Wadsworth.