Common Formulas
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The following formulas are in the order in which you learn about them in this textook. Use the Table of Contents to look for a specific equation.
Descriptive Statistics
Mean
ˉX=∑XN
Standard Deviation
s=√∑(X−¯X)2N−1
Which is also: s=√∑(X−¯X)2N−1=√SSdf
Some instructors prefer this formula because it is easier to calculate (but more difficult to see what's happening):
√(∑(X2)−(∑X)2N)(N−1)
z-score
To find the z-score when you have a raw score:
z=X−ˉXs
To find a raw score when you have a z-score:
x=zs+¯X
t-tests
One-Sample t-test
These are the same formulas, but formatted slightly differently.
t=(ˉX−μ)(s√n)
Confidence Interval
Margin of Error =t×(s√N)
Confidence Interval =¯X±(t×(s√N))
Independent Sample t-test
Unequal N
You can always use this formula:
t=(ˉX1−ˉX2)√[(n1−1)×s21+(n2−1)×s22n1+n2−2]×(1n1+1n2)
Equal N
You should only use this formula when your two independent groups are the same size (N), meaning the same number of people in each group.
(¯X1−¯X2)√(s21N1)+(s22N2)
Dependent Sample t-test
Conceptual Formula (symbols)
t=¯XD(sD√N)
Full Formula
t=(ΣDN)√(∑((XD−¯XD)2)(N−1))/√N
ANOVA
Sums of Squares for Between Groups Designs
Between Groups
SSB=∑EachGroup[(¯Xgroup−¯XT)2×(ngroup)]
Within Groups
SSW=∑EachGroup[∑((X−¯Xgroup)2)]
Total
SST=∑[(X−¯XT)2]
Tukey's HSD for Pairwise Comparison
HSD=q×√MSwngroup
Sums of Squares for Repeated Measures Designs
Between Groups
Same as above.
Participants
SSPs=[∑((∑XPs)2k)]−((∑X)2)N
Within Groups (Error)
SSWG=SST−SSBG−SP
Total
Same as above.
Pearson's r (Correlation)
The following formulas are the same. Use the first one when you already have the standard deviation calculated.
These are paired data, so N is the number of pairs.
SD Already Calculated:
r=(∑((xEach−¯Xx)×(yEach−¯Xy))(N−1))(sx×sy)
SD Not Calculated:
r=(∑((x−¯Xx)×(y−¯Xy))(N−1))(√∑((x−¯Xx)2)N−1)×(√∑((y−¯Xy)2)N−1)
Regression Line Equation
ˆY=a+(b×X)
a (intercept):
a=¯Xy−(bׯXx)
b (slope):
∑(Diffx×Diffy)∑(DiffX2)
In which "Diff" means the differences between each score and that variable's mean.
Pearson's χ2 (Chi-Square)
χ2=∑Each((E−O)2E)
Expected Frequencies
Goodness of Fit:
Nk
Test of Independence:
EEachCell=RT×CTN
In which RT = Row Total and CT = Column Total