Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Statistics LibreTexts

Common Formulas

( \newcommand{\kernel}{\mathrm{null}\,}\)

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)2N1

Which is also: s=(X¯X)2N1=SSdf

Some instructors prefer this formula because it is easier to calculate (but more difficult to see what's happening):

((X2)(X)2N)(N1)

z-score

To find the z-score when you have a raw score:

z=XˉXs

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=(ˉXμ)(sn)

Confidence Interval

Margin of Error =t×(sN)

 Confidence Interval =¯X±(t×(sN))

Independent Sample t-test

Unequal N

You can always use this formula:

t=(ˉX1ˉX2)[(n11)×s21+(n21)×s22n1+n22]×(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(sDN)

Full Formula

t=(ΣDN)(((XD¯XD)2)(N1))/N

SSB=EachGroup[(¯Xgroup¯XT)2×(ngroup)]

SSW=EachGroup[((X¯Xgroup)2)]

SST=[(X¯XT)2]

HSD=q×MSwngroup

Same as above.

SSPs=[((XPs)2k)]((X)2)N

SSWG=SSTSSBGSP

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))(N1))(sx×sy)

SD Not Calculated:

r=(((x¯Xx)×(y¯Xy))(N1))(((x¯Xx)2)N1)×(((y¯Xy)2)N1)

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((EO)2E)

Expected Frequencies

Goodness of Fit:

Nk

Test of Independence:

EEachCell=RT×CTN

In which RT = Row Total and CT = Column Total

Support Center

How can we help?