11.4 Chapter 11 Formulas

One-Way ANOVA

$H_{0} : μ_{1} = μ_{2} = μ_{3} = \dots = μ_{k}$

f $H sub 1:$ At least one mean is different.

Source	$S S$ = Sum of Squares	$d f$	$M S$ = Mean Square	F
Between (Factor)	$∑ n sub i open parenthesis x bar sub i minus x bar sub G M close parenthesis squared$	$k minus 1$	$M S B = \frac{S S B}{k - 1}$	$F = \frac{M S B}{M S W}$
Within (Error)	$∑ open parenthesis n sub i minus 1 close parenthesis s sub i squared$	$N - k$	$M S W = \frac{S S W}{N - k}$
Total	SST	$N minus 1$

${\bar{x}}_{i}$ = sample mean from the $i^{t h}$ group

$n_{i}$ = sample size of the $i^{t h}$ group

$k$ = number of groups

$s_{i}^{2}$ = sample variance from the $i^{t h}$ group

$N = n_{1} + n_{2} + \dots + n_{k}$

${\bar{x}}_{G M} = \frac{\sum x_{i}}{N}$

Bonferroni Test

$H sub 0: μ sub i equals μ sub j$

$H sub 1: μ sub i not equal to μ sub j$

Bonferroni test statistic: $t = \frac{{\bar{x}}_{i} - {\bar{x}}_{j}}{\sqrt{(M S W (\frac{1}{n_{i}} + \frac{1}{n_{j}}))}}$

Multiply p-value by $m equals k C sub 2$ , divide area for critical value by $m equals k C sub 2$ .

Two-Way ANOVA

Row Effect (Factor A):	$H sub 0:$ The row variable has no effect on the average ___________________. $H sub 1:$ The row variable has an effect on the average ___________________.
Column Effect (Factor B):	$H sub 0$ : The column variable has no effect on the average ___________________. $H sub 1$ : The column variable has an effect on the average ___________________.
Interaction Effect (A×B):	$H sub 0:$ There is no interaction effect between row variable and column variable on the average ___________________. $H sub 1:$ There is an interaction effect between row variable and column variable on the average ___________________.

Source	$S S$	$d f$	$M S$	F
$A$ (row factor)	$S S_{A}$	$a minus 1$	$M S_{A} = \frac{S S_{A}}{d f_{A}}$	$F_{A} = \frac{M S_{A}}{M S E}$
$B$ (column factor)	$S S_{B}$	$b minus 1$	$M S_{B} = \frac{S S_{B}}{d f_{B}}$	$F_{B} = \frac{M S_{B}}{M S E}$
$A \times B$ (interaction)	$S S_{A \times B}$	$(a - 1) (b - 1)$	$M S_{A \times B} = \frac{S S_{A \times B}}{d f_{A \times B}}$	$F_{A \times B} = \frac{M S_{A \times B}}{M S E}$
Error (within)	$S S E$	$a b open parenthesis n minus 1 close parenthesis$	$M S E = \frac{S S E}{d f_{E}}$
Total	$S S T$	$N minus 1$

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.