Calculating MS and F - Between subjects ANOVA

1. Compute the total sum of squared (SS) - square the sum of every individual score’s difference from the overall mean

• $SStotal=\sum (Xij-Xbar..)2SS\_\{total\}\; =\; \backslash sum(X\_\{ij\}\; -\; Xbar\_\{..\}\; )^2$
• SS contains within and between group differences

2. Find out what part can be attributed to the groups and what can be attributed to the individuals (within the group) - this calculates the between group differences

• $SSgroup=n\ast \sum (Xbar.j-Xbar..)2SS\_\{group\}\; =\; n*\; \backslash sum(Xbar\_\{.j\}\; -\; Xbar\_\{..\})^2$
• i.e. compute for each group the difference from the overall mean, and square that value and add them all together

3. Subtract SS$Group\_\{Group\}$ by SS$Total\_\{Total\}$ to get SS$error\_\{error\}$

4. Transform SS to MS by dividing SS values by df associates with them

• $dfGroup=k-1df\_\{Group\}\; =\; k-1$
• $dfError=k(n-1)df\_\{Error\}\; =\; k(n-1)$
• Divide SSgroup by df(group) and SSerror by df(error)
• We divide by df to correct for the SS being larger for larger sample sizes and number of groups
• $MSError=SSError/dfErrorMS\_\{Error\}\; =\; SS\_\{Error\}\; /\; df\_\{Error\}$
• $MSGroup=SSGroup/dfgroupMS\_\{Group\}\; =\; SS\_\{Group\}\; /\; df\_\{group\}$

5. Calculate F

• $F=MSGroup/MSErrorF\; =\; MS\_\{Group\}\; /\; MS\_\{Error\}$
• 
  • A = large between group differences / small within group differences = large F
  • B = small between group differences / small within group differences =
  • C = large between group differences / large within group differences =
  • D = small between group differences / large within group differences = very small F