ANOVA & Correlation - Stats

Between-Groups Variance (S²b)

This represents the variability between the means of the different groups.

SSB / dfB

dfB

number of groups minus one

dfW

number of scores minus number of groups

Within-Groups Variance (S²W)

This represents the variability inside each group

SSW / dfW

SSB

SUM { N (M-GM)² }

SSW

SUM (X - M)²

F

S²B / S²W

p-value

probability of getting this F or greater given these dfs

limitations correlation

1) not causation (could be third variable or coincidence)

2) only plots linear correlation (ex up and down not taken into account)

limits regression

1) prediction outside of range wildly innacurate (0$ income = -100 gpa)

2) only predicts linear

3) never 100% accurate

why correlation analysis ( r ) relies on Z scores

✅ Z-scores standardize different variables onto the same scale.
✅ Z-scores center the data and make multiplication meaningful.
✅ Correlation becomes unit-free, reflecting the true relationship strength.

Correlation​ properties

Strength & direction of relationship

Two continuous variables

r (correlation coefficient)

H0: r = 0

ANOVA properties

Difference between group means

One categorical (grouping) variable and one continuous outcome variable

F (F-ratio)

H0: μ₁ = μ₂ = μ₃

When to use Bonferroni and when to use Scheffe

Bonferroni is better for a few specific comparisons planned before the ANOVA is undergone

Scheffé is better when you want to explore many comparisons post hoc

Bonferroni Correction

• Divide the desired alpha level (e.g., 0.05) by the number of comparisons (m) to obtain a new, more stringent significance threshold for each individual test

• Planned contrast

Scheffe

• Adjusts the cutoff used in F-tests to account for the multiple comparisons, ensuring that the overall alpha level is maintained across all possible contrasts.

• Conducted after a sig result is found

• post hoc

Alpha Risk

• Risk of Type I error / false positive

• If you have three groups and do three independent means t tests, each with an alpha level of 0.05 (5% risk) than you have a 15% risk of Type I error

• ANOVA (as well as planned contrast and post hoc) allows for comparing three groups and maintaining the alpha level of 0.05