# Effects Size Estimates

## Cohen’s D

Cohen’s D is the number of standard deviations an effect is shifted above or below the populations mean stated by the null hypothesis.

### z test

When we have known population parameters, we use them to calculate $d = \frac{\bar{X} - \mu}{\sigma}$ Note that $\sigma$ is the standard deviation and NOT the standard error of the mean.

### One-sample t test

$d = \frac{\bar{X} - \mu}{s}$

### Independent sample t test

$d = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{s^2_P}}$ Consult a introductory statistics textbook for the calculation fo $s^2_p$.

### Cohen’s Effect Size Guidelines

d description
0.2 Small
0.2 - 0.8 Medium
0.8 Large

But don’t take these guidelines too seriously. It’s more important to compare the effect size to thos expected or typical within the field of study.

## Proportion of Variance Effect Size Measures

$\text{Proportion of Variance} = \frac{\text{variability explained}}{\text{total variability}}$

### Eta-Squared ($\eta^2$)

$\eta^2 = \frac{t^2}{t^2 + df}.$

### Omega-Squared ($\omega^2$)

While very popular, $\eta^2$ is biased and tends to overestimate effect size, particularly for studies with small sample size. Therefore, with small samples it is recommended that $\omega^2$ be used instead, which is a modified vesion of $\eta^2$. Note that $\omega^2$ is very similar to $\eta^2$, the only difference is that 1 is substraced from the numerator. This has the effect of making $\omega^2$ more conservative than $\eta^2$.

$\omega^2 = \frac{t^2 - 1}{t^2 + df}.$