## 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 \(\eta^2\).

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