Effect Size Cohen's d & r Reference Cheat Sheet

Key Facts

• Cohen's $d$ for two independent groups is $d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}$, where $s_p$ is the pooled standard deviation.
• The pooled standard deviation is $s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$.
• For a one-sample or paired comparison, Cohen's $d$ can be written as $d = \frac{\bar{x} - \mu_0}{s}$ or $d = \frac{\bar{d}}{s_d}$.
• A common rough guide is $d \approx 0.2$ small, $d \approx 0.5$ medium, and $d \approx 0.8$ large.
• The correlation coefficient satisfies $-1 \le r \le 1$, where the sign gives direction and $|r|$ gives strength.
• The coefficient of determination is $r^2$, which represents the proportion of variation in one variable explained by the linear relationship with the other variable.
• A common rough guide is $|r| \approx 0.1$ weak, $|r| \approx 0.3$ moderate, and $|r| \approx 0.5$ strong.
• Effect size should be reported with context because the same value of $d$ or $r$ can be small in one field and important in another.

Vocabulary

Effect size

A numerical measure of how large or strong a statistical result is.

Cohen's d

An effect size that measures the difference between means in standard deviation units.

Pooled standard deviation

A weighted average of two group standard deviations used when comparing independent group means.

Correlation coefficient

A value $r$ from $-1$ to $1$ that measures the direction and strength of a linear relationship.

Coefficient of determination

The value $r^2$ that describes the proportion of variance explained by a linear relationship.

Practical significance

The real-world importance of a result, judged by its size, context, and consequences.