Editing Effect size (section)

==== Standardized mean difference ====
[[File:Cohens d 4panel.svg|thumb|Plots of Gaussian densities illustrating various values of Cohen's d.]]

A (population) effect size ''θ'' based on means usually considers the standardized mean difference (SMD) between two populations<ref name="HedgesL1985Statistical">{{Cite book | author = [[Larry V. Hedges]] & [[Ingram Olkin]] | title = Statistical Methods for Meta-Analysis | publisher = [[Academic Press]] | year = 1985 | location = Orlando | isbn = 978-0-12-336380-0 }}</ref>{{Rp|p=78|date=November 2012}}
<math display="block">\theta = \frac{\mu_1 - \mu_2} \sigma,</math>
where ''μ''<sub>1</sub> is the mean for one population, ''μ''<sub>2</sub> is the mean for the other population, and σ is a [[standard deviation]] based on either or both populations.

In the practical setting the population values are typically not known and must be estimated from sample statistics.  The several versions of effect sizes based on means differ with respect to which statistics are used.

This form for the effect size resembles the computation for a [[t-test|''t''-test]] statistic, with the critical difference that the ''t''-test statistic includes a factor of <math>\sqrt{n}</math>.  This means that for a given effect size, the significance level increases with the sample size.  Unlike the ''t''-test statistic, the effect size aims to estimate a population [[parameter]] and is not affected by the sample size.

SMD values of 0.2 to 0.5 are considered small, 0.5 to 0.8 are considered medium, and greater than 0.8 are considered large.<ref name="Andrade2020">{{cite journal | last1 = Andrade | first1 = Chittaranjan | title = Mean Difference, Standardized Mean Difference (SMD), and Their Use in Meta-Analysis | journal = The Journal of Clinical Psychiatry | date = 22 September 2020 | volume = 81 | issue = 5 | eissn = 1555-2101 | doi = 10.4088/JCP.20f13681 | pmid = 32965803 | s2cid = 221865130 | url = | quote = SMD values of 0.2-0.5 are considered small, values of 0.5-0.8 are considered medium, and values > 0.8 are considered large. In psychopharmacology studies that compare independent groups, SMDs that are statistically significant are almost always in the small to medium range. It is rare for large SMDs to be obtained.| doi-access = free }}</ref>