<form id="esForm" method="post"> <table class="inline" id="info"> <tr> <th colspan="2" class="centeralign"> Group 1 </th> </tr> <tr> <td>Sample size</td> <td> <input type="text" name="n1" value="100"> </td> </tr> <tr> <td>Sample mean</td> <td> <input type="text" name="m1" value="0"> </td> </tr> <tr> <td>Sample variance</td> <td> <input type="text" name="v1" value="1"> </td> </tr> <tr> <th colspan="2" class="centeralign"> Group 2 </th> </tr> <tr> <td>Sample size</td> <td> <input type="text" name="n2" value="100"> </td> </tr> <tr> <td>Sample mean</td> <td> <input type="text" name="m2" value="1"> </td> </tr> <tr> <td>Sample variance</td> <td> <input type="text" name="v2" value="1"> </td> </tr> <tr> <th colspan="2" class="centeralign"> Upload data </th> </tr> <tr> <td>Select your file</td> <td> <input type="file" name="userData"> </td> </tr> <tr><td>Data information</td><td> <select name="data" id="data"> <option value="0">No variable names</option> <option value="1">With variable names</option> </select></td></tr> <tr> <th colspan="2" class="centeralign"> Options </th> </tr> <tr> <td>Confidence limit</td> <td> <input type="text" name="cl" value="0.95"> </td> </tr> <tr> <td>Number of bootstraps</td> <td> <input type="text" name="nboot" value="1000"> </td> </tr> <tr> <td>Type of CI</th> <td> <select name="type" id="type"> <option value="theory">Theoretical</option> <option value="bootstrap">Bootstrap</option> <option value="bayes">Bayesian</option> </select> </td> </tr> <tr><td>Correction</th> <td> <select name="correction" id="correction"> <option value="hedge">Hedge's g</option> <option value="none">No correction (Cohen's d)</option>     </select> </td> </tr> </table> <input type="hidden" name="note" value="independent two-sample t-test" /> <input type="submit" value="Calculate" /> </form> <br /> <br /> <div id=output> </div>   <script src="../../lib/jquery-1.9.1.js"></script> <script> // Save a reference to this version var jQueryWP = $.noConflict(true); </script>   <script type="text/javascript">   jQueryWP("#esForm").on('submit', (function(e) { e.preventDefault(); jQueryWP.ajax({ url: "/effectsize/n2ttest/run.php", type: "POST", data: new FormData(this), contentType: false, cache: false, processData: false, success: function(data) { jQueryWP("#output").html(data); }, error: function() {} }); })); </script>

In independent two-sample design, the most popular effect size measure is defined as

$$ \delta = \frac{\mu_1 - \mu_2}{\sigma} $$

where $\mu_1$ and $\mu_2$ are the population means of the two population and $\sigma$ is the population standard deviation which is assumed to be the same for the two populations.

Although Cohen's d is the most widely used effect size estimator in samples, it is biased. An unbiased estimator, often referred to Hedges' g, was developed by Hedges (1981). It corrects the Cohen's d in the following way:

$$ g = d\times \frac{\Gamma(m/2)}{\sqrt{(m/2)} \Gamma[(m-1)/2]} $$

where $m = n_1 + n_2 - 2$.

Hedges (1981) showed that the variance of $g$ is

$$ var(g) = \frac{m(1+\tilde{n} \delta^2)}{(m-2)\tilde{n}} - \frac{\delta^2}{c^2} $$

where $\tilde{n}=n_1 n_2/(n_1+n_2)$ and

$$c=\frac{\Gamma(m/2)}{\sqrt{(m/2)} \Gamma[(m-1)/2]}$$.

With it, a confidence interval for $g$ can be constructed as

$$g \pm \Phi^{-1}(1-\alpha/2) \sqrt{\widehat{var(g)}}$$

where $\sqrt{\widehat{var(g)}}$ is an estimate of $var(g)$ by replacing $\delta$ with Cohen's d.

• Normally distributed data
• Equal variances

1. Cohen's d
2. Effect size of independent two samples

Hedges, L. V. (1981). Distribution theory for Glass’s estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128.