Interpreting Cohen's d Effect Size

An Interactive Visualization

Created by Kristoffer Magnusson · Version 2.7.0 ·

The Cohen's d effect size is immensely popular in psychology. However, its interpretation is not straightforward and researchers often use general guidelines, such as small (0.2), medium (0.5) and large (0.8) when interpreting an effect. Moreover, in many cases it is questionable whether the standardized mean difference is more interpretable than the unstandardized mean difference.

In order to aid the interpretation of Cohen's d, this visualization offers these different representations of Cohen's d: visual overlap, Cohen's U₃, the probability of superiority, percentage of overlap, and the number needed to treat. It also lets you change the standard deviation and displays the unstandardized difference.

Cohen's d

Drag the chart to pan · double-click to recenter.

Cohen’s U₃

% Overlap

Probability of Superiority

Number Needed to Treat

A Common Language Explanation

With a Cohen's d of 0.80, 78.8% of the "Treatment" group will be above the mean of the "Control" group (Cohen's U₃), 68.9% of the two groups will overlap, and there is a 71.4% chance that a person picked at random from the "Treatment" group will have a higher score than a person picked at random from the "Control" group (probability of superiority). Moreover, in order to have one more favorable outcome in the "Treatment" group compared to the "Control" group, we need to treat 3.5 people on average. This means that if there are 100 people in each group, and we assume that 20 people have favorable outcomes in the "Control" group, then 20 + 28.3 people in the "Treatment" group will have favorable outcomes.¹

¹The values are averages, and it is assumed that 20 (CER) of the "Control" group have "favorable outcomes," i.e., their outcomes are below some cut-off. Change this using the Settings button above the slider. Go to the formula section for more information.

FAQ

How do I use this visualization?

Use the slider to change Cohen's d, or open the settings to change the parameters. The inputs can also be controlled with the keyboard arrows.

From the settings you can change the means, standard deviation, and the control group event rate (CER); the axis and distribution labels; and the slider's maximum and step size. Settings can be saved in your browser's localStorage so they persist across visits.

You can pan the x-axis by clicking and dragging the visualization. Double-click the visualization to recenter and rescale it.

What are the formulas?

Cohen's d is the standardized mean difference, where $\delta$ is the population Cohen's d, equal population variances ( $\sigma$ ) are assumed, and $\mu_i$ is each population mean:

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

Writing $\Phi$ for the standard-normal CDF and $\Psi$ for its inverse, Cohen's U₃ (Cohen, 1977), the overlapping coefficient (Reiser & Faraggi, 1999), the probability of superiority / common-language effect size (Ruscio, 2008), and the number needed to treat (Furukawa & Leucht, 2011) are:

U_3 = \Phi(\delta)

\text{OVL} = 2\,\Phi(-|\delta|/2)

\text{CL} = \Phi\!\left(\frac{\delta}{\sqrt{2}}\right)

\text{NNT} = \frac{1}{\Phi(\delta + \Psi(\text{CER})) - \text{CER}}

CER is the control group's event rate (20% by default; change it in the settings). The overlap statistic here intentionally differs from Cohen's original calculation — see the question below.

R code to calculate NNT from Cohen's d:

CER <- 0.2
d <- 0.2
1 / (pnorm(d + qnorm(CER)) - CER)

References

Cohen, J. (1977). Statistical power analysis for the behavioral sciences. Routledge.
Furukawa, T. A., & Leucht, S. (2011). How to obtain NNT from Cohen's d: comparison of two methods. PLoS ONE, 6(4), e19070.
Reiser, B., & Faraggi, D. (1999). Confidence intervals for the overlapping coefficient: the normal equal variance case. Journal of the Royal Statistical Society, 48(3), 413–418.
Ruscio, J. (2008). A probability-based measure of effect size: robustness to base rates and other factors. Psychological Methods, 13(1), 19–30.

How do I cite this page?

Cite this page according to your favorite style guide.

The current version is 2.7.0 (changelog).

APA 7

Magnusson, K. (2026). Interpreting Cohen's d effect size: An interactive visualization (Version 2.7.0) [Web App]. R Psychologist. https://rpsychologist.com/cohend/

BibTeX

@software{magnussonCohend,
  author = {Kristoffer Magnusson},
  title = {Interpreting Cohen's d effect size: An interactive visualization},
  url = {https://rpsychologist.com/cohend/},
  version = {2.7.0},
  date = {2026-10-24},
}

I found a bug/error/typo or want to make a suggestion!

Please open an issue on GitHub, or reach out on social media.

The overlap statistic differs from Cohen's calculations

This is intentional; you can read more about my reasons in this blog post: Where Cohen went wrong – the proportion of overlap between two normal distributions.

Can I include this visualization in my book/article/etc?

Yes, go ahead! I did not invent plotting two overlapping Gaussian distributions. This visualization is dedicated to the public domain (CC0), so you can copy, modify, and distribute it, even commercially, without asking permission. Attribution is not required but always appreciated. The source code is licensed MIT, and the text on this page is CC-BY 4.0.

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