Interpreting Cohen's d Effect Size

An Interactive Visualization

Created by Kristoffer Magnusson

Share

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

Loading visualization

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 by pressing the settings symbol to the right of the slider. Go to the formula section for more information.

Written by Kristoffer Magnusson, a researcher in clinical psychology. You should follow him on Twitter and come hang out on the open science discord Git Gud Science.

FAQ

Change Cohen’s d

Use the slider to change Cohen’s d, or open the settings drawer and change the parameters. The inputs can also be controlled using the keyboard arrows.

Settings

You can change the following settings by clicking on the settings icon to the right of the slider.

  • Parameters
    • Mean 1
    • Mean 2
    • SD
    • Control group event rate (CER)
  • Labels
    • X axis
    • Distribution 1
    • Distribution 2
  • Slider settings
    • Slider Max
    • Slider Step: Controls the step size of the slider

Save settings

The settings can be saved in your browser’s localStorage and will thus persist across visits.

Pan and rescale

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

Offline use

This site is cached using a service worker and will work even when you are offline.

Cohen’s d

Cohen’s d is simply the standardized mean difference,

,

where is the population parameter of Cohen’s d. Where it is assumed that , i.e., homogeneous population variances. And is the mean of the respective population.

Cohen’s U3

Cohen (1977) defined U3 as a measure of non-overlap, where “we take the percentage of the A population which the upper half of the cases of the Β population exceeds”. Cohen’s d can be converted to Cohen’s U3 using the following formula

where is the cumulative distribution function of the standard normal distribution, and the population Cohen’s d.

Overlap

Generally called the overlapping coefficient (OVL). Cohen’s d can be converted to OVL using the following formula (Reiser and Faraggi, 1999)

where is the cumulative distribution function of the standard normal distribution, and the population Cohen’s d.

Probability of superiority

This is effect size with many names: common language effect size (CL), Area under the receiver operating characteristics (AUC) or just A for its non-parametric version (Ruscio & Mullen, 2012). It is meant to be more intuitive for persons without any training in statistics. The effect size gives the probability 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. Cohen’s d can be converted CL using the following formula (Ruscio, 2008)

where is the cumulative distribution function of the standard normal distribution, and the population Cohen’s d.

Number Needed to Treat

NNT is the number of patients we would need to treat with the intervention to achieve one more favorable outcome compared to the control group. Furukawa and Leucht (2011) gives the following formula for converting Cohen’s d into NNT

where is the cumulative distribution function of the standard normal distribution and its inverse, CER is the control group’s event rate and the population Cohen’s d. N.B. CER is set to 20 % in the visualization above. You can change this be pressing the settings symbol to the right of the slider. The definition of an “event” or a “response” is arbitrary and could be defined as the proportion of patients who are in remission, e.g. bellow some cut-off on a standardized questionnaire. It is possible to convert Cohen’s d into a version of NNT that is invariant to the event rate of the control group. The interested reader should look at Furukawa and Leucht (2011) where a convincing argument is given to why this complicates the interpretation of NNT.

R code to calculate NNT from Cohen’s d

Since many have asked about R code for the formula above, here it is

References

  • Baguley, T. (2009). Standardized or simple effect size: what should be reported? British journal of psychology, 100(Pt 3), 603–17.
  • Cohen, J. (1977). Statistical power analysis for the behavioral sciencies. Routledge.
  • Furukawa, T. A., & Leucht, S. (2011). How to obtain NNT from Cohen’s d: comparison of two methods. PloS one, 6(4).
  • 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.
  • Ruscio, J., & Mullen, T. (2012). Confidence Intervals for the Probability of Superiority Effect Size Measure and the Area Under a Receiver Operating Characteristic Curve. Multivariate Behavioral Research, 47(2), 201–223.

Cite this page according to your favorite style guide. The references below are automatically generated and contain the correct information.

APA 7

Magnusson, K. (2021). Interpreting Cohen's d effect size: An interactive visualization (Version 2.5.1) [Web App]. R Psychologist. https://rpsychologist.com/cohend/

BibTex

Please report errors or suggestions by opening an issue on GitHub, if you want to ask a question use GitHub discussions

No, it will be fine. The app runs in your browser so the server only needs to serve the files.

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

Yes, go ahead! I did not invent plotting two overlapping Gaussian distributions. This visualization is dedicated to the public domain, which means “you can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission” (see Creative common’s CC0-license). Although, attribution is not required it is always appreciated!

The source code for this page is licensed using MIT, and the text on the page is CC-BY 4.0.

Contribute/Donate

There are many ways to contribute to free and open software. If you like my work and want to support it you can:

A huge thanks to the 100 supporters who've bought me a 225 coffees!

Jason Rinaldo bought ☕☕☕☕☕☕☕☕☕☕ (10) coffees

I've been looking for applets that show this for YEARS, for demonstrations for classes. Thank you so much! Students do not need to tolarate my whiteboard scrawl now. I'm sure they'd appreciate you, too.l

@LinneaGandhi bought ☕☕☕☕☕ (5) coffees

This is awesome! Thank you for creating these. Definitely using for my students, and me! :-)

@ICH8412 bought ☕☕☕☕☕ (5) coffees

very useful for my students I guess

@KelvinEJones bought ☕☕☕☕☕ (5) coffees

Preparing my Master's student for final oral exam and stumbled on your site. We are discussing in lab meeting today. Coffee for everyone.

Someone bought ☕☕☕☕☕ (5) coffees

What a great site

@Daniel_Brad4d bought ☕☕☕☕☕ (5) coffees

Wonderful work!

David Loschelder bought ☕☕☕☕☕ (5) coffees

Terrific work. So very helpful. Thank you very much.

@neilmeigh bought ☕☕☕☕☕ (5) coffees

I am so grateful for your page and can't thank you enough!  

@giladfeldman bought ☕☕☕☕☕ (5) coffees

Wonderful work, I use it every semester and it really helps the students (and me) understand things better. Keep going strong.

Dean Norris bought ☕☕☕☕☕ (5) coffees

Sal bought ☕☕☕☕☕ (5) coffees

Really super useful, especially for teaching. Thanks for this!

dde@paxis.org bought ☕☕☕☕☕ (5) coffees

Very helpful to helping teach teachers about the effects of the Good Behavior Game

@akreutzer82 bought ☕☕☕☕☕ (5) coffees

Amazing visualizations! Thank you!

@rdh_CLE bought ☕☕☕☕☕ (5) coffees

So good!

Someone bought ☕☕☕ (3) coffees

@PhysioSven bought ☕☕☕ (3) coffees

Amazing illustrations, there is not enough coffee in the world for enthusiasts like you! Thanks!

Cheryl@CurtinUniAus bought ☕☕☕ (3) coffees

🌟What a great contribution - thanks Kristoffer!

vanessa moran bought ☕☕☕ (3) coffees

Wow - your website is fantastic, thank you for making it.

Someone bought ☕☕☕ (3) coffees

mikhail.saltychev@gmail.com bought ☕☕☕ (3) coffees

Thank you Kristoffer This is a nice site, which I have been used for a while. Best Prof. Mikhail Saltychev (Turku University, Finland)

Someone bought ☕☕☕ (3) coffees

Ruslan Klymentiev bought ☕☕☕ (3) coffees

@lkizbok bought ☕☕☕ (3) coffees

Keep up the nice work, thank you!

@TELLlab bought ☕☕☕ (3) coffees

Thanks - this will help me to teach tomorrow!

SCCT/Psychology bought ☕☕☕ (3) coffees

Keep the visualizations coming!

@elena_bolt bought ☕☕☕ (3) coffees

Thank you so much for your work, Kristoffer. I use your visualizations to explain concepts to my tutoring students and they are a huge help.

A random user bought ☕☕☕ (3) coffees

Thank you for making such useful and pretty tools. It not only helped me understand more about power, effect size, etc, but also made my quanti-method class more engaging and interesting. Thank you and wish you a great 2021!

@hertzpodcast bought ☕☕☕ (3) coffees

We've mentioned your work a few times on our podcast and we recently sent a poster to a listener as prize so we wanted to buy you a few coffees. Thanks for the great work that you do!Dan Quintana and James Heathers - Co-hosts of Everything Hertz 

Cameron Proctor bought ☕☕☕ (3) coffees

Used your vizualization in class today. Thanks!

eshulman@brocku.ca bought ☕☕☕ (3) coffees

My students love these visualizations and so do I! Thanks for helping me make stats more intuitive.

Someone bought ☕☕☕ (3) coffees

Adrian Helgå Vestøl bought ☕☕☕ (3) coffees

@misteryosupjoo bought ☕☕☕ (3) coffees

For a high school teacher of psychology, I would be lost without your visualizations. The ability to interact and manipulate allows students to get it in a very sticky manner. Thank you!!!

Chi bought ☕☕☕ (3) coffees

You Cohen's d post really helped me explaining the interpretation to people who don't know stats! Thank you!

Someone bought ☕☕☕ (3) coffees

You doing useful work !! thanks !!

@ArtisanalANN bought ☕☕☕ (3) coffees

Enjoy.

@jsholtes bought ☕☕☕ (3) coffees

Teaching stats to civil engineer undergrads (first time teaching for me, first time for most of them too) and grasping for some good explanations of hypothesis testing, power, and CI's. Love these interactive graphics!

@notawful bought ☕☕☕ (3) coffees

Thank you for using your stats and programming gifts in such a useful, generous manner. -Jess

Mateu Servera bought ☕☕☕ (3) coffees

A job that must have cost far more coffees than we can afford you ;-). Thank you.

@cdrawn bought ☕☕☕ (3) coffees

Thank you! Such a great resource for teaching these concepts, especially CI, Power, correlation.

Julia bought ☕☕☕ (3) coffees

Fantastic work with the visualizations!

@felixthoemmes bought ☕☕☕ (3) coffees

@dalejbarr bought ☕☕☕ (3) coffees

Your work is amazing! I use your visualizations often in my teaching. Thank you. 

@PsychoMouse bought ☕☕☕ (3) coffees

Excellent!  Well done!  SOOOO Useful!😊 🐭 

Dan Sanes bought ☕☕ (2) coffees

this is a superb, intuitive teaching tool!

@whlevine bought ☕☕ (2) coffees

Thank you so much for these amazing visualizations. They're a great teaching tool and the allow me to show students things that it would take me weeks or months to program myself.

Someone bought ☕☕ (2) coffees

@notawful bought ☕☕ (2) coffees

Thank you for sharing your visualization skills with the rest of us! I use them frequently when teaching intro stats. 

Someone bought ☕ (1) coffee

Michael Hansen bought ☕ (1) coffee

ALEXANDER VIETHEER bought ☕ (1) coffee

mather bought ☕ (1) coffee

Someone bought ☕ (1) coffee

Bastian Jaeger bought ☕ (1) coffee

Thanks for making the poster designs OA, I just hung two in my office and they look great!

@ValerioVillani bought ☕ (1) coffee

Thanks for your work.

Someone bought ☕ (1) coffee

Great work!

@YashvinSeetahul bought ☕ (1) coffee

Someone bought ☕ (1) coffee

Angela bought ☕ (1) coffee

Thank you for building such excellent ways to convey difficult topics to students!

@inthelabagain bought ☕ (1) coffee

Really wonderful visuals, and such a fantastic and effective teaching tool. So many thanks!

Someone bought ☕ (1) coffee

Someone bought ☕ (1) coffee

Yashashree Panda bought ☕ (1) coffee

I really like your work.

Ben bought ☕ (1) coffee

You're awesome. I have students in my intro stats class say, "I get it now," after using your tool. Thanks for making my job easier.

Gabriel Recchia bought ☕ (1) coffee

Incredibly useful tool!

Shiseida Sade Kelly Aponte bought ☕ (1) coffee

Thanks for the assistance for RSCH 8210.

@Benedikt_Hell bought ☕ (1) coffee

Great tools! Thank you very much!

Amalia Alvarez bought ☕ (1) coffee

@noelnguyen16 bought ☕ (1) coffee

Hi Kristoffer, many thanks for making all this great stuff available to the community!

Eran Barzilai bought ☕ (1) coffee

These visualizations are awesome! thank you for creating it

Someone bought ☕ (1) coffee

Chris SG bought ☕ (1) coffee

Very nice.

Gray Church bought ☕ (1) coffee

Thank you for the visualizations. They are fun and informative.

Qamar bought ☕ (1) coffee

Tanya McGhee bought ☕ (1) coffee

@schultemi bought ☕ (1) coffee

Neilo bought ☕ (1) coffee

Really helpful visualisations, thanks!

Someone bought ☕ (1) coffee

This is amazing stuff. Very slick. 

Someone bought ☕ (1) coffee

Sarko bought ☕ (1) coffee

Thanks so much for creating this! Really helpful for being able to explain effect size to a clinician I'm doing an analysis for. 

@DominikaSlus bought ☕ (1) coffee

Thank you! This page is super useful. I'll spread the word. 

Someone bought ☕ (1) coffee

Melinda Rice bought ☕ (1) coffee

Thank you so much for creating these tools! As we face the challenge of teaching statistical concepts online, this is an invaluable resource.

@tmoldwin bought ☕ (1) coffee

Fantastic resource. I think you would be well served to have one page indexing all your visualizations, that would make it more accessible for sharing as a common resource.

Someone bought ☕ (1) coffee

Fantastic Visualizations! Amazing way to to demonstrate how n/power/beta/alpha/effect size are all interrelated - especially for visual learners! Thank you for creating this?

@jackferd bought ☕ (1) coffee

Incredible visualizations and the best power analysis software on R.

Cameron Proctor bought ☕ (1) coffee

Great website!

Someone bought ☕ (1) coffee

Hanah Chapman bought ☕ (1) coffee

Thank you for this work!!

Someone bought ☕ (1) coffee

Jayme bought ☕ (1) coffee

Nice explanation and visual guide of Cohen's d

Bart Comly Boyce bought ☕ (1) coffee

thank you

Dr. Mitchell Earleywine bought ☕ (1) coffee

This site is superb!

Florent bought ☕ (1) coffee

Zampeta bought ☕ (1) coffee

thank you for sharing your work. 

Mila bought ☕ (1) coffee

Thank you for the website, made me smile AND smarter :O enjoy your coffee! :)

Deb bought ☕ (1) coffee

Struggling with statistics and your interactive diagram made me smile to see that someone cares enough about us strugglers to make a visual to help us out!😍 

Someone bought ☕ (1) coffee

@exerpsysing bought ☕ (1) coffee

Much thanks! Visualizations are key to my learning style! 

Someone bought ☕ (1) coffee

Sponsors

You can sponsor my open source work using GitHub Sponsors and have your name shown here.

Backers ✨❤️

Pull requests are also welcome, or you can contribute by suggesting new features, add useful references, or help fix typos. Just open a issues on GitHub.

Webmentions

João Peça
João Peça 2021-08-27
What a wonderful way of visualizing Cohen's d! Also, check out the other amazing tools for power analysis. Great work by @krstoffr! #statistics #neuroscience #greatfortheclassroom rpsychologist.com/cohend/
Sarah Morris
Sarah Morris 2021-07-18
Just saw a link to this from another thread. What a great way to build intuition around effect sizes rpsychologist.com/cohend/
Prof Courtenay Norbury
here is what d=.26 looks like - improvement, but not ameliorated (from excellent website: rpsychologist.com/cohend/)
Brian Pulling
Brian Pulling 2021-04-12
awesome visualisation for interpreting cohen's d (also demonstrates why it can be so hard to interpret effect sizes!) #rstats rpsychologist.com/cohend/
MRCPsych Twit Revision
Again, fantastic resource to help visualise Cohen’s d further. Interpreting Cohen's d - an interactive visualization: rpsychologist.com/cohend/ #MRCPsychTR #MRCPsychTRPaperB #MCRPsychPaperB
R posts you might have missed!
✍️ Interpreting Cohen's d | R Psychologist 👤 Kristoffer Magnusson @krstoffr 🔗 rpsychologist.com/cohend/ #rstats #datascience
Mara Averick
Mara Averick 2021-02-17
Wooot!! I've loved this for so long…
günlük
günlük 2021-02-17
İlginçmiş. rpsychologist.com/likelihood/
Kristoffer Magnusson
Absolutely! The translation process is documented here, DM me if you need any help! github.com/rpsychologist/…
Kristoffer Magnusson
Great! You can find instructions here, DM me if you need any help! github.com/rpsychologist/…
Daniel Alcalá López
Count me in for a Spanish translation!
Arthur Sá Ferreira 😷
If you want Portuguese-Brazil translation I am interest 🇧🇷

(Webmentions sent before 2021 will unfortunately not show up here.)

More Visualizations

Understanding p-values Through Simulations

An interactive simulation to help explain p-values

Maximum Likelihood

An interactive post covering various aspects of maximum likelihood estimation.

Cohen's d

An interactive app to visualize and understand standardized effect sizes.

Statistical Power and Significance Testing

An interactive version of the traditional Type I and II error illustration.

Confidence Intervals

An interactive simulation of confidence intervals

Bayesian Inference

An interactive illustration of prior, likelihood, and posterior.

Correlations

Interactive scatterplot that let's you visualize correlations of various magnitudes.

Equivalence and Non-Inferiority Testing

Explore how superiority, non-inferiority, and equivalence testing relates to a confidence interval

P-value distribution

Explore the expected distribution of p-values under varying alternative hypothesises.

t-distribution

Interactively compare the t- and normal distribution.