Statistical significance means that a result may not be the cause of random variations within the data. But not every significant result refers to an effect with a high impact, resp. it may even describe a phenomenon that is not really perceivable in everyday life. Statistical significance mainly depends on the sample size, the quality of the data and the power of the statistical procedures. If large data sets are at hand, as it is often the case f. e. in epidemiological studies or in large scale assessments, very small effects may reach statistical significance. In order to describe, if effects have a relevant magnitude, effect sizes are used to describe the strength of a phenomenon. The most popular effect size measure surely is Cohen's d (Cohen, 1988).
Here you will find a number of online calculators for the computation of different effect sizes and an interpretation table at the bottom of this page:
If the two groups have the same n, then the effect size is simply calculated by subtracting the means and dividing the result by the pooled standard deviation. The resulting effect size is called d_{Cohen} and it represents the difference between the groups in terms of their common standard deviation. It is used f. e. for calculating the effect for pre-post comparisons in single groups.
In case of relevant differences in the standard deviations, Glass suggests not to use the pooled standard deviation but the standard deviation of the control group. He argues that the standard deviation of the control group should not be influenced, at least in case of non-treatment control groups. This effect size measure is called Glass' Δ ("Glass' Delta"). Please type the data of the control group in column 1 for the correct calculation of Glass' Δ.
Group 1 | Group 2 | |
Mean | ||
Standard Deviation | ||
Effect Size d_{Cohen} | ||
Effect Size Glass' Δ |
Analogously, the effect size can be computed for groups with different sample size, by adjusting the calculation of the pooled standard deviation with weights for the sample sizes. This approach is overall identical with d_{Cohen} with a correction of a positive bias in the pooled standard deviation. In the literature, usually this computation is called Cohen's d as well. Please have a look at the remarks bellow the table.
Group 1 | Group 2 | |
Mean | ||
Standard Deviation | ||
Sample Size (N) | ||
Effect Size d_{Cohen}, g_{Hedges} ^{*} |
Intervention studies usually compare at least an intervention and a control group, as well as a measurement prior and post to the intervention. The following calculator follows the suggestions of Klauer (2001), who controlled for different sample sizes and pre test differences. The downside to this approach: The pre-post-tests are not treated as repeated measures but as independent data. For dependent tests, you can use calculator 4 or transform eta square from repeated measures in order to account for dependences between measurement points.
Group 1 | Group 2 | |||
Pre | Post | Pre | Post | |
Mean | ||||
Standard Deviation | ||||
Sample Size (N) | ||||
Effect Size d_{corr} ^{*} |
Effect sizes can be obtained by using the tests statistics from hypothesis tests, like Student t tests, as well. In case of independent samples, the result is essentially the same as in effect size calculation #2.
Dependent testing usually yields a higher power, because the interconnection between data points of different measurements are kept. This may be relevant f. e. when testing the same persons repeatedly, or when analyzing test results from matched persons or twins. Accordingly, more information may be used when computing effect sizes.
Please choose the mode of testing (dependent vs. independent) and specify the t statistic. In case of an dependent t test, please type in the number of cases and the correlation between the two variables. In case of independent samples, please specify the number of cases in each group. The calculation is based on the formulas reported by Borenstein (2009, pp. 228).
Mode of testing | |
Student t Value | |
n_{1} | |
n_{2} | |
r | |
Effektgröße d |
^{*} Wie used the formula t_{c} described in Dunlop, Cortina, Vaslow & Burke (1996, S. 171) in order to calculate d from dependend t-tests. Simulations proved it to have the least distortion in estimating d. We would like to thank Frank Aufhammer for pointing us to this publication. In case, the correlation is unknown, please fill in 0. The results will be a conservative estimation in this case, because standard errors will not be controlled then.
A very easy to interpret effect size from analyses of variance (ANOVAs) is η^{2} that reflects the explained proportion variance of the total variance. This proportion may be transformed directly into d. If η^{2} is not available, the F value of the ANOVA can be used as well, as long as the sample size is known. The following computation only works for ANOVAs with two distinct groups (df1 = 1; Thalheimer & Cook, 2002):
F-Value | |
Sample Size of the Treatment Group | |
Sample Size of the Controll Group | |
Effect Size d |
In case, the groups means are known from ANOVAs with multiple groups, it is possible to compute the effect sizes f and d (Cohen, 1988, S. 273 ff.). Prior to computing the effect size, you have to determine the minimum and maximum mean and to calculate the deviation of means manually (a. compute the differences between the single means, b. square the differences and sum them up, c. divide the sum by the number of means, d. draw the square root).
Additionally, you have to decide, which scenario fits the data best:
Highest Mean (m_{max}) | |
Lowest Mean (m_{min}) | |
Deviation of Means | |
Number of Groups | |
Distribution of Means | |
Effect Size f | |
Effect Size d |
Measures of effect size like d or correlations can be hard to communicate, e. g. to patients. If you use r^{2} f. e., effects seem to be really small and when a person does not know or understand the interpretation guidelines, even effective interventions could be seen as futile. And even small effects can be very important, as Hattie (2007) underlines:
Rosenthal and Rubin (1982) suggest another way of looking on the effects of treatements by considering the increase of success through interventions. The approach is suitable for 2x2 contingency tables with the different treatment groups in the rows and the number of cases in the columns. The BESD is computed by subtracting the probability of success from the intervention an the controll group. The resulting percentage can be transformed into d_{Cohen}.
Another measure, that is widely used in evidence based medicine, is the so called Number Needed to Treat. It shows, how many people are needed in the treatment group in order to obtain at least one additional favourable outcome. In case of a negative value, it is called Number Needed to Harm.
Please fill in the number of cases with a fortunate and unfortunate outcome in the different cells:
Success | Failure | Probability of Success | |
Intervention group | |||
Control Group | |||
Binomial Effect Size Display (BESD) (Increase of Intervention Success) |
|||
Number Needed to Treat | |||
r_{Phi} | |||
Effect Size d_{cohen} | |||
A conversion between NNT and other effect size measures liken Cohen's d is not easily possible. Concerning the example above, the transformation is done via the point-biserial correlation r_{phi} which is nothing but an estimation. Alternative approaches (comp. Furukawa & Leucht, 2011) allow to convert between d and NNT. Within a certain interval (f. e. 1.0 ≤ d ≤ 1.0) the results of the conversion overall correspond to the calculation based on the raw data above:
Cohen's d | Number Needed to Treat (NNT) |
Studies, investigating if specific incidences occur (e. g. death, healing, academic success ...) on a binary basis (yes versus no), and if two groups differ in respect to these incidences, usually Odds Ratios, Risk Ratios and Risk Differences are used to quantify the differences between the groups (Borenstein et al. 2009, chap. 5). These forms of effect size are therefore commonly used in clinical research and in epidemiological studies:
Incidence | no Incidence | N | |
Teatment | |||
Controll | |||
| |||
Risk Ratio | Odds Ratio | Risk Difference | |
Result | |||
Log | |||
Estimated Variance V | |||
Estimated Standard Error SE | |||
Yule's Q |
Cohen (1988, S. 109) suggests an effect size measure with the denomination q that permits to interpret the difference between two correlations. The two correlations are transformed with Fisher's Z and subtracted afterwards. Cohen proposes the following categories for the interpretation: <.1: no effect; .1 to .3: small effect; .3 to .5: intermediate effect; >.5: large effect.
Correlation r_{1} | |
Correlation r_{2} | |
Cohen's q | |
Interpretation |
Especially in metaanalytic research, it is often necessery to average correlations or to perform significance tests on the difference between correlations. Please have a look at our page Testing the Significance of Correlations for online calculators on these subjects.
In order to compute Conhen's d, it is necessary to determine the mean (pooled) standard deviation. Here, you will find a small tool that does this for you. Different sample sizes are corrected as well:
Group 1 | Group 2 | |
Standard Deviation | ||
Sample size (N) | ||
Pooled Standard Deviation s_{pool} |
Please choose the effect size, you want to transform, in the drop-down menu. Specify the magnitude of the effect size afterwards. The transformation is done according to Cohen (1988), Rosenthal (1994, S. 239) and Borenstein, Hedges, Higgins, und Rothstein (2009; transformation of d in Odds Ratios).
Effektstärke | ||
d | ||
r | ||
η^{2} | ||
f | ||
Odds Ratio | ||
Number Needed to Treat (NNT) |
The χ^{2} and z test statistics from hypothesis tests can be used to compute d and r(Rosenthal & DiMatteo, 2001, p. 71; comp. Elis, 2010, S. 28). The calculation is however only correct for χ^{2} tests with one degree of freedom. Please choose the tests static measure from the drop-down menu und specify the value and N. The transformation from d to r and η^{2} is based on the formulas used in the prior section.
Test Statistic | ||
N | ||
d | ||
r | ||
η^{2} |
Here, you can see the suggestions of Cohen (1988) and Hattie (2009 S. 97) for interpreting the magnitude of effect sizes. Hattie refers to real educational contexts and therefore uses a more benignant classification, compared to Cohen. We slightly adjusted the intervals, in case, the interpretation did not exactly match the categories of the original authors.
d | r^{*} | η^{2} | Interpretation sensu Cohen (1988) | Interpretation sensu Hattie (2007) |
< 0 | < 0 | - | Adverse Effect | |
0.0 | .00 | .000 | No Effect | Developmental effects |
0.1 | .05 | .003 | ||
0.2 | .10 | .010 | Small Effect | Teacher effects |
0.3 | .15 | .022 | ||
0.4 | .2 | .039 | Zone of desired effects | |
0.5 | .24 | .060 | Intermediate Effect | |
0.6 | .29 | .083 | ||
0.7 | .33 | .110 | ||
0.8 | .37 | .140 | Large Effect | |
0.9 | .41 | .168 | ||
≥ 1.0 | .45 | .200 |
Borenstein (2009). Effect sizes for continuous data. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta analysis (pp. 221-237). New York: Russell Sage Foundation.
Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to Meta-Analysis, Chapter 7: Converting Among Effect Sizes . Chichester, West Sussex, UK: Wiley.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2. Auflage). Hillsdale, NJ: Erlbaum.
Dunlap, W. P., Cortina, J. M., Vaslow, J. B., & Burke, M. J. (1996). Meta-analysis of experiments with matched groups or repeated measures designs. Psychological Methods, 1, 170-177.
Elis, P. (2010). The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results. Cambridge: Cambridge University Press.
Furukawa, T. A., & Leucht, S. (2011). How to obtain NNT from Cohen's d: comparison of two methods. PloS one, 6, e19070.
Hattie, J. (2009). Visible Learning. London: Routledge.
Klauer, K. J. (2001). Handbuch kognitives Training. Göttingen: Hogrefe.
Rosenthal, R. (1994). Parametric measures of effect size. In H. Cooper & L. V. Hedges (Eds.), The Handbook of Research Synthesis (231-244). New York, NY: Sage.
Rosenthal, R. & DiMatteo, M. R. (2001). Meta-Analysis: Recent Developments in Quantitative Methods for Literature Reviews. Annual Review of Psychology, 52(1), 59-82. doi:10.1146/annurev.psych.52.1.59
Thalheimer, W., & Cook, S. (2002, August). How to calculate effect sizes from published research articles: A simplified methodology. Retrieved March 9, 2014 from http://work-learning.com/effect_sizes.htm.