How to Use the Effect Size Calculator
This calculator allows you to calculate effect sizes for common hypothesis tests using summary statistics. It supports t-tests, z-tests, ANOVA, chi-square tests, correlation, regression, and several nonparametric tests.
To use the calculator:
- Select the statistical test from the dropdown menu.
- Enter the required summary statistics
- Click Calculate.
The calculator will instantly return the appropriate effect size, its magnitude classification where applicable, and an interpretation of the result. Below the answer, you will find a step-by-step solution showing the formula, the meaning of each symbol, and how the values were substituted to obtain the answer.
Tip. If you have already performed your statistical analysis using SPSS, R, Excel, or another statistical software package, you can enter the relevant summary statistics directly into the calculator. You do not need to upload or enter your original dataset.
What Is Effect Size in Statistics?
Effect size is a statistical measure that describes the magnitude of a difference, relationship, or association. It goes beyond telling us whether the result is statistically significant by allowing us to understand how large an observed effect is.
For example, suppose a researcher compares the mean examination scores of students taught using two different teaching methods. After performing an independent-samples t-test, the researcher obtains a statistically significant result.
Although this tells us that the observed difference is unlikely under the null hypothesis, it does not tell us how large the difference is. To determine the magnitude of the difference, the researcher can calculate an effect size such as Cohen’s d.
If Cohen’s d = 0.5, the two group means differ by half a pooled standard deviation. Under conventional guidelines, this represents a medium effect. However, Cohen’s d is not appropriate for every statistical analysis. The effect size measure you should use depends on the type of hypothesis test and the research design.
Which Effect Size Should You Use?
Different hypothesis tests use different effect size measures. For example, t-tests commonly use Cohen’s d, while ANOVA uses measures such as eta squared and omega squared. Similarly, Cramér’s V is commonly used for chi-square tests of independence.
The table below summarizes the effect size measures supported by this calculator for different hypothesis tests.
| Statistical Test | Common Effect Size Measures |
|---|---|
| One-sample t-test | Cohen’s d |
| One-sample z-test (mean) | Cohen’s d using the known population SD |
| Independent-samples t-test | Cohen’s d, Hedges’ g, Glass’s delta |
| Welch’s t-test | RMS-standardized mean difference |
| Paired-samples t-test | Cohen’s d_z |
| Two-sample z-test (means) | Standardized mean difference |
| One- and two-proportion z-tests | Cohen’s h |
| One- and two-sample variance tests | Variance ratio, log variance ratio |
| One-way ANOVA | Eta squared (η²), omega squared (ω²), Cohen’s f |
| Two-way ANOVA | Eta squared and partial eta squared for each factor and interaction |
| Repeated-measures ANOVA | Partial eta squared, generalized eta squared |
| ANCOVA | Partial eta squared, partial Cohen’s f |
| Regression F-test | R², Cohen’s f² |
| Chi-square test of independence | Cramér’s V, phi for 2 × 2 tables |
| Chi-square goodness-of-fit test | Cohen’s w |
| Fisher’s exact test | Odds ratio |
| McNemar’s test | Cohen’s g, matched-pairs odds ratio |
| Pearson correlation test | Pearson’s r, r² |
| Mann–Whitney U test | Rank-biserial correlation |
| Wilcoxon signed-rank test | Matched rank-biserial correlation |
| Kruskal–Wallis test | Rank-based eta squared and epsilon squared |
| Friedman test | Kendall’s W |
As shown in the table, some statistical tests have more than one effect size measure. For example, an independent-samples t-test can be used to calculate Cohen’s d, Hedges’ g, and Glass’s delta. Although these measures describe differences between means, they use different standardization methods.
Similarly, a two-way ANOVA produces separate effect sizes for Factor A, Factor B, and their interaction. Therefore, you should report the effect size for each effect you are interested in, rather than treating the entire analysis as one effect.
Our effect size calculator automatically selects and calculates the appropriate measures based on the statistical test you choose.
Effect Size Formulas for Common Hypothesis Tests
There is no single formula for calculating effect size across all statistical tests. The formula depends on what is being measured and the summary statistics available.
Below are some of the most commonly used effect size formulas.
1. Cohen’s d Formula
Cohen’s d measures the standardized difference between means. It is the most common effect size metric for t-tests. The cohen’s d formulas for the most common t-tests are discussed below:
One Sample T-Test
Cohen’s d formula for a one-sample t-test is:
d = (x̄ − μ₀) / s
Where:
- d = Cohen’s d
- x̄ = Sample mean
- μ₀ = Hypothesized population mean
- s = Sample standard deviation
Independent Samples T-Test
Cohen’s d formula for an independent-samples t-test, assuming equal variances, is:
d = (x̄₁ − x̄₂) / sₚ
Where sₚ is the pooled standard deviation, which you can quickly compute using our pooled standard deviation calculator or the pooled variance calculator.
Paired T-Test
For paired-samples data, Cohen’s d formula is:
d_z = d̄ / sd
Where d̄ is the mean of the paired differences, and sd is their standard deviation.
2. Eta Squared Formula
Eta squared (η²) is commonly used to measure effect size in ANOVA. It represents the proportion of total variation in the dependent variable associated with the effect being studied.
For a one-way ANOVA, the effect size formula is:
Eta squared, η² = SS_between / SS_total
Where:
- SS_between = Between-groups sum of squares
- SS_total = Total sum of squares
However, for a two-way ANOVA, repeated-measures ANOVA, and ANCOVA, we use the partial eta squared.
The formula is: Partical eta squared, η²ₚ = SS_effect / (SS_effect + SS_error)
Already know the F statistic and its degrees of freedom? You can calculate the partial eta squared using the following alternative formula
Partial Eta Squared, η²ₚ = (F × df_effect) / (F × df_effect + df_error)
Note that eta squared and partial eta squared are not always equal. Partial eta squared measures an effect relative to that effect and its associated error variation, while ordinary eta squared uses the corrected total variation.
3. Cramér’s V Formula
Cramér’s V measures the strength of association between categorical variables in a chi-square test of independence.
The formula is:
V = √[χ² / (n × min(r − 1, c − 1))]
Where:
- χ² = Pearson chi-square statistic
- n = Total number of observations
- r = Number of rows in the contingency table
- c = Number of columns in the contingency table
Cramér’s V ranges from 0 to 1, where 0 indicates no association and larger values indicate stronger association. When interpreting this value, you should also consider the dimensions of the contingency table.
4. Other Common Effect Size Formulas
Other effect size measures supported by the calculator include:
- Cohen’s h: h = 2asin(√p₁) − 2asin(√p₂), used for comparing proportions.
- Cohen’s w: w = √(χ² / n), used for chi-square goodness-of-fit tests.
- Pearson’s r: r = t / √(t² + df), when the t statistic comes from a Pearson correlation test.
- Rank-biserial correlation: r_rb = 2U₁ / (n₁ × n₂) − 1, using the Mann–Whitney U statistic for Group 1.
- Kendall’s W: W = χ²_F / [N(k − 1)], used for the Friedman test.
When using our effect size calculator, you don’t have to master all the formulas. Simply select the test, enter the values and the calculator will apply the correct formula and show you the workings.
How to Calculate Effect Size by Hand
To calculate an effect size manually:
- Identify the hypothesis test
- Choose an appropriate effect size measure.
- Obtain the summary statistics required by the formula and substitute the values.
The following examples demonstrate how to calculate Cohen’s d and eta squared.
Example 1. Calculating Cohen’s d for an Independent-Samples t-Test
Suppose a researcher compares examination scores for two groups of students. The following summary statistics are obtained:
- Group 1: n₁ = 20, x̄₁ = 78, s₁ = 8
- Group 2: n₂ = 20, x̄₂ = 73, s₂ = 9
Calculate Cohen’s d.
Solution
Since the two groups are independent, we first need to calculate the pooled standard deviation.
Step 1: Calculate the pooled standard deviation.
By definition, the pooled standard deviation formula is:
sₚ = √[((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2)]
Substituting the values:
sₚ = √[((20 − 1) × 8² + (20 − 1) × 9²) / (20 + 20 − 2)]
= √[(19 × 64 + 19 × 81) / 38]
= √[(1216 + 1539) / 38]
sₚ = √(2755 / 38)
= √72.5
= 8.514693
Step 2: Calculate Cohen’s d.
The formula is:
d = (x̄₁ − x̄₂) / sₚ
Substituting the values:
d = (78 − 73) / 8.514693
= 5 / 8.514693
= 0.587220
Therefore, Cohen’s d = 0.587220.
Using conventional Cohen’s d benchmarks, this represents a medium effect. This means that the mean examination score for Group 1 is approximately 0.59 pooled standard deviations higher than that of Group 2.
Example 2. Calculating Eta Squared for a One-Way ANOVA
Suppose a researcher performs a one-way ANOVA to compare examination scores across three teaching methods. The ANOVA results give:
- Between-groups sum of squares (SS_between) = 45
- Within-groups sum of squares (SS_error) = 135
Calculate eta squared.
Solution
The eta squared formula is:
η² = SS_between / SS_total
Step 1: Calculate the total sum of squares.
SS_total = SS_between + SS_error
= 45 + 135
= 180
Step 2: Calculate eta squared.
η² = 45 / 180
= 0.25
Therefore, η² = 0.25.
This means that 25% of the observed variation in examination scores is associated with differences between the teaching methods. Based on conventional eta-squared benchmarks, this represents a large effect.
For a one-way ANOVA, our calculator can also calculate omega squared and Cohen’s f when the required summary statistics are available.
How to Interpret Effect Size
The interpretation of an effect size depends on the measure used. Some effect sizes describe differences in standard deviation units, while others indicate the strength of an association or the proportion of variation explained.
Researchers commonly use small, medium, and large benchmarks when describing effect sizes. However, these values are general guidelines and should not replace the research context.
The table below shows common reference values for selected effect size measures.
| Effect Size Measure | Small | Medium | Large |
|---|---|---|---|
| Cohen’s d, Hedges’ g, Cohen’s h | 0.20 | 0.50 | 0.80 |
| Pearson’s r | 0.10 | 0.30 | 0.50 |
| Eta squared (η²) | 0.01 | 0.06 | 0.14 |
| Cohen’s f | 0.10 | 0.25 | 0.40 |
| Cohen’s w | 0.10 | 0.30 | 0.50 |
Note. The values represent conventional benchmarks rather than strict cutoffs for practical importance. Also, for signed measures such as Cohen’s d and Pearson’s r, the magnitude is determined using the absolute value. A negative sign indicates direction, not a smaller effect.
Effect Size vs. Statistical Significance
Effect size and statistical significance are related, but they answer different questions.
Statistical significance helps us determine whether the observed data provide sufficient evidence against the null hypothesis at a chosen significance level. Effect size describes the magnitude of the observed difference or relationship.
For example, suppose a researcher obtains a p-value of 0.002 and Cohen’s d = 0.15 when comparing two groups.
Since the p-value is less than 0.05, the result is statistically significant at the 5% level. However, Cohen’s d = 0.15 is below the conventional small-effect benchmark.
This means that although the data provide evidence of a difference, the observed standardized difference between the groups is relatively small.
A small effect can be statistically significant when the sample is large enough. Conversely, a potentially important effect may fail to reach statistical significance when the data are too imprecise.
Therefore, reporting both the p-value and an appropriate effect size gives a more complete description of the statistical result. Where possible, an effect-size confidence interval also helps show the uncertainty surrounding the estimate.
Why Use This Effect Size Calculator?
This calculator is designed for students and researchers who have already performed a statistical test and want to calculate or verify the associated effect size.
Some of its main features include:
- Multiple hypothesis tests. Calculate effect sizes for t-tests, z-tests, ANOVA, chi-square tests, correlation, regression, and several nonparametric tests.
- Summary data input. Enter statistics such as group means, standard deviations, sample sizes, F statistics, degrees of freedom, or sums of squares without entering the original data.
- Multiple effect size measures. Where appropriate, the calculator returns more than one measure, such as Cohen’s d, Hedges’ g, and Glass’s delta for an independent-samples t-test.
- Step-by-step calculations. See the formula, definition of symbols, substitution of values, and final answer.
- Effect size interpretation. Get an interpretation of the calculated value, including a magnitude classification where suitable.
- Related statistical tests. Access the corresponding hypothesis-test calculator when available.
You can also copy the calculated results and working for use in your notes or to verify calculations in statistical software.
Frequently Asked Questions
An effect size calculator is an online tool used to measure the magnitude of a statistical difference, relationship, or association. Our calculator supports several common statistical tests and uses the summary statistics you provide to calculate the appropriate effect size. It also shows the formulas, calculations, and an interpretation of the results.
Yes. If you already have the required summary statistics, you can calculate an effect size without the original dataset.
There is no single effect size that is considered good for every study. The interpretation depends on the effect size measure, research area, and practical importance of the result.
No. Cohen’s d is one type of effect size, commonly used to measure standardized differences between means. Other effect sizes include eta squared for ANOVA, Cramér’s V for categorical associations, Pearson’s r for correlation, and Cohen’s h for differences between proportions.
Yes, some effect size measures can be greater than 1. For example, Cohen’s d can exceed 1 when the difference between means is larger than the standard deviation used to standardize it. Odds ratios and variance ratios can also exceed 1.
For a one-way ANOVA, eta squared and omega squared are commonly reported. However, for two-way ANOVA and repeated-measures ANOVA, partial eta squared is frequently used.
