Binomial Effect Size Display: A Tool for Interpreting Effect Sizes

The binomial effect size display (BESD) is a method for converting a correlation coefficient into a pair of success rates that show the practical impact of an effect. If a study reports an effect size of r = 0.30, the BESD translates that value into a 65% success rate for the treatment group and a 35% success rate for the control group. This article explains how the BESD works, how to calculate it from common effect size statistics, and where it fits within the broader practice of interpreting research findings. The intended readers are students, researchers, life-science professionals, and informed general readers who need to judge whether a statistically significant result matters in practical terms.

What the Binomial Effect Size Display Shows

The BESD answers a simple question: if a treatment or intervention works, how many more people succeed with it than without it? The display takes a correlation coefficient and presents it as two percentages that always sum to 100. The treatment group success rate equals 50 plus half the correlation expressed as a percentage, and the control group success rate equals 50 minus half the correlation expressed as a percentage. For a correlation of r = 0.20, the treatment success rate is 60% and the control success rate is 40%. The difference between those rates, 20 percentage points, is the success rate difference.

This presentation was introduced as a general purpose display of experimental effect magnitude in the Journal of Educational Psychology in 1982. The original article described the BESD as a simple way to show the magnitude of an experimental effect without requiring readers to interpret abstract statistical units. The method gained wider attention through its use in meta-analytic work and its inclusion in textbooks on research methods and effect size interpretation.

The BESD is not a new statistical test. It is a reexpression of an existing effect size, usually Pearson's r, in a format that non-specialists can grasp. A correlation of r = 0.30 does not tell most readers whether an intervention is worth adopting. A statement that 65 out of 100 treated individuals succeed compared with 35 out of 100 untreated individuals carries immediate practical meaning.

Why Effect Sizes Need Interpretation Tools

Statistical significance tells a researcher whether an observed effect is likely due to chance. It does not tell the researcher whether the effect is large enough to matter in practice. A study with a very large sample can produce a statistically significant result for an effect that is trivially small. Conversely, a study with a small sample may fail to reach significance for an effect that would be clinically or practically important.

Standardized effect sizes such as Cohen's d and Pearson's r provide a numerical index of magnitude, but their units are not intuitive. Cohen's d is expressed in standard deviation units, and r is a correlation coefficient that ranges from negative one to positive one. Neither unit translates directly into decisions about whether to adopt a treatment, change a policy, or invest in an intervention.

The BESD addresses this problem by converting the effect size into success rates. This conversion is especially useful when communicating findings to audiences that include non-researchers, such as clinicians, educators, policymakers, and the public. A 2021 article in Clinical Psychology in Europe noted that the standardized mean difference is difficult to explain to non-researchers and suggested the BESD and the number needed to treat as more accessible alternatives. The same article cautioned that small effect sizes may have considerable clinical meaning while large effect sizes may not, which means that interpretation tools must be used alongside domain knowledge instead of as replacements for it.

At a Glance: BESD Conversion Table

The table below shows how selected correlation values convert to BESD success rates. The treatment success rate is calculated as 50 plus 50 times the correlation, and the control success rate is 50 minus 50 times the correlation. The success rate difference is the difference between the two rates.

The table shows a consistent pattern: the success rate difference is always double the correlation expressed as a percentage. A correlation of 0.10 corresponds to a 10 percentage point difference, and a correlation of 0.50 corresponds to a 50 percentage point difference. This linear relationship makes the BESD easy to compute without specialized software.

Calculating the BESD from Different Effect Size Statistics

Researchers encounter effect sizes in several forms. The most common are Pearson's r, Cohen's d, and Hedges's g. Each can be converted to the BESD, but the conversion path differs.

Converting Pearson's r Directly

When the effect size is already expressed as Pearson's r, the BESD calculation is straightforward. The treatment success rate is 0.50 plus r divided by 2, and the control success rate is 0.50 minus r divided by 2. These values are then multiplied by 100 to express them as percentages. For example, with r = 0.18, the treatment success rate is 0.50 plus 0.09, which equals 0.59 or 59%. The control success rate is 0.50 minus 0.09, which equals 0.41 or 41%.

This exact calculation appeared in a 1990 meta-analysis of neuroleptic treatment in dementia published in the Journal of the American Geriatrics Society. The authors reported an effect size of r = 0.18 and used the BESD to show that treatment changed the improvement rate from 41% to 59%. They interpreted this as 18 out of 100 dementia patients benefiting from treatment beyond what placebo achieved.

Converting Cohen's d to r

When the effect size is reported as Cohen's d, the researcher must first convert d to r before applying the BESD formula. The conversion formula for two equal-sized groups is r equals d divided by the square root of d squared plus 4. For unequal group sizes, the formula requires adjustment. A 2000 article in Psychological Science reviewed expressions for calculating g and d from t statistics and for transforming between g or d and r, noting that the formulas must be adjusted when sample sizes are unequal.

The conversion from d to r is not a simple ratio. A Cohen's d of 0.20 converts to approximately r = 0.10, which yields a BESD success rate difference of 10 percentage points. A Cohen's d of 0.50 converts to approximately r = 0.24, which yields a success rate difference of about 24 percentage points. A Cohen's d of 0.80 converts to approximately r = 0.37, which yields a success rate difference of about 37 percentage points.

Using the BESD with Multiple Correlation Coefficients

The BESD was originally developed for zero-order correlation coefficients, which describe the relationship between two variables without adjustment for other factors. Researchers have explored extending the BESD to multiple correlation coefficients obtained from hierarchical regression analyses. A 2026 tutorial in Psychological Methods showed that BESD and gain-probability interpretations of hierarchical regression findings can imply different conclusions from each other and from the traditional change in R squared. The authors argued that multiple interpretations provide a more thorough understanding of the data than any single index.

This extension matters because many research questions involve predictors that are entered in steps. A researcher might first enter demographic variables, then add a treatment indicator, and then add interaction terms. The change in R squared at each step indicates how much additional variance the new variables explain. The BESD can be applied to the multiple correlation at each step to show the practical impact in success rate terms.

Practical Steps for Using the BESD

Applying the BESD to research findings requires a systematic approach. The steps below outline the process from initial data analysis to final interpretation.

Step 1: Identify the Effect Size

Determine which effect size statistic is appropriate for the research question and study design. For two-group comparisons, Cohen's d or Hedges's g are common choices. For correlational studies, Pearson's r is standard. The choice depends on the nature of the variables and the analytic approach used in the original study.

Step 2: Convert to Pearson's r

If the effect size is not already expressed as r, convert it using the appropriate formula. The conversion from d to r requires knowing whether the groups are equal in size. Unequal group sizes require adjusted formulas, as described in the Psychological Science article on contrasts and correlations in effect size estimation.

Step 3: Apply the BESD Formula

Calculate the treatment success rate as 0.50 plus r divided by 2. Calculate the control success rate as 0.50 minus r divided by 2. Express both as percentages. The difference between the two percentages is the success rate difference.

Step 4: Interpret in Context

Place the BESD values within the context of the research area. A success rate difference of 20 percentage points may be substantial for a condition with no effective treatments but trivial for a condition with several highly effective options. The interpretation requires knowledge of the specific field and the outcomes being measured.

Step 5: Report Both the Effect Size and the BESD

Report the original effect size alongside the BESD values. This practice allows readers who prefer traditional effect size metrics to use those values while giving other readers an accessible interpretation. The 2021 article on the standardized mean difference recommended reporting binary outcomes and the BESD alongside traditional effect sizes to improve communication with non-researchers.

Options and Tradeoffs in Effect Size Interpretation

The BESD is one of several methods for making effect sizes more interpretable. Each method has strengths and limitations, and the choice depends on the audience and the research context.

The BESD Compared with Cohen's Guidelines

Cohen's guidelines classify effect sizes as small, medium, or large based on conventional thresholds. For Pearson's r, the thresholds are 0.10, 0.30, and 0.50. For Cohen's d, the thresholds are 0.20, 0.50, and 0.80. These guidelines are widely taught but were not derived from empirical data. A 2019 study in gerontology found that Cohen's guidelines appear to overestimate effect sizes in that field and recommended lower thresholds of r = 0.10, 0.20, and 0.30 for individual differences research. A 2020 study in rehabilitation research similarly established empirically based guidelines from a large sample of meta-analyses, with small effects ranging from 0.08 to 0.15 in Cohen's d units.

The BESD does not rely on fixed thresholds. It presents the effect in absolute success rate terms, allowing readers to judge practical importance for themselves. This feature makes the BESD more flexible than threshold-based interpretation, but it also places more interpretive responsibility on the reader.

The BESD Compared with the Number Needed to Treat

The number needed to treat (NNT) is another method for expressing treatment effects. It represents the number of patients who must receive the treatment for one additional patient to benefit. The NNT is calculated as the reciprocal of the absolute risk reduction. For a treatment that improves success rates from 40% to 60%, the absolute risk reduction is 20 percentage points, and the NNT is 5.

The NNT and the BESD are closely related. Both express the treatment effect in terms of success rates. The 2021 Clinical Psychology in Europe article discussed both methods as alternatives to the standardized mean difference. The choice between them depends on the audience. The NNT is common in clinical medicine, while the BESD is more common in psychology and education research.

The BESD Compared with Percent of Nonoverlapping Data

In single case research, effect size interpretation has historically relied on the percent of nonoverlapping data (PND). A 2007 article in Behavior Therapy explored five alternative interpretations of Cohen's d and R squared for single case research, including the BESD. The authors evaluated each method on intuitive appeal, relevance to visual analysis, ease of calculation, and technical adequacy. They found that three of the five methods appeared to be improvements over prevailing practices, though the article did not identify the BESD as the single best option.

The BESD may be less familiar to single case researchers than PND, but it offers the advantage of being applicable across research designs. PND is specific to single case designs, while the BESD can be used with any study that reports a correlation or an effect size convertible to a correlation.

Limitations and Known Biases of the BESD

The BESD has important limitations that researchers must understand before using it. The most significant limitation is that BESD success rates generally overestimate real-world success rate differences. A 2004 article in Psychological Methods demonstrated that success rate differences displayed in BESDs overestimate the actual success rate differences implied by correlations for dichotomous variables, point-biserial correlations, and continuous variable correlations. The overestimation bias is larger for continuous variables than for point-biserial correlations.

This bias means that a BESD showing a 30 percentage point success rate difference does not necessarily mean that 30 more people out of 100 will succeed with the treatment. The actual difference may be smaller. The magnitude of the bias depends on the type of correlation and the underlying distributions of the variables.

The same 2004 article recommended the stochastic difference index as an alternative to the BESD. This index, developed by Cliff and extended by Vargha and Delaney, provides a different measure of the difference between two groups that may be less biased than the BESD. Researchers who need precise estimates of success rate differences should consider this alternative.

Another limitation is that the BESD assumes a base success rate of 50% in the control group. This assumption is built into the formula: the control success rate is always 50 minus half the correlation. If the actual control group success rate is much higher or lower than 50%, the BESD may present a misleading picture. For example, if the control group success rate is 90%, a treatment that improves outcomes to 95% represents a meaningful improvement, but the BESD would show a different pattern because it anchors the control rate at 50%.

Common Failure Patterns in Using the BESD

Researchers and readers encounter several recurring problems when applying or interpreting the BESD. Recognizing these patterns helps avoid misinterpretation.

Treating BESD Values as Observed Rates

The most common error is treating BESD success rates as if they were directly observed in the study. The BESD is a reexpression of a correlation coefficient, not a report of actual success rates. The actual success rates in the study may differ substantially from the BESD values, especially when the control group success rate is not near 50%. The 2004 Psychological Methods article documented this overestimation bias and warned that BESD success rate differences reported for different types of correlations are not directly comparable.

Applying the BESD Without Checking the Base Rate

The BESD formula assumes a 50% base success rate. When the actual base rate is much higher or lower, the BESD can mislead. A researcher who applies the BESD to a study with a 90% control success rate will produce values that do not reflect the study's actual conditions. Checking the base rate before applying the BESD is essential.

Using the BESD for Non-Binary Outcomes

The BESD is designed for binary outcomes, where each individual either succeeds or fails. Applying the BESD to continuous outcomes requires dichotomizing the outcome, which involves choosing a cutoff point. Different cutoff points can produce different BESD values. The 2021 Clinical Psychology in Europe article noted that it is not clear what the best binary outcome is for continuous outcomes, which limits the applicability of the BESD in these situations.

Ignoring the Difference Between Correlation Types

The BESD was originally developed for Pearson's r with continuous variables. Applying it to phi coefficients, point-biserial correlations, or other correlation types introduces different biases. The 2004 Psychological Methods article showed that overestimation biases are larger for continuous variable correlations than for point-biserial correlations. Researchers should be cautious when comparing BESD values across studies that used different correlation types.

Records and Measurements for BESD Reporting

When reporting a BESD in a research paper, presentation, or policy document, certain information should accompany the display to ensure transparency and reproducibility.

Report the Original Effect Size

Always report the original effect size statistic alongside the BESD. This allows readers to verify the conversion and to apply their own interpretation frameworks. A BESD without the underlying effect size is incomplete.

Report the Correlation Type

Specify whether the correlation is Pearson's r, a point-biserial correlation, a phi coefficient, or another type. Because the BESD bias differs by correlation type, this information is essential for accurate interpretation.

Report the Sample Size

The BESD does not incorporate sample size, but sample size affects the precision of the effect size estimate. A BESD based on a study with 20 participants is less reliable than one based on 2,000 participants. Reporting the sample size allows readers to judge the stability of the displayed values.

Report the Base Rate

If the study provides information about the control group success rate, report it alongside the BESD. This allows readers to assess whether the BESD assumption of a 50% base rate is reasonable for the specific context.

Report the Conversion Formula

When converting from Cohen's d or another statistic to r, report the formula used. This is especially important when group sizes are unequal, because the conversion formula differs from the equal group size version.

Quality Controls and Verification

Several quality control measures can improve the accuracy and usefulness of BESD reporting.

Verify the Conversion

Before presenting a BESD, verify the conversion from the original effect size to r and then to success rates. A simple check is that the treatment and control success rates must sum to 100%. Another check is that the success rate difference must equal the correlation expressed as a percentage multiplied by 2.

Cross-Check with the Number Needed to Treat

When the outcome is binary and the base rate is known, calculate the number needed to treat as a cross-check on the BESD. The NNT should be consistent with the success rate difference. Discrepancies may indicate an error in the BESD calculation or an inappropriate application of the method.

Consult Field-Specific Guidelines

Effect size interpretation guidelines vary by field. The 2019 gerontology study and the 2020 rehabilitation study both found that field-specific effect sizes differ from Cohen's original guidelines. Researchers should consult field-specific guidelines when available and should not rely solely on generic thresholds.

Use Reporting Standards

Several organizations provide reporting standards for research. The EQUATOR Network maintains a collection of reporting guidelines for health research, and the National Institute of Standards and Technology supports the Research Data Framework for managing and sharing research data. The NC3Rs Experimental Design Assistant helps researchers design experiments with appropriate sample sizes and statistical power. These resources support transparent and reproducible research reporting, which includes clear presentation of effect sizes and their interpretations.

Welfare and Safety Context for Life-Science Applications

In life-science research, the interpretation of effect sizes has direct implications for human and animal welfare. A treatment that shows a statistically significant effect but a small practical impact may not justify its costs, risks, or side effects. Conversely, a treatment with a modest effect size may be highly valuable if the condition is serious and no better alternatives exist.

The 1990 meta-analysis of neuroleptic treatment in dementia illustrates this point. The effect size of r = 0.18 is small by conventional standards, and the BESD showed that treatment improved the improvement rate from 41% to 59%. The authors described this as consistent with the modest efficacy noted in previous qualitative reviews. For clinicians deciding whether to prescribe neuroleptics to agitated dementia patients, the BESD provided a clearer picture of the tradeoff between potential benefits and risks than the correlation coefficient alone.

In animal research, the same principles apply. The NC3Rs Experimental Design Assistant supports researchers in designing experiments that use the minimum number of animals necessary to achieve reliable results. Understanding effect sizes and their practical significance is part of this process, because it helps researchers determine appropriate sample sizes and interpret the magnitude of treatment effects.

Professional Escalation Criteria

Researchers and practitioners should seek additional expertise when certain conditions are present.

When Effect Sizes Are Inconsistent Across Studies

If different studies of the same intervention produce widely varying effect sizes, the interpretation becomes more complex. A 2026 article in the Journal of Clinical Epidemiology used meta-regression to examine effect moderation in surgical site infection risk factors and found that population size moderated the effect estimates. When effect sizes vary by study characteristics, a simple BESD based on a pooled effect size may be misleading. Consultation with a statistician or meta-analyst is appropriate.

When the Outcome Definition Is Unclear

A 2026 methodological review of perioperative bleeding trials found 36 unique definitions of bleeding across 115 trials. When outcome definitions vary, effect sizes and their BESD conversions may not be comparable across studies. Researchers should consult the original study protocols and consider whether the outcomes are sufficiently similar to warrant combined interpretation.

When the Base Rate Is Extreme

If the control group success rate is very high or very low, the BESD may not be appropriate. The BESD assumes a 50% base rate, and extreme base rates can produce misleading displays. A statistician can recommend alternative methods, such as the stochastic difference index or the number needed to treat.

When Reporting to Regulatory or Policy Audiences

Regulatory submissions and policy documents may have specific requirements for effect size reporting. The EQUATOR Network provides reporting guidelines for health research, and the National Library of Medicine's PubMed database indexes studies that follow these standards. Researchers preparing materials for regulatory or policy audiences should consult the relevant guidelines and may need to involve experts in research reporting standards.

Frequently Asked Questions

What exactly does the binomial effect size display show?

The BESD shows the practical impact of a correlation coefficient by converting it into two success rates. The treatment group success rate is 50 plus half the correlation expressed as a percentage, and the control group success rate is 50 minus half the correlation expressed as a percentage. For r = 0.30, the display shows a 65% treatment success rate and a 35% control success rate.

How do I calculate the BESD from Cohen's d?

First convert Cohen's d to Pearson's r using the formula r equals d divided by the square root of d squared plus 4 for equal group sizes. Then apply the BESD formula to the resulting r value. A Cohen's d of 0.50 converts to approximately r = 0.24, which yields a treatment success rate of 62% and a control success rate of 38%.

Why does the BESD overestimate real-world success rate differences?

The BESD assumes a 50% base success rate and a specific relationship between the correlation and the success rate difference. A 2004 article in Psychological Methods showed that these assumptions generally lead to overestimation of actual success rate differences, with larger biases for continuous variable correlations than for point-biserial correlations.

When should I use the BESD instead of Cohen's d?

Use the BESD when communicating with audiences that are not familiar with standardized effect size units. The BESD is especially useful for clinicians, educators, policymakers, and the public. Use Cohen's d when the audience expects traditional effect size reporting or when the analysis requires a standardized mean difference for meta-analysis.

Can the BESD be used with multiple regression results?

Researchers have extended the BESD to multiple correlation coefficients from hierarchical regression analyses. A 2026 tutorial in Psychological Methods showed that BESD interpretations of hierarchical regression findings can differ from traditional change in R squared interpretations. The extension is possible but requires careful attention to the type of correlation being displayed.

What is the relationship between the BESD and the number needed to treat?

The BESD and the number needed to treat are closely related. The NNT is the reciprocal of the absolute risk reduction. If the BESD shows a 20 percentage point success rate difference, the NNT is 5, meaning five patients must receive the treatment for one additional patient to benefit. Both methods express treatment effects in terms of success rates.

Does the BESD work for continuous outcomes?

The BESD is designed for binary outcomes. Applying it to continuous outcomes requires dichotomizing the outcome, which involves choosing a cutoff point. Different cutoff points can produce different BESD values, and it is not always clear what the best binary outcome is for a continuous measure.

What should I report alongside the BESD?

Report the original effect size, the correlation type, the sample size, the base rate if known, and the conversion formula used. This information allows readers to verify the conversion and to assess whether the BESD is appropriate for the specific context.

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References and Further Reading

• Research Data Framework. National Institute of Standards and Technology.
• EQUATOR Network. EQUATOR Network.
• Experimental Design Assistant. NC3Rs.
• NCBI Literature Resources. National Center for Biotechnology Information.
• PubMed. National Library of Medicine.
• Just How Effective is Direct Instruction?. Perspectives on behavior science, 2021.
• Contrasts and correlations in effect-size estimation.. Psychological science, 2000.
• Binomial effect size displays and gain-probability: Alternative ways to interpret hierarchical regression findings, with tutorial.. Psychological methods, 2026.
• Has the Time Come to Stop Using the "Standardised Mean Difference"?. Clinical psychology in Europe, 2021.
• Useful effect size interpretations for single case research.. Behavior therapy, 2007.
• A metaanalysis of controlled trials of neuroleptic treatment in dementia.. Journal of the American Geriatrics Society, 1990.
• Biases of success rate differences shown in binomial effect size displays.. Psychological methods, 2004.
• How to teach genetic drift.. Yi chuan = Hereditas, 2020.
• Novel Effect Size Interpretation Guidelines and an Evaluation of Statistical Power in Rehabilitation Research.. 2020.
• Effect Size Interpretation, Sample Size Calculation, and Statistical Power in Gerontology. 2019.
• Effect moderation and heterogeneity in meta-analysis: a meta-regression-based reinterpretation of surgical site infection risk factors after craniotomy and cranioplasty.. 2026.
• Definition, analysis, reporting, and interpretation of perioperative bleeding in randomized controlled trials: a methodological systematic review.. 2026.
• Using the binomial effect size display (BESD) to present the magnitude of effect sizes to the evaluation audience. Practical Assessment Research and Evaluation, 2005.
• A simple, general purpose display of magnitude of experimental effect. Journal of Educational Psychology, 1982.

This article is educational and does not replace institutional policy, professional advice, or applicable safety and regulatory requirements.