Calculator Inputs
Use the responsive calculator grid below. Large screens show three columns, smaller screens show two, and mobile shows one.
Example Data Table
Use this sample to test the calculator quickly.
| Scenario | Group 1 Mean | Group 1 SD | Group 1 n | Group 2 Mean | Group 2 SD | Group 2 n | Prior Mean | Prior SD | ROPE | Credible Level |
|---|---|---|---|---|---|---|---|---|---|---|
| Worked Example | 82.0 | 10.5 | 36 | 76.0 | 11.2 | 34 | 0.0 | 0.5 | 0.10 | 95 |
Formula Used
This calculator estimates a Bayesian standardized mean difference for two independent groups.
1) Pooled standard deviation
sp = √[ ((n₁ - 1)s₁² + (n₂ - 1)s₂²) / (n₁ + n₂ - 2) ]
2) Observed standardized effect
d = (x̄₁ - x̄₂) / sp
3) Small-sample correction
g = J × d, where J = 1 - 3 / (4(n₁ + n₂) - 9)
4) Approximate sampling uncertainty
SE(g) ≈ √[(n₁ + n₂)/(n₁n₂) + g²/(2(n₁ + n₂ - 2))]
5) Bayesian update
Prior: δ ~ Normal(μ₀, τ₀²)
Likelihood approximation: g | δ ~ Normal(δ, SE(g)²)
6) Posterior distribution
Posterior variance = 1 / (1/τ₀² + 1/SE(g)²)
Posterior mean = Posterior variance × (μ₀/τ₀² + g/SE(g)²)
7) Credible interval and probabilities
The credible interval uses the chosen normal quantile. Posterior probabilities such as P(δ > 0) and P(|δ| < ROPE) come from the posterior normal distribution.
8) Approximate Bayes factor
The calculator compares a point-null model at zero effect against the alternative model using the marginal density ratio at the observed effect size.
How to Use This Calculator
- Enter the mean, standard deviation, and sample size for each group.
- Set the prior mean and prior standard deviation for the expected effect size.
- Choose a ROPE half-width to define practical equivalence around zero.
- Pick a credible level, such as 95%.
- Press the calculate button.
- Review the posterior mean, credible interval, directional probabilities, and Bayes factors.
- Use the chart to compare prior, likelihood, and posterior distributions.
- Download the summary as CSV or PDF for reporting.
Frequently Asked Questions
1) What does this calculator estimate?
It estimates a Bayesian standardized mean difference for two independent groups. It combines observed data with a normal prior to produce a posterior effect size distribution, interval estimates, and decision-support probabilities.
2) Why are Cohen’s d and Hedges’ g both shown?
Cohen’s d is the raw standardized difference. Hedges’ g applies a small-sample correction, so it is usually preferred when sample sizes are modest. The Bayesian update here is based on the corrected effect size.
3) What is the role of the prior mean and prior SD?
The prior mean expresses your central expectation before seeing the data. The prior SD controls how strongly that belief is spread out. Larger prior SD values make the prior weaker and more data-driven.
4) What does P(effect > 0) tell me?
It gives the posterior probability that the effect is positive. Values near 50% suggest uncertainty, while values near 100% or 0% indicate strong directional evidence under the chosen model and prior.
5) What is ROPE and why use it?
ROPE stands for region of practical equivalence. It defines a small interval around zero where effects are treated as practically negligible. The calculator reports the posterior probability that the true effect falls inside that interval.
6) How should I read the approximate Bayes factor?
BF10 compares evidence for the alternative against the null. Values above 1 favor the alternative, while values below 1 favor the null. Larger departures from 1 indicate stronger evidence, not necessarily larger effects.
7) Is this an exact Bayesian t-test?
No. This page uses a normal-likelihood approximation centered on the corrected standardized effect size. It is practical and interpretable, but it is still an approximation rather than a full exact sampling-model solution.
8) When is this calculator most useful?
It is useful when you want more than a single effect-size point estimate. It helps combine prior beliefs, summarize uncertainty, evaluate practical equivalence, and generate clean reporting outputs from two-group summary statistics.