Credible Interval Width Calculator

What This Calculator Measures

A credible interval is the Bayesian counterpart to a confidence interval: given a posterior distribution over a parameter, it's the range that contains a stated proportion (say 95%) of the posterior probability. Its width tells you how precisely your data and prior beliefs together pin down the parameter — a narrow interval means the posterior is concentrated, a wide one means real uncertainty remains even after seeing the data.

This calculator estimates that width from a normal-approximation posterior using your posterior standard deviation, sample size, confidence (credibility) level, and two optional adjustments — a prior weight, which tightens the interval to reflect the equivalent number of "extra observations" your prior belief contributes, and a design effect, which widens it to account for non-simple-random sampling such as clustering or stratification.

How to Use This Well

Enter the posterior standard deviation and the confidence (credibility) level.
Add sample size and the prior's equivalent weight in observations.
Set effect size and design effect.
Review the interval width and half-width.
Adjust sample size or prior weight if the interval is too wide for your decision.

Formula Breakdown

Effective n = Sample size ÷ Design effect
Posterior n = Effective n + Prior weight
Width = 2 × z × Posterior std dev ÷ √(Posterior n)
Half-width = Width ÷ 2
Signal score = Effect size ÷ Width

z: the standard-normal critical value for your confidence level (z ≈ 1.96 for 95%), found by inverting the normal CDF.

Prior weight: how many equivalent data points your prior is worth — a prior weight of 20 acts like adding 20 more observations' worth of information before computing the width.

Design effect: a multiplier (typically 1.0-2.0) that inflates variance for clustered or stratified samples; dividing sample size by it gives the "effective" simple-random-sample-equivalent size.

Signal score: a rough ratio of the effect size you care about to the interval width — values well above 1 suggest the interval is narrow enough to distinguish the effect from noise.

Worked Example

Using the calculator's own defaults — posterior standard deviation 1.8, 95% confidence, sample size 150, prior weight 20, effect size 0.6, and design effect 1.1:

z for 95% confidence ≈ 1.96.
Effective n: 150 ÷ 1.1 ≈ 136.4.
Posterior n: 136.4 + 20 = 156.4.
Width: 2 × 1.96 × 1.8 ÷ √156.4 ≈ 0.564.
Half-width: 0.564 ÷ 2 ≈ 0.282.
Signal score: 0.6 ÷ 0.564 ≈ 1.06.

A width of 0.564 falls in the "moderate" band below — standard precision, with the effect size only slightly larger than the width itself, so the interval is informative but not by a wide margin.

Interpretation Guide

Width	Meaning	Action
Under 0.5	Tight — high precision.	Current settings are working well.
0.5-1.0	Moderate — standard precision.	Usable for most decisions; consider more data for close calls.
1.0-1.5	Wide.	Increase sample size or strengthen the prior.
1.5+	Very wide.	Refine the model or gather substantially more data before deciding.

Common Mistakes to Avoid

Treating a credible interval like a confidence interval: a 95% credible interval means there's a 95% probability the parameter lies in that range given your prior and data — it does not carry the frequentist "95% of intervals built this way would contain the true value" interpretation, even though the two are often numerically close.
Forgetting the design effect for clustered data: survey or panel data collected in clusters (classrooms, households, clinics) has less independent information than its raw sample size suggests; skipping the design effect will understate your interval width.
Treating prior weight as free precision: a large prior weight tightens the interval only if the prior is actually justified — an unjustified strong prior produces a falsely narrow, overconfident interval.
Ignoring the signal score: a tight interval around an effect size close to zero can still mean "no meaningful effect" — always compare the interval width to the effect size you actually care about, not just to zero.

Implementation Checklist

Estimate the posterior standard deviation from your model or prior analysis.
Pick a confidence (credibility) level appropriate for your decision.
Set sample size and account for clustering via the design effect.
Review the width against your minimum effect size of interest.

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Frequently Asked Questions

How accurate are the results?

The Credible Interval Width applies a standard normal-approximation formula to your inputs — accuracy depends on how precisely you measure those inputs and how close your posterior is to normal. For planning and estimation, results are reliable. For high-stakes or professional decisions, cross-check the output with a domain expert or primary source.

What sample size do I need for reliable results?

It depends on the desired confidence level, margin of error, and population variance. For a typical survey (95% confidence, ±5% margin), n ≈ 385 for a large population. Smaller samples are fine for exploratory analysis, but don't over-interpret the results — widen your confidence intervals to reflect the uncertainty.

How does prior weight affect the interval width?

Prior weight is treated as an equivalent number of extra observations added to your actual sample size (after adjusting for design effect). A larger prior weight narrows the interval the same way collecting more real data would — but only if that prior information is genuinely reliable, since an unjustified prior produces a falsely confident, overly narrow interval.

What is a design effect and when do I need one?

A design effect (often written DEFF) adjusts for sampling designs that aren't simple random sampling — cluster sampling, stratification, or unequal selection probabilities all change how much independent information your sample actually contains. A design effect of 1.0 means no adjustment; values above 1.0 are typical for clustered survey data and reduce your effective sample size.