Example data table
| Scenario | n | Observed mean | Observed SD | Prior mean | Prior SD | Future size | Level | Predictive interval |
|---|---|---|---|---|---|---|---|---|
| Base case | 30 | 12.4 | 2.1 | 11.8 | 3.5 | 1 | 95% | Approx. 8.20 to 16.45 |
| Tighter prior | 30 | 12.4 | 2.1 | 12.0 | 1.5 | 1 | 95% | Narrower than base case |
| Future average of 5 | 30 | 12.4 | 2.1 | 11.8 | 3.5 | 5 | 95% | Much tighter average forecast band |
Formula used
Prior for the mean: μ ~ Normal(μ0, τ02)
Observed data model: x̄ | μ ~ Normal(μ, σ2/n)
Posterior variance: τn2 = 1 / (1/τ02 + n/σ2)
Posterior mean: μn = τn2 ( μ0/τ02 + n x̄ /σ2 )
Predictive SD for a future average of m observations: √(τn2 + σ2/m)
The interval is built as μn ± z × predictive SD, where z comes from the chosen interval level. Here, the entered sample standard deviation acts as σ, the observation spread proxy.
How to use this calculator
- Enter the observed sample mean, standard deviation, and sample size.
- Enter your prior mean and prior standard deviation for the unknown true mean.
- Choose how many future observations you want to average in the forecast horizon.
- Set the interval confidence level, such as 90%, 95%, or 99%.
- Optionally enter a target and lower or upper specifications for probability checks.
- Press Calculate interval to show results above the form and the graph below.
- Use the CSV or PDF buttons to export the summary once the result appears.
FAQs
1) What does a posterior predictive interval show?
It shows a likely range for a future value or future average after combining prior beliefs with observed data. It reflects both remaining uncertainty about the mean and the natural variability of future observations.
2) Why does the calculator ask for a prior mean and prior standard deviation?
Those inputs describe your Bayesian prior for the unknown mean. The prior mean sets the starting center, while the prior standard deviation controls how strongly that prior influences the posterior result.
3) What happens when the future sample size is larger than one?
The calculator forecasts the average of that many future observations. Because averages smooth random noise, the predictive standard deviation shrinks and the interval usually becomes narrower.
4) Why can a strong prior change the interval noticeably?
A small prior standard deviation means you trust the prior strongly. That increases prior weight in the posterior mean and can pull the predictive interval toward the prior center.
5) Is this interval the same as a classical confidence interval?
No. A confidence interval usually targets an unknown parameter. This tool gives a Bayesian predictive interval for future outcomes, which includes future observation noise as well as parameter uncertainty.
6) Why is the sample standard deviation used as the observation spread proxy?
This single-file tool uses the entered sample standard deviation as an estimate of the underlying spread. It is practical for fast forecasting, though a fuller Bayesian variance model can be more rigorous.
7) What do the target and specification probabilities mean?
The target probability estimates how likely the future average is to exceed a chosen threshold. The specification probability estimates how much predictive mass lies within your acceptable lower and upper limits.
8) When should I use a higher interval level?
Use a higher level when missing future extremes is costly and you prefer a wider, more conservative range. Lower levels give tighter bands but leave more probability outside the interval.