What is a truncated normal?
A truncated normal is a normal distribution conditioned to stay within bounds such as a≤X≤b. It is also known as a bounded normal or truncated Gaussian.
- Common use cases: non-negative values, scores bounded to [0,100], probability-like values in [0,1], and rare-event tail sampling.
- Truncation changes the mean/variance and can create mass near boundaries — check the PDF/CDF and sample stats.
How it works: it uses the inverse CDF method (sample a uniform random number and map it through the truncated CDF). This avoids rejection-sampling slowdowns when truncation is extreme.
Use this tool as a learning reference for high-stakes domains (medical/financial/legal), and verify final decisions with qualified sources. You don’t need to enter personal information.
Presets
Pick a practical preset (you can tweak values after applying). It regenerates instantly.
Tip: presets are meant as starting points.
Generator
Set μ/σ and bounds, then generate samples and export results.
Sample stats
Samples (first 20)
How to use this tool
Use this page when a normal model is useful but values outside a lower or upper bound are impossible. For broader distribution browsing open the Distributions hub, use Distribution sampler for side-by-side models, and switch to Normal distribution calculator when you need untruncated CDF or quantile work.
Use in 3 steps
- Set the original normal mean and standard deviation, then define the lower and upper truncation bounds.
- Generate a sample and compare the histogram with the theoretical truncated mean, variance, and retained mass.
- Move one bound at a time to see whether the change comes from stronger truncation or from the original normal parameters.
How to read the result
The displayed mean and variance belong to the truncated distribution, not the original unconstrained normal. The retained-mass term shows how much of the original distribution survives between the two bounds.
Boundary checks
- If the retained mass is small, truncation is strong and the resulting mean can move a lot.
- A clipped normal and a truly truncated normal are not the same model.
- Keep track of whether a shift comes from changing the bounds or changing the original standard deviation.