Statistical Tolerance Limits and Prediction Intervals

Statistical tolerance limits and prediction intervals provide two complementary frameworks for quantifying uncertainty in statistical inference. A tolerance interval is constructed from sample data to assert, with a specified confidence level, that at least a given proportion of the population lies within the interval. By contrast, a prediction interval aims to capture the range for a future individual observation with a stated probability. While confidence intervals estimate fixed parameters of a population distribution, tolerance and prediction intervals address variability in population values and future measurements respectively. The choice between these intervals depends on the inferential goal: ensuring adequate coverage of population quantiles for quality control demands tolerance limits, whereas forecasting or quality assurance for a single future datum employs prediction limits. Technological advances in mixed-effects modelling, non-parametric approaches and extreme-value theory have expanded the toolkit available for constructing these intervals under complex designs, small sample sizes or heavy-tailed distributions. These intervals have found wide application in clinical laboratory reference ranges, manufacturing quality control, environmental risk assessment and beyond, underscoring their global significance and practical relevance for regulatory and research settings alike.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Research from all publishers

Recent methodological advances have addressed the generalisation and practical implementation of tolerance and prediction intervals under varied modelling frameworks. In linear mixed models, a unified reformulation of two-sided prediction intervals leverages the observed Fisher information and Satterthwaite degrees of freedom to yield accurate coverage across nested, crossed and unbalanced designs; parallel constructions for tolerance intervals demonstrate robust performance in simulation studies and real-world applications in clinical and vaccine research. Critiques of prediction intervals as reference ranges in clinical laboratories have led to the promotion of (P, γ) tolerance intervals, which ensure that a specified proportion of the population is encompassed at a prespecified confidence, overcoming undercoverage issues inherent to prediction-interval-based reference limits. More recently, extensions of extreme-value theory to mixture normal distributions have facilitated the construction of tolerance intervals tailored to small-sample regimes and heavy-tailed components, showing superior accuracy relative to traditional methods in both simulated and empirical datasets.

Statistical Tolerance Limits and Prediction Intervals publication trend

The graph below shows the total number of articles in statistical tolerance limits and prediction intervals across all publications each year (not limited to Nature Index journals).

Technical terms

Statistical tolerance limit (Tolerance Interval): An interval, derived from sample data, that with a specified confidence contains at least a given proportion of the population.

Prediction interval: An interval that, with a specified probability, is expected to include a single future observation drawn from the same population.

Reference range: A data-derived range intended to encompass a predefined percentage of healthy or typical values in a population, often used in clinical settings.

References

1. Confidence, prediction, and tolerance in linear mixed models. Statistics in Medicine (2019).
2. Reference range: Which statistical intervals to use?. Statistical Methods in Medical Research (2020).
3. Tolerance Interval for the Mixture Normal Distribution Based on Generalized Extreme Value Theory. Mathematics (2024).