Concept Architecture
Concept
Theoretically, Tolerance Interval is a statistical interval constructed to contain a specified proportion of a population with a stated level of confidence. Unlike confidence intervals, which quantify uncertainty about a population parameter, or prediction intervals, which predict a single future observation, tolerance intervals are designed to estimate the range within which a predetermined percentage of the underlying population is expected to lie. The concept is founded on statistical tolerance theory and is widely applied in quality control, laboratory medicine and clinical measurement, although its use in health economics is generally confined to uncertainty assessment and simulation outputs.
Mathematically, a tolerance interval is defined using sample statistics together with a tolerance factor that depends on the desired population coverage, confidence level and sample size. For normally distributed data, the interval is centred on the sample mean and extends by a multiple of the sample standard deviation. The tolerance factor is derived from the joint distribution of the sample mean and sample variance to ensure the required probabilistic coverage.
In practice, tolerance intervals are estimated after selecting the desired population coverage (for example, 95%) and confidence level (for example, 95%). Parametric methods are commonly used for normally distributed data, whereas non-parametric methods are available when distributional assumptions are inappropriate. In health economics, tolerance intervals may be applied to model outputs, patient-level cost distributions or simulated clinical outcomes when assessing the range expected to contain most future observations.
Purpose
Used to estimate an interval expected to contain a specified proportion of an underlying population with a stated level of confidence, supporting uncertainty assessment, model validation and interpretation of variability in health economic analyses.
Mathematical Formulae
Primary Formula
TI = x? � k ? s
where:
- x? = sample mean
- s = sample standard deviation
- k = tolerance factor determined by sample size, confidence level and required population coverage
Supporting Formulae
P(L � X � U) = p
Pr(P(L � X � U) � p) = ?
where:
- L, U = lower and upper tolerance limits
- p = required population coverage
- ? = confidence level
Related Mathematical Methods
- Statistical Tolerance Theory
- Normal Distribution
- Non-Parametric Tolerance Intervals
- Order Statistics
- Confidence Interval
- Prediction Interval
Example
A health economist analyses annual healthcare costs for 150 patients receiving a new intervention. The sample mean cost is �9,800 and the standard deviation is �1,200. A two-sided 95%/95% normal tolerance interval has a tolerance factor of approximately 2.31.
TI = 9,800 � (2.31 ? 1,200)
= 9,800 � 2,772
= (�7,028, �12,572)
The analyst concludes, with 95% confidence, that at least 95% of the underlying patient cost distribution lies between �7,028 and �12,572.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B151) | Calculate the sample mean of patient costs. |
| STDEV.S | =STDEV.S(B2:B151) | Estimate the sample standard deviation. |
| COUNT | =COUNT(B2:B151) | Determine the sample size for selecting the tolerance factor. |
| LOWER LIMIT | =AVERAGE(B2:B151)-2.31*STDEV.S(B2:B151) | Calculate the lower tolerance limit. |
| UPPER LIMIT | =AVERAGE(B2:B151)+2.31*STDEV.S(B2:B151) | Calculate the upper tolerance limit. |
VBA (Optional)
Automate calculation of normal tolerance intervals for multiple health economic outcomes using user-specified population coverage and confidence levels.
Sources
- Krishnamoorthy K, Mathew T. Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley.
- Hahn GJ, Meeker WQ. Statistical Intervals: A Guide for Practitioners. Wiley.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a tolerance interval?
An interval calculated to contain a specified proportion of an entire population's values with a given confidence, unlike a confidence or prediction interval.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What proportion of a population does a tolerance interval aim to cover?
A tolerance interval is calculated to contain a specified proportion of an entire population's values, with a stated confidence that it really does. Rather than bracketing a parameter such as a mean, as a confidence interval does, or a single future observation, as a prediction interval does, it aims to enclose, say, ninety-five per cent of all individual values. This makes it useful for setting reference ranges, where the goal is to span most of the population. Covering most of the population's values is its aim. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How does a tolerance interval differ from a confidence interval?
A tolerance interval is designed to contain a specified proportion of the population's individual values with a given confidence, while a confidence interval is designed to contain a population parameter, such as a mean, with a given confidence. The tolerance interval concerns the spread of individual values across the population, whereas the confidence interval concerns the location of a parameter. So the two differ in what they cover, with the tolerance interval addressing a proportion of the population's values and the confidence interval a parameter, which is why tolerance intervals are used for questions about where most individuals fall and confidence intervals for questions about summary quantities.
Source: Casella & Berger 2002
How does a tolerance interval differ from a prediction interval?
A tolerance interval is designed to contain a specified proportion of the whole population's values with a given confidence, while a prediction interval is designed to contain a single future observation with a given confidence. The tolerance interval concerns a fraction of the entire population, whereas the prediction interval concerns one new individual value. So the two differ in scope, with the tolerance interval covering a proportion of the population and the prediction interval a single future value, which is why tolerance intervals suit setting ranges expected to encompass most of a population and prediction intervals suit predicting where an individual new observation will fall.
Source: Casella & Berger 2002
When is a tolerance interval used?
A tolerance interval is used when the interest is in a range expected to contain a specified proportion of a population's values, such as establishing reference ranges for laboratory measurements, setting product specifications, or defining limits within which most individuals fall. So a tolerance interval is used for questions about the spread of individual values across a population, which is common in quality control and in defining normal ranges, since it provides limits covering a stated fraction of the population with stated confidence, distinguishing it from confidence and prediction intervals that address parameters and single future observations respectively.
Source: Casella & Berger 2002
What determines the width of a tolerance interval?
The width of a tolerance interval is determined by the proportion of the population to be covered, the confidence level, the variability of the data, and the sample size: covering a larger proportion or requiring higher confidence widens it, greater variability widens it, and larger samples narrow it. So the width of a tolerance interval reflects both the demanding requirement of covering a fraction of the population with confidence and the uncertainty from estimating from a sample, which is why tolerance intervals are typically wider than confidence intervals, since they must account for the spread of individual values as well as the uncertainty in estimating the population's distribution.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/tolerance-interval
- Term code
- HE-ES-SA-220
Stable URI · Machine-readable · Resolvable · CC BY 4.0