Tolerance interval

A tolerance interval is a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls. Tolerance intervals are used in both descriptive statistics and inferential statistics to describe the spread of data and to make predictions about future data samples. They are particularly useful in quality control and reliability engineering.

Definition

A tolerance interval for a population is an interval that covers a specified proportion of the population, with a certain level of confidence. It is calculated from a sample of data and provides bounds within which we can expect to find a given percentage of the population data. Unlike confidence intervals, which estimate a population parameter, or prediction intervals, which predict future observations, tolerance intervals cover a fixed proportion of the population.

Calculation

The calculation of a tolerance interval depends on the underlying distribution of the data, the size of the sample, and the desired proportion and confidence level. For normally distributed data, the tolerance interval can be calculated using specific formulas that incorporate the sample mean, sample standard deviation, and factors from the normal distribution.

Applications

Tolerance intervals are widely used in various fields such as:

• Manufacturing, to ensure that a process is within specified bounds.
• Environmental science, to assess compliance with environmental standards.
• Pharmaceutical industry, for validating manufacturing processes and ensuring consistent drug efficacy and safety.

See also

• Confidence interval
• Prediction interval
• Statistical hypothesis testing