As a subject matter expert in statistical quality control, I can explain the concept of a tolerance limit. In statistics, a tolerance limit is a statistical estimate of the range within which a certain proportion of a population lies, given a sample of data from that population. It is used when the goal is to understand the variability of a process or to set specifications for a product or process.

Tolerance limits are often calculated using the sample mean and standard deviation, and they can be one-sided or two-sided, depending on the context. For instance, a common use of tolerance limits is to estimate the range within which 95% of all items produced by a process will fall. This is known as a 95% one-sided tolerance limit.

The calculation of tolerance limits can be complex and depends on the distribution of the data. For normally distributed data, the formula for a two-sided tolerance limit at a confidence level \( C \) is:

\[ T = \bar{x} \pm T_{n-1}(C) \times s \times \sqrt{\frac{1}{n}} \]

Where:
- \( \bar{x} \) is the sample mean,
- \( T_{n-1}(C) \) is the tolerance factor derived from the Student's t-distribution with \( n-1 \) degrees of freedom and confidence level \( C \),
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.

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A tolerance interval is a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls. ... "A (p, 1?��) upper tolerance limit (TL) is simply an 1?�� upper confidence limit for the 100 p percentile of the population."read more >>