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Tolerance Interval¶
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QXL Stat Tools Tab > Analysis Tools > Tolerance Interval
A tolerance interval is a range, calculated from a sample, that captures at least a specified proportion of the entire population with a stated level of confidence. For example, a 95%/95% two-sided tolerance interval means: "we are 95% confident that at least 95% of all future values from this population will fall within these bounds."
This is different from a confidence interval. A confidence interval brackets a single population parameter (such as the mean); a tolerance interval brackets a proportion of the individual values in the population. Tolerance intervals are wider, because they must account for both the spread of the population and the uncertainty in estimating it from a sample.
Quantum XL produces both a normal tolerance interval (assuming the data follow a normal distribution) and a nonparametric tolerance interval (making no distributional assumption), together with a normality diagnostic.
Step #1: Select data source¶
Select one or more columns of continuous data. Each column is analyzed independently; Group By columns can be used to split the analysis by category.
Step #2: Set options¶
| Option | Description |
|---|---|
| Confidence Level (%) | The probability (e.g., 95%) that the interval truly captures the specified proportion of the population. Written \(C\) in the equations. |
| Percent of Population (%) | The minimum proportion of the population (e.g., 95%) the interval is required to contain. Written \(P\) in the equations. |
| Interval Side | Two-Sided — lower and upper bounds together contain the proportion. One-Sided Lower — a lower bound such that at least the stated proportion lies above it. One-Sided Upper — an upper bound such that at least the stated proportion lies below it. |
Step #3: Press Finish¶
Quantum XL calculates the tolerance interval and produces a statistics table and two charts.
Statistics Table¶
The table is organized into labeled sections:
| Section | Row | Description |
|---|---|---|
| Statistics | N | Sample size. |
| Mean | Sample mean. | |
| Std Dev | Sample standard deviation (n − 1 denominator). | |
| Normal Bounds | Lower | Lower bound of the normal tolerance interval (omitted for one-sided upper). |
| Upper | Upper bound of the normal tolerance interval (omitted for one-sided lower). | |
| Nonparametric Bounds | Lower | Lower bound of the nonparametric tolerance interval (omitted for one-sided upper). |
| Upper | Upper bound of the nonparametric tolerance interval (omitted for one-sided lower). | |
| Attained Confidence | The confidence level actually achieved by the nonparametric interval (see note below). | |
| Goodness of Fit | AD statistic | Anderson-Darling test statistic for normality. |
| AD p-value | p-value for the Anderson-Darling test. A p-value greater than 0.05 is generally consistent with normality. |
Attained Confidence (Nonparametric)
Because nonparametric bounds must be actual observed data values, the interval usually cannot reach the exact requested confidence at a given sample size. The Attained Confidence is the confidence level the returned nonparametric interval actually achieves. It will often be close to, but not exactly equal to, the requested confidence level, especially for small samples.
Normal Probability Plot¶
The first chart is a normal probability plot of the data. Points that fall close to the reference line indicate that a normal distribution is a reasonable model. Significant departures from the line suggest non-normality, in which case the nonparametric interval may be more appropriate.
Tolerance Interval Line Plot¶
The second chart compares the Normal and Nonparametric tolerance intervals side by side as horizontal line segments. The mean is shown as a center marker. This visual comparison makes it easy to see whether the two methods agree and how wide each interval is relative to the data spread.
Methods¶
Quantum XL computes the interval two ways and reports both:
- Normal Method — assumes the data come from a normal distribution. Bounds have the form \(\bar{x} \pm k s\), where the tolerance factor \(k\) is computed exactly.
- Nonparametric Method — makes no assumption about the shape of the distribution. Bounds are order statistics (sorted data values) chosen so the interval attains the requested confidence.
Notation¶
Both method pages use this notation:
| Symbol | Meaning |
|---|---|
| \(n\) | number of (finite) data values in the sample |
| \(P\) | proportion of the population to capture (Percent of population ÷ 100) |
| \(C\) | confidence level as a fraction (Confidence level ÷ 100) |
| \(\alpha\) | \(1 - C\) |
| \(\bar{x},\ s\) | sample mean and sample standard deviation |
| \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\) | the sample sorted in ascending order (order statistics) |
Data requirements¶
- At least two finite data values are required.
- The values must not all be identical — zero spread leaves nothing to estimate, so a sample with no variation is reported as an error rather than a zero-width interval.
- Missing values and non-numeric entries are ignored; all calculations use only the finite values actually measured.