Tolerance Interval

Tolerance intervals estimate a range that is likely to contain a specified proportion of future product measurements, given a desired confidence level. These intervals define upper and/or lower bounds where, at least, a certain percentage of the process output is expected to fall, combining both the target population coverage and statistical certainty.

The population distribution will be assumed as normal. Calculation for non-normal distributions is not available yet.

To perform the study choose Quality > Tolerance Interval

../_images/ti1.png

  • Sample Values: Select the column containing the sample values from the population to be studied. The values have to be numerical and continuous.

  • Summerized Data: Input the number of sample points, their mean and standard deviation instead of providing the sample data set. The input will override the sample data column selection.

  • Specify Proportion to Cover: The percentage of the population you want to capture. For example, to calculate at least 80% of the process output will fall into the interval, at given confidence level, input 0.8.

  • Alpha: Define the confidence level in the calculation. For example, to calculate at least a given potion of the process output to fall into the interval, at 80% confidence level, input 0.2. While for the most commmon 95% confidence level, keep the default 0.05 input.

The result aligns with Minitab 22 and JMP 17.

Unlike confidence intervals (which estimate where a population parameter lies) or prediction intervals (which predict where future observations will fall), tolerance intervals aim to capture a specific percentage of the entire population.

Comparison of Statistical Intervals

Characteristic

Confidence Interval (CI)

Prediction Interval (PI)

Tolerance Interval (TI)

Purpose

Estimates where a population parameter lies

Predicts where future individual observations will fall

Defines a range containing a specified proportion of the population

Answers question

Where is the true population mean, variance, or other parameter?

Where will the next observation(s) fall?

Where do X% of all population values lie?

Key components

  • Confidence level (1-α)

  • Confidence level (1-α)

  • Coverage proportion (p)

  • Confidence level (1-α)

Interpretation

We are (1-α)% confident that the interval contains the true parameter

We are (1-α)% confident that a future observation will fall in this interval

We are (1-α)% confident that at least p% of the population falls within this interval

Width characteristics

Narrows as sample size increases

Always wider than a CI; affected by both parameter uncertainty and individual variation

Widest of the three; affected by sample size, coverage proportion, and confidence level

Common uses

  • Estimating population parameters

  • Hypothesis testing

  • Research reporting

  • Forecasting

  • Process monitoring

  • Identifying outliers

  • Quality control

  • Manufacturing specs

  • Regulatory compliance

  • Reference ranges

Example

“We are 95% confident the true mean weight is between 9.8-10.2 kg”

“We are 95% confident the next measurement will be between 9.5-10.5 kg”

“We are 99% confident that 95% of all products will weigh between 9.2-10.8 kg”

Here below is the sample output of the calculation. It includes two-side and one-side intervals, and is pretty self explaining.

---- Tolerance Interval ----
95.00% confident that at least 90.00% of the
population's values for this characteristic will
fall between -2.32 and 2.32
+------------+----------+----------+---------+
| Proportion | Lower TI | Upper TI | 1-Alpha |
+------------+----------+----------+---------+
|    0.90    |  -2.32   |   2.32   |  0.950  |
+------------+----------+----------+---------+

One-Sided Tolerance Interval
+------------+----------+----------+---------+
| Proportion | Lower TI | Upper TI | 1-Alpha |
+------------+----------+----------+---------+
|    0.90    |  -1.93   |    -     |  0.950  |
|    0.90    |    -     |   1.93   |  0.950  |
+------------+----------+----------+---------+