Methods and formulas for the probability plot in Tolerance Intervals (Nonnormal Distribution)

Select the method or formula of your choice.

Probability plot

The probability plots include:

  • Points, which are the estimated percentiles for corresponding probabilities of an ordered data set.
  • Middle lines, which are the expected percentile from the distribution based on maximum likelihood parameter estimates. If the distribution is a good fit for the data, the points fall along the middle line.

Estimated probabilities

Minitab estimates the probability (P) that is used to calculate the plot points using the following methods.

  • Median rank (Benard's method)
  • Mean Rank (Herd-Johnson estimate)
  • Modified Kaplan-Meier (Hazen)
  • Kaplan-Meier product limit estimate


nNumber of observations
iRank of the ith ordered observation x(i), where x(1), x(2),...x(n) are the order statistics, or the data ordered from smallest to largest

Plot points

The middle line of the probability plot is constructed using the x and y coordinate calculations in this table.

Distribution x coordinate y coordinate
Smallest extreme value x ln(–ln(1 – p))
Largest extreme value x ln(–ln p)
Weibull ln(x) ln(–ln(1 – p))
Exponential ln(x) ln(–ln(1 – p))
Lognormal ln(x) Φ–1norm
Logistic x
Loglogistic ln(x)
Gamma x Φ–1gamma

Because the plot points do not depend on any distribution, they are the same (before being transformed) for any probability plot. However, the fitted line differs depending on the parametric distribution chosen.


pThe estimated probability
Φ-1normValue returned for p by the inverse CDF for the standard normal distribution
Φ-1gammaValue returned for p by the inverse CDF for the incomplete gamma distribution
ln(x)The natural log of x