Methods and formulas for percentiles in Distribution ID Plot (Right Censoring)

Percentiles and standard errors of percentiles

Percentiles are estimates of the times at which a certain percent of the population has failed. By default, Minitab displays tables of percentiles for parametric distribution analysis for common percentiles.

The standard errors for the percentile estimates are the square root of the variances.

, , , , , , , , , , and denote the variances and covariances of the MLEs of μ, σ, α, β, θ, and λ, which are taken from the appropriate element of the inverse of the Fisher information matrix.

The formulas used for percentile and variance estimates for each distribution are as follows:

Smallest extreme value

Percentile
Variance
where zp is the pth percentile of the standard smallest extreme value distribution

Weibull

Percentile
Variance
where zp is the pth percentile of the standard smallest extreme value distribution

3-parameter Weibull

Percentile
Variance
where zp is the pth percentile of the standard smallest extreme value distribution

Exponential

Percentile
Variance

2-parameter exponential

Percentile
Variance

Normal

Percentile
Variance
where zp is the pth percentile of the standard normal distribution

Lognormal

Percentile
Variance
where zp is the pth percentile of the standard normal distribution

3-parameter lognormal

Percentile
Variance
where zp is the pth percentile of the standard normal distribution

Logistic

Percentile
Variance
where zp is the pth percentile of the standard logistic distribution

Loglogistic

Percentile
Variance
where zp is the pth percentile of the standard logistic distribution

3-parameter loglogistic

Percentile
Variance
where zp is the pth percentile of the standard logistic distribution

Notation

TermDescription
zp

the inverse cdf of the standard distribution evaluated at p (the pth percentile of the standard distribution)

Confidence limits for percentiles

Distribution Confidence limits

Smallest extreme value

Normal

Logistic

Weibull

Exponential

Lognormal

Loglogistic

where (for Weibull distribution)

(for exponential distribution)

(for lognormal and loglogistic distributions)

3-parameter Weibull

2-parameter exponential

3-parameter lognormal

3-parameter loglogistic

If λ < 0:

If λ 0:

where (for 3-parameter Weibull distribution)

(for 2-parameter exponential distribution)

(for 3-parameter lognormal and loglogistic distributions)

For the calculations of the variance of the estimated xp, see the section "Percentiles and standard error of percentiles".

Notation

TermDescription
zα the upper critical value for the standard normal distribution where 100α % is the confidence level.
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