Page 152 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

August 31, 2006
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Illustration of the Two Methodologies Using a Case Study Data set 137
Table 10.1 Common distributions.
Distribution Parameters
Smallest extreme value
Normal μ = location −∞ <μ< ∞
Logistic σ = scale σ> 0
Lognormal μ = location μ> 0
Loglogistic σ = scale σ> 0
Three-parameter lognormal μ = location μ> 0
Three-parameter loglogistic σ = scale σ> 0
λ = threshold −∞ <λ< ∞
Weibull α = scale α = exp(μ)
β = shape β = 1/σ
Three-parameter Weibull α = scale α = exp(μ)
β = shape β = 1/σ
λ = threshold −∞ <λ< ∞
Exponential θ = mean θ> 0
Two-parameter exponential θ = scale θ> 0
λ = threshold −∞ <λ< ∞

Skewness can be calculated using the following formula:
3

N x i − ¯x

(N − 1)(N − 2) s
where the symbols have the same meanings as in the kurtosis formula above.
Table 10.1 shows the commonly used distributions and their parameters to be
estimated. If the distribution is skewed, try the lognormal, loglogistic, exponen-
tial, weibull, and extreme value distributions. If the distribution is not skewed,
depending on the kurtosis, try the uniform (if the kurtosis is negative), normal
(if the kurtosis is close to zero), Laplace or logistic (if the kurtosis is positive).
2. By nature of the data.

(a) Cycle time and reliability data typically follow either an exponential or Weibull
distribution.
(b) If the data is screened from a highly incapable process, it is likely to be uniformly
distributed.
(c) If the data is generated from selecting the highest or lowest value of multiple
measurements, it is likely to following the extreme value distribution.

10.2.2.2 Parameter estimation
After short-listing the possible distributions that are likely to fit the data well, the
parameters for the distribution need to be estimated. The two common estimation
methods are least squares and maximum likelihood.