Contents - Index

Method of Moments

An approach to fitting a distribution to a set of data that involves matching the moments of the data to the distribution.  The number of moments that are matched depends on the number of parameters of the distribution.  For the normal distribution, with 2 parameters, the average and standard deviation are matched.  For the Johnson family of distributions, with 4 parameters, the average, standard deviation, skewness and kurtosis are all matched.

The method of moments approach has the disadvantage that the resulting estimates can be inconsistent with the data set.  This can occur for bounded distributions.  It is possible that the distribution fit to the data has data values below the lower bound or above the upper bound of the distribution.  When this occurs, Distribution Analyzer reduces the number of moments that are matched and instead matches the bounds.

For example, the Beta distribution has 4 parameters and both upper and lower bounds.  Initially Distribution Analyzer attempts to match all four moments.  If a value occurs below the lower bound, the program will match the lower bound, average, standard deviation and skewness.  If a value occurs above the upper bound, the program will match the upper bound, average, standard deviation and skewness.  If values occur both below the lower bound above the upper bound, the program will match the lower bound, upper bound, average and standard deviation.  As a result, the data will fall within the range of the fitted distribution.  This approach has not been implemented for the Johnson distribution so it is still possible that data be outside the range of the fitted distribution.

This approach has been further adapted so that the user can specify a range of values that must be in the range of the selected distribution.  This range can be specified using the Select Distribution to Fit Data dialog box.  When the Find Best Distribution button is clicked in the Data window, the required range is automatically specified as at least 1 standard deviation beyond any spec limits.  This assures the spec limits are also within the range of the distribution and can be transformed.

An alternative approach for fitting data is the maximum likelihood method.