Cumulative Frequency Analysis - Fitting of Probability Distributions - Continuous Distributions

Continuous Distributions

To present the cumulative frequency distribution as a continuous mathematical equation instead of a discrete set of data, one may try to fit the cumulative frequency distribution to a known cumulative probability distribution,.
If successful, the known equation is enough to report the frequency distribution and a table of data will not be required. Further, the equation helps interpolation and extrapolation.However, care should be taken with extrapolating a cumulative frequency distribution, because this may be a source of errors. One possible error is that the frequency distribution does not follow the selected probability distribution any more beyond the range of the observed data.

Any equation that gives the value 1 when integrated from a lower limit to an upper limit agreeing well with the data range, can be used as a probability distribution for fitting. A sample of probability distributions that may be used can be found in probability distributions.

Probability distributions can be fitted by several methods, for example:

• the parametric method, determining the parameters like mean and standard deviation from the X data using the method of moments, the maximum likelihood method and the method of probability weighted moments.
• the regression method, linearizing the probability distribution through transformation and determining the parameters from a linear regression of the transformed Pc (obtained from ranking) on the transformed X data.

Application of both types of methods using for example

• the normal distribution, the lognormal distribution, the logistic distribution, the loglogistic distribution, the exponential distribution, the Fréchet distribution, the Gumbel distribution, the Pareto distribution, the Weibull distribution and other

often shows that a number of distributions fit the data well and do not yield significantly different results, while the differences between them may be small compared to the width of the confidence interval. This illustrates that it may be difficult to determine which distribution gives better results.