Fit Distributions

Use the Continuous or Discrete Fit options to fit a distribution to a continuous or discrete variable.

A curve is overlaid on the histogram, and a Parameter Estimates report is added to the report window. A red triangle menu contains additional options. See Fit Distribution Options.

Note: The Life Distribution platform also contains options for distribution fitting that might use different parameterizations and allow for censoring. See the Quality and Process Methods book.

Continuous Fit

Use the Continuous Fit options to fit the following distributions to a continuous variable.

•	The Normal distribution is often used to model measures that are symmetric with most of the values falling in the middle of the curve. JMP uses the unbiased estimate when determining the parameters for the Normal distribution.

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The LogNormal distribution is often used to model values that are constrained by zero but have a few very large values. The LogNormal distribution can be obtained by exponentiating the Normal distribution. JMP uses the maximum likelihood estimation when determining the parameters for the LogNormal distribution.

•	The Weibull, Weibull with threshold, and Extreme Value distributions often provide a good model for estimating the length of life, especially for mechanical devices and in biology.

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The Exponential distribution is especially useful for describing events that randomly occur over time, such as survival data. The exponential distribution might also be useful for modeling elapsed time between the occurrence of non-overlapping events, such as the time between a user’s computer query and response of the server, the arrival of customers at a service desk, or calls coming in at a switchboard.

•	The Gamma distribution is bound by zero and has a flexible shape.

•	The Beta distribution is useful for modeling the behavior of random variables that are constrained to fall in the interval 0,1. For example, proportions always fall between 0 and 1.

•	The Normal Mixtures distribution fits a mixture of normal distributions. This flexible distribution is capable of fitting multi-modal data. You can also fit two or more distributions by selecting the Normal 2 Mixture, Normal 3 Mixture, or Other options.

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The Smooth Curve distribution... A smooth curve is fit using nonparametric density estimation (kernel density estimation). The smooth curve is overlaid on the histogram and a slider appears beneath the plot. Control the amount of smoothing by changing the kernel standard deviation with the slider. The initial Kernel Std estimate is calculated from the standard deviation of the data.

•	The Johnson Su, Johnson Sb, and Johnson Sl Distributions are useful for its data-fitting capabilities because it supports every possible combination of skewness and kurtosis.

•	The Generalized Log (Glog) distribution is useful for fitting data that are rarely normally distributed and often have non-constant variance, like biological assay data.

Comparing All Distributions

The All option fits all applicable continuous distributions to a variable. The Compare Distributions report contains statistics about each fitted distribution. Use the check boxes to show or hide a fit report and overlay curve for the selected distribution. By default, the best fit distribution is selected.

The Show Distribution list is sorted by AICc in ascending order.

If your variable contains negative values, the Show Distribution list does not include those distributions that require data with positive values. Only continuous distributions are fitted by this command. Distributions with threshold parameters, like Beta and Johnson Sb, are not included in the list of possible distributions.

Related Information

•	Statistical Details for Continuous Fit Distributions

•	Statistical Details for Fitted Quantiles

•	Statistical Details for Fit Distribution Options

Discrete Fit

Use the Discrete Fit options to fit a distribution (such as Poisson or Binomial) to a discrete variable. The available distributions are as follows:

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Poisson

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Binomial

•	Gamma Poisson

•	Beta Binomial

Related Information

•	Statistical Details for Discrete Fit Distributions

•	Statistical Details for Fitted Quantiles

•	Statistical Details for Fit Distribution Options

Fit Distribution Options

Each fitted distribution report has a red triangle menu that contains additional options.

Description of Fit Distribution Options
Diagnostic Plot	Creates a quantile or a probability plot. See Diagnostic Plot.
Density Curve	Uses the estimated parameters of the distribution to overlay a density curve on the histogram.
Goodness of Fit	Computes the goodness of fit test for the fitted distribution. See Goodness of Fit.
Fix Parameters	Enables you to fix parameters and re-estimate the non-fixed parameters. An Adequacy LR (likelihood ratio) Test report also appears, which tests your new parameters to determine whether they fit the data.
Quantiles	Returns the un-scaled and un-centered quantiles for the specific lower probability values that you specify.
Set Spec Limits for K Sigma	Use this option when you do not know the specification limits for a process and you want to use its distribution as a guideline for setting specification limits. Usually specification limits are derived using engineering considerations. If there are no engineering considerations, and if the data represents a trusted benchmark (well behaved process), then quantiles from a fitted distribution are often used to help set specification limits. See Statistical Details for Fit Distribution Options.
Spec Limits	Computes generalizations of the standard capability indices, based on the specification limits and target you specify. See Spec Limits.
Save Fitted Quantiles	Saves the fitted quantile values as a new column in the current data table. See Statistical Details for Fitted Quantiles.
Save Density Formula	Creates a new column in the current data table that contains fitted values that have been computed by the density formula. The density formula uses the estimated parameter values.
Save Spec Limits	Saves the specification limits as a column property. See Statistical Details for Fit Distribution Options.
Save Transformed	Creates a new column and saves a formula. The formula can transform the column to normality using the fitted distribution. This option is available only when one of the Johnson distributions or the Glog distribution is fit.
Remove Fit	Removes the distribution fit from the report window.

Diagnostic Plot

The Diagnostic Plot option creates a quantile or a probability plot. Depending on the fitted distribution, the plot is one of four formats.

Descriptions of Plot Formats

Plot Format

Applicable Distributions

The fitted quantiles versus the data

•	Weibull with threshold

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Gamma

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Beta

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Poisson

•	GammaPoisson

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Binomial

•	BetaBinomial

The fitted probability versus the data

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Normal

•	Normal Mixtures

•	Exponential

The fitted probability versus the data on log scale

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Weibull

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LogNormal

•	Extreme Value

The fitted probability versus the standard normal quantile

•	Johnson Sl

•	Johnson Sb

•	Johnson Su

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Glog

Descriptions of the Diagnostic Plot Options describes the options in the red triangle menu next to Diagnostic Plot.

Descriptions of the Diagnostic Plot Options
Rotate	Reverses the x- and y-axes.
Confidence Limits	Draws Lilliefors 95% confidence limits for the Normal Quantile plot, and 95% equal precision bands with a = 0.001 and b = 0.99 for all other quantile plots (Meeker and Escobar (1998)).
Line of Fit	Draws the straight diagonal reference line. If a variable fits the selected distribution, the values fall approximately on the reference line.
Median Reference Line	Draws a horizontal line at the median of the response.

Goodness of Fit

The Goodness of Fit option computes the goodness of fit test for the fitted distribution. The goodness of fit tests are not Chi-square tests, but are EDF (Empirical Distribution Function) tests. EDF tests offer advantages over the Chi-square tests, including improved power and invariance with respect to histogram midpoints.

•	For Normal distributions, the Shapiro-Wilk test for normality is reported when the sample size is less than or equal to 2000, and the KSL test is computed for samples that are greater than 2000.

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For discrete distributions (such as Poisson distributions) that have sample sizes less than or equal to 30, the Goodness of Fit test is formed using two one-sided exact Kolmogorov tests combined to form a near exact test. For details, see Conover 1972. For sample sizes greater than 30, a Pearson Chi-squared goodness of fit test is performed.

Related Information

•	Statistical Details for Fit Distribution Options

Spec Limits

The Spec Limits option launches a window requesting specification limits and target, and then computes generalizations of the standard capability indices. This is done using the fact that for the normal distribution, 3σ is both the distance from the lower 0.135 percentile to median (or mean) and the distance from the median (or mean) to the upper 99.865 percentile. These percentiles are estimated from the fitted distribution, and the appropriate percentile-to-median distances are substituted for 3σ in the standard formulas.

Related Information

•	Statistical Details for Fit Distribution Options