One-Sample Kolmogorov-Smirnov Test

The One-Sample Kolmogorov-Smirnov Test procedure compares the observed cumulative distribution function for a variable with a specified theoretical distribution, which may be normal, uniform, Poisson, or exponential. The Kolmogorov-Smirnov Z is computed from the largest difference (in absolute value) between the observed and theoretical cumulative distribution functions. This goodness-of-fit test tests whether the observations could reasonably have come from the specified distribution.

Starting with version 27.0, the Lilliefors test statistic can be used to estimate the p-value by using the Monte Carlo sampling for testing against a normal distribution with estimated parameters (this functionality was previously possible only through the Explore procedure).

Example: Many parametric tests require normally distributed variables. The one-sample Kolmogorov-Smirnov test can be used to test that a variable (for example, income) is normally distributed.
Statistics: Mean, standard deviation, minimum, maximum, number of non-missing cases, quartiles, Lilliefors test, and Monte Carlo simulation.

One-Sample Kolmogorov-Smirnov test data considerations

Data: Use quantitative variables (interval or ratio level of measurement).
Assumptions: The Kolmogorov-Smirnov test assumes that the parameters of the test distribution are specified in advance. This procedure estimates the parameters from the sample. The sample mean and sample standard deviation are the parameters for a normal distribution, the sample minimum and maximum values define the range of the uniform distribution, the sample mean is the parameter for the Poisson distribution, and the sample mean is the parameter for the exponential distribution. The power of the test to detect departures from the hypothesized distribution may be seriously diminished.; When certain parameters of the distribution have to be estimated from the sample, the Kolmogorov-Smirnov test no longer applies. In these instances, the Lilliefors test statistic can be used to estimate the p-value by using the Monte Carlo sampling for testing normality with mean and variance unknown. The Lilliefors test applies to the three continuous distributions (Normal, Exponential, and Uniform). Note that the test does not apply if the underlying distribution is discrete (Poisson). The test is only defined for one-sample inference when the corresponding distribution parameters are not specified.

Obtaining a One-Sample Kolmogorov-Smirnov test

This feature requires the Statistics Base option.

From the menus choose:
Analyze > Nonparametric Tests > Legacy Dialogs > 1-Sample K-S...
Select one or more numeric test variables. Each variable produces a separate test.
Optionally, select a test distribution method:

Normal

When selected, you can specify whether distribution parameter(s) are estimated from sample data (the default setting) or from custom settings. When Use sample data is selected, both the existing asymptotic results and Lilliefors significance correction based on the Monte Carlo sampling are used. When Custom is selected, provide values for both Mean and Std Dev.

Uniform

When selected, you can specify whether distribution parameter(s) are estimated from sample data (the default setting) or from custom settings. When Use sample data is selected, the Lilliefors test is used. When Custom is selected, provide values for both Min and Max.

Poisson

When selected, specify a Mean parameter value.

Exponential

When selected, you can specify whether distribution parameter(s) are estimated from the sample mean (the default setting) or from custom settings. When Use sample data is selected, the Lilliefors test is used. When Custom is selected, provide a Mean parameter value.
Optionally, click Simulation to specify Monte Carlo simulation parameters, click Exact to specify exact test parameters, or click Options for descriptive statistics, quartiles, and control of the treatment of missing data.

This procedure pastes NPAR TESTS command syntax.