One-Sample Kolmogorov-Smirnov goodness-of-fit test

BRIEF DESCRIPTION:

The Kolmogorov-Smirnov (K-S) test is a goodness-of-fit measure for continuous scaled data. It tests whether the observations could reasonably have come from the specified distribution, such as the normal distribution (or poisson, uniform, or exponential distribution, etc.), so it most frequently is used to test for the assumption of univariate normality. The categorical data counterpart is the Chi-Square (χ²) goodness-of-fit test. The K-S test is a non-parametric procedure. 
 
SIMILAR STATISTICAL PROCEDURES:
  1. Adjusted Kolmogorov-Smirnov Lilliefors test (null hypothesis does not specify which normal distribution so no expected values and variance of the distribution)
  2. Shapiro-Wilke (for smaller samples and continuous scaled data)
  3. Chi-Square (χ²) goodness-of-fit test (for dichotomous or nominal data)
  4. Binomial goodness-of-fit test (for binary data)
  5. Anderson-Darling goodness-of-fit test (a modified Kolmogorov-Smirnov test for continuous data that gives more weight to the tails of the distribution)

CHARACTERISTICS OF THE VARIABLES
  • Dependent variable: continuous scaled variable (or a strong ordinal scale such as a 5-point scale or above).
  • Independent variable: not applicable
DATA ASSUMPTIONS
  1. The Kolmogorov-Smirnov test makes no assumption about the data distribution as it is non-parametric and thus distribution free.
  2. It assumes random sampling, continuous scaled data, and the hypothetical distribution specified in advance such as for the normal distribution with the expected sample mean and standard deviation.
 
WHERE TO FIND IN SPSS?
  1. ANALYZSE / NON-PARAMETRIC / 1 SAMPLE K-S
  2. ANALYZE / DESCRIPTIVE STATS / EXPLORE / PLOTS select “normality plots with test”” (both the K-S and the Shapiro-Wilke are produced in the output)
 
HOW TO REPORT THE FINDINGS?
If the Kolmogorov-Smirnov test is significant (e.g. p<.05) then it indicates that the distribution of our sample is significantly different from the distribution against which it is being compared, e.g. a normal distribution (therefore the sample distribution does not fit the assumption of normality).
  • Reporting example: “The customer satisfaction scores of D(100) = 0.10, p<.05, and does significantly deviate from normality”. NOTE: If p>.05 you don’t report the value (e.g. p>.05 or p=.129) but only indicate it as NS (non-significant)
  • Explanation of the above example: The D refers to the Kolmogorov-Smirnov D test statistic, the 100 in brackets is the degrees of freedom (df), the 0.10 is the actual K-S test statistic, and the p<.05 indicates an actual p-value which is less (e.g. .023) than our chosen confidence level of .05. 
 
FURTHER COMMENTS.
The Kolmogorov-Smirnov test is at its best for samples less than 100. For larger samples, it is safer to rely on histograms and other graphics because in large samples even very small differences easily show up as significant which lead to a Type I error.

_____________________________________________
/zza71

Additional Reading