Manuals/calci/KSTESTNORMAL

KSTESTNORMAL(XRange,ObservedFrequency,Mean,Stdev,Test,Logicalvalue)


  • is the array of x values.
  • is the frequency of values to test.
  • is the mean of set of values.
  • is the standard deviation of the set of values.
  • is the type of the test.
  • is either TRUE or FALSE.

Description

  • This function gives the test statistic of the K-S test.
  • K-S test is indicating the Kolmogorov-Smirnov test.
  • It is one of the non parametric test.
  • This test is the equality of continuous one dimensional probability distribution.
  • It can be used to compare sample with a reference probability distribution or to compare two samples.
  • This test statistic measures a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples.
  • The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples.
  • It is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.
  • This test can be modified to serve as a goodness of fit test.
  • The assumption of the KS test is:
  • Null Hypothesis(H0):The sampled population is normally distributed.
  • Alternative hypothesis(Ha):The sampled population is not normally distributed.
  • The Kolmogorov-Smirnov test to compare a data set to a given theoretical distribution is as follows:
  • 1.Data set sorted into increasing order and denoted as  , where i=1,...,n.
  • 2.Smallest empirical estimate of fraction of points falling below  , and computed as   for i=1,...,n.
  • 3.Largest empirical estimate of fraction of points falling below   and computed as   for i=1,...,n.
  • 4.Theoretical estimate of fraction of points falling below   and computed as  , where F(x) is the theoretical distribution function being tested.

5.Find the absolute value of difference of Smallest and largest empirical value with the theoretical estimation of points.

  • This is a measure of "error" for this data point.
  • 6.From the largest error, we can compute the test statistic.
  • The Kolmogorov-Smirnov test statitic for the cumulative distribution F(x) is: ,where   is the supremum of the set of distances.
  •   is the empirical distribution function for n,with the observations   is defined as: ,where   is the indicator function, equal to 1 if   and equal to 0 otherwise.