# Manuals/calci/KSTESTNORMAL

KSTESTNORMAL (XRange,ObservedFrequency,Confidence,DoMidPointOfIntervals,NewTableFlag)

• is the array of x values.
• is the frequency of values to test.
• is the mean Value.
• is the standard deviation of the set of values.
• 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 statistic 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.

## Example

A B
1 15 20
2 17 14
3 19 16
4 21 25
5 23 27
• =KSTESTNORMAL(A1:A5,B1:B5,19,3.16)
KOLMOGOROV-SMIRNOV TEST
DATA OBSERVED FREQUENCY CUMULATIVE OBSERVED FREQUENCY SN Z-SCORE F(X) DIFFERENCE
15 20 20 0.19608 -0.74915 0.22688 0.03081
17 14 34 0.33333 -0.07293 0.47093 0.1376
19 16 50 0.4902 0.6033 0.72684 0.23665
21 25 75 0.73529 1.27952 0.89964 0.16435
23 27 102 1 1.95574 0.97475 0.02525
TEST STATISTICS
ANALYSIS
MEAN 17.21569
STANDARDDEVIATION 2.95761
COUNT 5
D 0.23665
D-CRITICAL #ERROR

KS TEST TYPE NORMALDIST

• CONCLUSION - THE DATA IS NOT A GOOD FIT WITH THE DISTRIBUTION.

K-S Test