Manuals/calci/KSTESTNORMAL

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

• $xRange$ is the array of x values.
• $ObservedFrequency$ is the frequency of values to test.
• $Mean$ is the mean of set of values.
• $Stdev$ is the standard deviation of the set of values.
• $Test$ is the type of the test.
• $Logicalvalue$ 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 $x_{i}$, where i=1,...,n.
• 2.Smallest empirical estimate of fraction of points falling below $x_{i}$, and computed as ${\frac {(i-1)}{n}}$ for i=1,...,n.
• 3.Largest empirical estimate of fraction of points falling below $x_{i}$ and computed as ${\frac {i}{n}}$ for i=1,...,n.
• 4.Theoretical estimate of fraction of points falling below $x_{i}$ and computed as $F(x_{i})$, 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:$D_{n}=Sup_{x}|F_{n}(x)-F(x)|$where $sup_{x}$ is the supremum of the set of distances.
• $F_{n}(x)$ is the empirical distribution function for n,with the observations $X_{i}$ is defined as:$F_{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}I_{X_{i}\leq x}$where $I_{X_{i}\leq x}$ is the indicator function, equal to 1 if $X_{i}\leq x$ and equal to 0 otherwise.

Example

Spreadsheet

	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.

Related Videos

K-S Test

See Also

• LEVENESTEST
• MOODSMEDIANTEST
• RIEMANNZETA
• MANNWHITNEYUTEST

References

• Kolmogorov-Smirnov Goodness of Fit test