Overview

Introduction

This toolbox provides elementary tests for the evaluation of univariate measurement data that are typically recorded by scientists and engineers. These data have to be normally distributed for SampleSTAT's routines. SampleSTAT is focused on very small sample sizes (<10), but also offers routines for larger distributions (up to 50 values). It provides functions for calculating the range of values and the mean value with respect to a given statistical confidence level. It also provides tests for outliers and a method to test the data for normal distribution as well as a sophisticated Individual Value Plot.

These routines are applicable for normally distributed data ONLY.

Measures of Variation:

Calculates the stray areas of values and mean of normally distributed data. That gives you more information of the quality of your data as the sample standard deviation (s) of samples can do. s has an statistical confidence of 68%. Stray area (of values) can provide a confidence level of 95%, 99% and 99.9%. It determines the range in which the values are expected to fall within the specified statistical confidence level.

The trust area (stray area of the mean) gives you information of the confidence of your mean with a probability of 95%, 99% and 99.9%. It determines the range in which the mean stray within the specified statistical confidence level.

Outlier Tests:

Provides test to determine whether an extreme value is an outlier or not. Test are available for small and bigger distributions.

Tests for Normal Distribution:

All routines are applicable for normally distributed data only. To test for normal distribution, tests are provided.

These functions are good to extend the built-in functions mean(), stdev(), max(), min(), median() and can be extended by routines of the toolboxes "Stixbox" and "Distfun".

Functions - Measures of Variation

ST_strayarea:

Calculates the stray area (range of dispersion of the values) of univariate data for a statistical confidence level (95%, 99%, 99.9%) and level of significance (0.5, 0.01, 0.001), resp. It defines the range around the mean within which the values may vary at a given significance level.

ST_trustarea:

Calculates the trust area (range of dispersion of the mean) of univariate data and for a statistical confidence level (95%, 99%, 99.9%) and level of significance (0.5, 0.01, 0.001), resp. It defines the range around the mean may vary at a given significance level.

ST_studentfactor:

Determines the student factor for an amount of numbers and for a statistical confidence level (95%, 99%, 99.9%) and level of significance (0.5, 0.01, 0.001), resp.- service function for ST_strayarea and ST_trustarea.

Functions - Tests on Outliers

ST_grubbs:

The Grubbs test for outliers is eminently suitable for small sample sizes and widly used. It can detect just one outlier per data set. For samples having more than 30 observations the Generalized Extreme Studentized Deviate (ESD) test can be used. An iterative version of Grubbs (ST_grubbs_iterat) is also available. It identifies more than one outlier per data set. This version is sometimes used, but is not recommended as the statistical confidence deteriorates with each iteration step.

ST_esd:

The Generalized Extreme Studentized Deviate (ESD) test for outliers is eminently suitable for medium to big sample sizes (>25) and also widly used and accepted. It can detect more than one outlier per data set.

ST_nalimov:

This test on outliers provides a quick hint for all sample sizes and is often used in German and East European publications. This version can find more than one outlier per data set.

There is some scientific discussion about this test. Especially in the iterative version implemented here. It is thus recommended to use other tests instead of the Nalimov test (Grubbs for small or ESD for bigger sample sizes).

ST_deandixon:

This test for outliers is eminently suitable for small sample sizes. It can find just one outlier (on each side). It is not used frequently, today. For samples having more than 30 observations and more expected outliers the Generalized Extreme Studentized Deviate (ESD) test is useful.

ST_pearsonhartley:

For random samples larger than 30 objects possible outliers may be identified by using the significance thresholds of Pearson and Hartley. It can detect more than one outlier buit is not widley used anymore, today.

ST_outlier:

Provide two basic and standard tests on outliers based on population standard deviation (S.D.) or inter-quartile range (IQR). It should be applied to sample sizes larger than 10 better more than 25. For small sample sizes the Grubbs test is recommended. The iqr-options are robust against non-normally distributed data. ST_outlier with one of the iqr-option is the only test in sampleSTAT which can be used with skewed data.

Functions - Distribution tests

ST_shapirowilk:

The Shapiro-Wilk test is a statistical significance test that tests the hypothesis that the underlying population of a sample is normally distributed. It tests for normality exhibiting high power, leading to good results even with a small number of observations, especially in comparison to Chi-Square or Kolmogorov-Smirnov tests.

ST_ivplot

Individual value plot to examine and compare the distributions of sample data. In a scatter plot, a point is plotted for the actual value of each observation in a group. The spread of the distribution can be clearly seen.

Authors

Hani A. Ibrahim - hani.ibrahim@gmx.de

Bibliography

R. Kaiser, G. Gottschalk; "Elementare Tests zur Beurteilung von Meßdaten", BI Hochschultaschenbücher, Bd. 774, Mannheim 1972.

Lohringer, H., "Grundlagen der Statistik", Oct, 10th, 2012, http://www.statistics4u.info/

Shapiro, Wilk: "An Analysis of Variance Test for Normality", Biometrika, Vol. 52, No. 3/4. (Dec., 1965), pp. 591-611.

Grubbs, F. E. (1950). Sample criteria for testing outlying observations. Annals of Mathematical Statistics, 21(1), 27-58.

Grubbs, F. E. (1969). Procedures for detecting outlying observations in samples. Technometrics, 11(1), 1-21.

NIST/SEMATECH e-Handbook of Statistical Methods.

Rosner, B. (1983). Percentage Points for a Generalized ESD Many-Outlier Procedure. Technometrics, 25(2), 165-172.