Peter Filzmoser, Karel Hron, Clemens Reimann

Received 8 May 2009

Received in revised form 24 July 2009

Accepted 5 August 200

**abstract**

For almost 30 years it has been known that compositional (closed) data have special geometrical properties.In environmental sciences, where the concentration of chemical elements in different sample materials isinvestigated, almost all datasets are compositional. In general, compositional data are parts of a whole whichonly give relative information. Data that sum up to a constant, e.g. 100wt.%, 1,000,000 mg/kg are the bestknown example. It is widely neglected that the “closure” characteristic remains even if only one of all possible elements is measured, it is an inherent property of compositional data. No variable is free to vary independent of all the others. Existing transformations to “open” closed data are seldom applied. They are more complicated than a logtransformation and the relationship to the original data unit is lost. Results obtained when using classical statistical techniques for data analysis appeared reasonable and the possible consequences of working with closed data were rarely questioned. Here the simple univariate case of data analysis is investigated. It can be demonstrated that data closure must be overcome prior to calculating even simple statistical measures like mean or standard deviation or plotting graphs of the data distribution, e.g. a histogram. Some measures like the standard deviation (or the variance) make no statistical sense with closed data and all statistical tests building on the standard deviation (or variance) will thus provide erroneous results if used with the original data.

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