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Scale Invariance (of Measures)

Statistical Glossary

Scale Invariance (of Measures):

Scale invariance is a property of descriptive statistics . If a statistic Math image is scale-invariant, it has the following property for any sample Math image and any non-negative value Math image :

Glossary Math (1)

or, in mathematically equivalent form

Glossary Math

In other words, if a statistic Math image is scale-invariant, then multiplication of all elements Math image of the sample by an arbitrary non-negative value Math image results in multiplication of the resultant value Math image of the statistic Math image by the same value Math image .

Measures of central tendency and measures of dispersion are normally scale-invariant, as well as most other measures that have values in the same units as the initial data Math image .

Scale invariance is an important practical requirement imposed on many classes of statistical measures. For example, consider data Math image on the selling price of a particular car model at 1000 ( Math image ) locations across the country. A market researcher computes the values of two statistics, say, Math image and Math image for these data. Statistic Math image is a measure of central tendency and Math image is a measure of dispersion. In other words, Math image reflects a “typical” value of the price, and Math image reflects the magnitude of variation of the prices Math image around the typical values. The researcher obtained Math image and Math image US dollars. Now, he is asked to report these values in Euro (suppose that 1 dollar = 0.8 Euro). There are two reasonable methods of transition from the initial units (dollars) to new units (euro):

  • (i) convert every price Math image from dollars to Euros – i.e. to multiply it by 0.8 ( Math image ), then compute values of the two statistics Math image and Math image for the new data Math image ;
  • (ii) multiply values of Math image and Math image (which are in dollars) by 0.8.

Scale invariance of statistics Math image and Math image , as defined by expression (1) , guarantees that in both cases the result will be the same.

A general practical recommendation: if a statistic Math image takes on values in the same units as the initial data Math image , it is strongly recommended to use only scale invariant statistics. Scale invariance of a statistic Math image may be checked either from analytical considerations or, if this is difficult, at least numerically – by computing values Math image and Math image for a few test samples and checking the property (1) .

See also the online short course Basic Concepts in Probability and Statistics

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