Statistical Glossary
Scale Invariance (of Measures):
Scale invariance is a property of descriptive statistics . If a statistic 
is scale-invariant, it has the following property for any sample 
and any non-negative value 
:
|
(1) |
or, in mathematically equivalent form
|
In other words, if a statistic 
is scale-invariant, then multiplication of all elements 
of the sample by an arbitrary non-negative value results in multiplication of the resultant value 
of the statistic by the same value .
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 .
Scale invariance is an important practical requirement imposed on many classes of statistical measures. For example, consider data 
on the selling price of a particular car model at 1000 ( 
) locations across the country. A market researcher computes the values of two statistics, say, 
and 
for these data. Statistic is a measure of central tendency and is a measure of dispersion. In other words, reflects a "typical" value of the price, and reflects the magnitude of variation of the prices around the typical values. The researcher obtained 
- (i) convert every price
from dollars to Euros - i.e. to multiply it by 0.8 ( 
and for the new data 
; - (ii) multiply values of and (which are in dollars) by 0.8.
;Scale invariance of statistics and , as defined by expression (1) , guarantees that in both cases the result will be the same.
A general practical recommendation: if a statistic takes on values in the same units as the initial data , it is strongly recommended to use only scale invariant statistics. Scale invariance of a statistic may be checked either from analytical considerations or, if this is difficult, at least numerically - by computing values and 
See also the online short course Basic Concepts in Probability and Statistics