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 
and 
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
from dollars to Euros – i.e. to multiply it by 0.8 ( 
), then compute values of the two statistics 
and for the new data 
; - (ii) multiply values of and (which are in dollars) by 0.8.
), then compute values of the two statistics 
;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 
for a few test samples and checking the property (1) .
See also the online short course Basic Concepts in Probability and Statistics