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Shift Invariance (of Measures)

Shift Invariance (of Measures):

Shift invariance is a property of descriptive statistics . If a statistic Math image is shift-invariant, it possesses the following property for any data set Math image :

or, in equivalent form

In other words, if a statistic Math image is shift-invariant, then addition of an arbitrary value Math image , positive or negative, to all elements of the sample results in the increase/decrease of S by the same amount Math image .

Measures of central location – like the mean , the median , the trimmed mean , the mode , the weighted mean – are shift invariant.

Shift invariance is an important requirement imposed on many classes of statistical measures, e.g. measures of central tendency and measures of dispersion . If the beginning of the scale used to measure the primary data Math image is quite arbitrary (e.g. time, or a spatial coordinate), then it is strongly recommended to use only shift-invariant measures.

For example, consider Math image values Math image of the difference between a reference price Math image of a particular car model (e.g. the suggested retail price of that car model) and the actual sale prices of this model for 1000 recent sales across the country. The question of interest is the typical value of the price differences Math image .

Suppose a researcher has chosen a measure Math image of central tendency (like the trimmed mean or mode ), calculated its value Math image for the data at hand, and obtained the following result: Math image (dollars) – i.e. price Math image is higher than the prices at other dealers by a typical value of 500 dollars. The owner decides to set a new price Math image , say, 200 dollars lower ( Math image , or Math image ). What about the value of Math image now? There are two reasonable methods to calculate the value of statistic Math image for new data Math image :

(i) Subtract 200 (dollars) from each Math image price difference: Math image and, then, compute the new value Math image of statistic Math image for these new data Math image :

(ii) Subtract 200 (dollars) from the old value of Math image i.e. Math image .

For a measure that is shift invariant, the result in both cases is always the same – i.e. for any data set and any value of the shift Math image .

A general practical recommendation: if the data at hand are measured on a scale with the zero point chosen quite arbitrarily (normally such data may take on both negative and positive values), and the quantity of interest is expressed in the same units as the data Math image themselves, then it is preferable to use statistics that are shift invariant.

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