Centering (statistics)

As mean centering , or short centering in which is statistics a transformation with shifting the values of a variable by the value of the average value understood this variable. A series of observations is called centered if . If the empirical standard deviation also applies , one speaks of a standardized observation series. To center a series of observations, subtract from each sample value. A typical example in the social sciences is the shift in the age of people by the mean of the age of a population . The advantage is that below-average values ​​are given negative signs. Furthermore, when using multivariate methods, the interpretation is easier because then it is not estimated at an unrealistic zero point. A special case of centering is studentization . ${\ displaystyle {\ overline {x}} = 0}$ ${\ displaystyle s = 1}$${\ displaystyle {\ overline {x}}}$

Application examples

The calculation of the empirical variance requires that the arithmetic mean is first determined from the observation series and then the observation series must be used again in order to form the deviations of the feature values from the arithmetic mean. This process is called centering the series of observations. However, the empirical variance can also be represented in a non-centered form by means of the displacement theorem. ${\ displaystyle {\ tilde {s}} ^ {2}: = {\ frac {1} {n}} \ sum \ nolimits _ {i = 1} ^ {n} \ left (x_ {i} - {\ overline {x}} \ right) ^ {2}}$ ${\ displaystyle {\ overline {x}}}$${\ displaystyle \ nu = \ left (x_ {i} - {\ overline {x}} \ right)}$

Analogously, a centered random variable is obtained if the expected value of a random variable is subtracted, i.e. H. forms. A centered random variable always has an expected value of zero, da . The term is also called the first central moment . ${\ displaystyle X}$ ${\ displaystyle \ operatorname {E} (X)}$${\ displaystyle (X- \ operatorname {E} (X))}$${\ displaystyle \ operatorname {E} (X- \ operatorname {E} (X)) = \ operatorname {E} (X) - \ operatorname {E} (X) = 0}$${\ displaystyle \ operatorname {E} (X- \ operatorname {E} (X))}$