Algebraic formula for the variance

The shift theorem (also known as Steiner's theorem ) is a calculation rule for determining the sum of squared deviations from the arithmetic mean.

In short, he says:

The shift theorem for example, facilitates the calculation of the sample variance when the measured values ​​is charged continuously. It is then not necessary to store all (memory), yet again all summands go through ( computing time ). Using this formula with limited computational accuracy, it may cause a certain numerical extinction if significantly greater than the variance.

  • 2.1 random variable 2.1.1 variance
  • 2.1.2 covariance

Explanation on the case of a finite sequence of numbers: the sample mean

The shift theorem is presented first at the simplest case: There is a sequence of real numbers xi, where, for example, a random sample. It is formed from the arithmetic mean of these values, the sum Q of the squared deviations of the individual values ​​xi:

In which

The arithmetic mean of the numbers.

The displacement rate is given by

Example

As part of the quality assurance coffee packs are weighed continuously. For the first four packets to the values ​​obtained (in g) xi

Is the weight average

It is

For the application of the shift theorem, we can calculate

And

One can thus, for example, determine the corrected sample variance:

In the example

Now comes another bubble in the sample, it is sufficient to recalculate the sample variance using the shifting theorem, only the values ​​and recalculate. In the fifth parcel weight will measure 510 g. Then:

The sampling variance of the new, larger sample is then

Applications

Random variable

Variance

The variance of a random variable

Can be specified with the shift theorem as

This result is also referred to as a set of king -Huygens. It follows from the linearity of expectation:

  • Is then obtained for a discrete random variable X with the characteristics and the associated probability of
  • For a continuous random variable X with the forms Ω = { x | x ∈ R} and the corresponding density function f ( x) is

Obtained here with the shift theorem

Covariance

The sample covariance of two random variables X and Y can be stated as E ( (Y )) ( XE ( X) ) · ( YE).

For discrete random variables is obtained for

According to the above

With f ( xj, yk ) as a joint probability that X = Y is = xj and yk.

For continuous random variables resulting f ( x, y) as the common density function of X and Y at the position x and y for the covariance

According to the above

Sample covariance

For the sample covariance of two features x and y are required

This results in the shift theorem

The corrected sample covariance is then calculated as

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Sample (statistics) Christiaan Huygens
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