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Central Moments and Raw Moments in Statistics

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The raw moments and central moments in statistics are quantities that help us to determine the shape of a distribution. They can be used to calculate the skewness and kurtosis for a given set of data values.

The raw moments measure the deviation of the data values from 0 whereas, the central moments measure the deviation of the data values from the mean.

Raw Moments:

The rth raw moment is defined as E[Xr] which is the expectation of the random variable raised to the rth power. We use the symbol μr‘ to denote the rth raw moment. Therefore we have the formula,

Formula for Raw Moments

where f(x) is the probability distribution/probability mass function of the random variable X.

Central Moments:

The rth central moment is defined as E[(X-E[X])r which is the expectation of the deviation random variable from the mean(=E[X]) raised to the rth power.

We use the symbol μr to denote the rth central moment. Therefore we have the formula,

Formula for Central Moments

where f(x) is the probability distribution/probability mass function of the random variable X.

Relationship between the raw and central moments:

We have the following identities which give the values of the central moments in terms of the raw moments:

μ2 = μ2‘ – (μ1‘)2.

μ3 = μ3‘ – 3μ2μ1‘+ 2(μ1‘)3.

μ4 = μ4‘ – 4μ3μ1‘ + 6μ2‘(μ1‘)2 – 3(μ1‘)4.

Notice that the first raw moment is nothing but the mean of the distribution and the second central moment is the variance of the distribution.

Relationship between raw and central moments

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