# Central Moments and Raw Moments (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[X^r] that is the expectation of the random variable raised to the rth power. We use the symbol \mu_r' to denote the rth raw moment. Therefore we have the formula,

\mu_r' = \begin{cases} \int x^r f(x) dx,& \text{if the variable is continuous } \\ \sum x^r f(x) & \text{if the variables is discrete} \end{cases}

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

### Central Moments:

The rth central moment is defined as E[(X-E[X])^r] that is the expectation of the deviation random variable from the mean(=E[X]) raised to the rth power. We use the symbol \mu_r to denote the rth central moment. Therefore we have the formula,

\mu_r = \begin{cases} \int (x-E[x])^r f(x) dx,& \text{if the variable is continuous } \\ \sum (x-E[x])^r f(x) & \text{if the variables is discrete} \end{cases}

where f(x) is the probabilitity 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:

\mu_2 = \mu_2' - (\mu_1')^2 \mu_3 = \mu_3' - 3\mu_2'\mu_1' + 2(\mu_1')^3 \mu_4 = \mu_4' - 4\mu_3'\mu_1' + 6\mu_2'(\mu_1')^2 - 3(\mu_1')^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.

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