The raw and central moments in statistics are greatly useful in helping us understand how a given distribution looks like. Using the moments we can obtain an understanding of the spread, skewness, and shape of any given data set. The raw moments (denoted as \mu_r') can be calculated using the formula, \mu_r' = \frac{\sum x_i^r}{n}. The central moments (denoted as \mu_r) can be calculated using the formula, \mu_r = \frac{\sum (x_i-\bar{x})^r}{n}. We now list out some of the applications of moments in statistics.

**Uses/Applications of Moments:**

- The
*first raw moment is the arithmetic mean*which is highly useful as a measure of the central tendency of a given set of data values.It gives us an idea about the concentration of the values in the central part of the distribution. The average of a statistical series is the value of the variable that is representative of the entire distribution. - The
*second central moment*is another important constant called as the*variance*of the distribution. The variance of a set of data values gives us important information about the “spread” of the data values. It measures the degree of deviation of the data values from the mean of the distribution. A greater value of variance indicates a greater spread in the values of the data points. - The
*third central moment*is useful since it helps us to*calculate the skewness*of a distribution. The skewness of a distribution tells us whether the data is symmetrical or not. We say that the distribution is skewed if the curve drawn with the help of the given data is not symmetrical but stretched more to one side than to the other. The Skewness can be calculated using the third central moment using the formula, \text{Skewness }= \frac{\mu_3^2}{\mu_2^3} - The
*fourth central moment*is helpful since it helps us in*calculating the kurtosis*of the distribution. The kurtosis gives us an idea of the shape of a given distribution. It tells us whether the data is relatively “peaked” or “flat”. The Kurtosis can be calculated using the formula, \text{Kurtosis }= \frac{\mu_4}{\mu_2^2} - We can define a moment generating function which can allow us to uniquely identify a particular distribution.