Descriptive Statistics – Absolute Moments

In the earlier notes of descriptive statistics, we have seen raw and central moments and how raw and central moments are related to each other. In this note, we will cover absolute moments.

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1 Absolute Moments

1.1 Absolute moment for discrete data

1.2 Absolute moment for continuous data

2 References

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Absolute Moments

Data are usually categorized into two categories, discrete and continuous, and both these types of data are handled differently. In the below sections, we will see how to deal with discrete and continuous data. The $r^{th}$ moment of a variable X around mean is obtained as follows:

Absolute moment for discrete data

The $r^{th}$ absolute moment based on observations $x_1, x_2, x_3, ...., x_n$ for ungrouped or discrete data around any arbitrary value is defined as follows:

$|\mu|_r$ = $\frac{1}{n} \sum_{i = 1}^n | x_i - \bar{x}|^r$

$\bar{x} = \sum_{i=0}^n x_i$ , where i = 0 to n.

For r = 1, it give absolute deviation around the mean. It is defined as follows:

$|\mu|_1$ = $\frac{1}{n} \sum_{i = 1}^n | x_i - \bar{x}|$

For r = 2, it gives absolute mean deviation and it is defined as follows:

$|\mu|_2$ = $\frac{1}{n} \sum_{i = 1}^n | x_i - \bar{x}|^2$

Absolute moment for continuous data

Suppose we have observations on a variable X and having k class intervals such as $e_1 - e_2, e_2 - e_3, .... e_{k-1} - e_k$ in a frequency distribution table. The midpoint value is obtained for each interval is as follows:

$x_i = \frac{e_i + e_j}{2}$ , where i < j

and associated absolute frequency is $f_i$ for the class interval $e_i - e_j$ . The $f_i$ represents a number of observations belong to the class interval $e_i - e_j$ . The sum of all the absolute frequencies must be n = $\sum_{i=1}^k {f_i}$ .