Descriptive Statistics – Absolute Moments

By / February 27, 2022

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.

Table of Contents hide
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}.

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

n = \sum_{i=1}^k f_i, and \bar{x} = \frac {1}{n} \sum_{i=1}^k f_i x_i

References

  1. Descriptive Statistic, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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