ML | Raw and Central Moments

Last Updated : 25 May, 2022

Moments are a set of statistical parameters which are used to describe different characteristics and feature of a frequency distribution i.e. central tendency, dispersion, symmetry, and peakedness (hump) of the frequency curve. For Ungrouped data i.e. discrete data, observations on a variable X are obtained as $x_1, x_2, x_3, ...., x_n$ , For Grouped data i.e. continuous data, observations on a variable X are obtained and tabulated in K class intervals in a frequency table. The mid points of the intervals are denoted by $x_1, x_2, x_3, ...., x_n$ which occur with frequencies $f_1, f_2, f_3, ...., f_n$ respectively and $n=f_1, f_2, f_3, ...., f_n$ .

Class Intervals	Mid Points ( $x_i$ )	Absolute Frequency ( $f_i$ )
$c_1 - c_2$	$x_1 = (c_1 + c_2)/2$	$f_1$
$c_2 - c_3$	$x_2 = (c_2 + c_3)/2$	$f_2$
$c_3 - c_4$	$x_3 = (c_3 + c_4)/2$	$f_3$
…	…	…
…	…	…
$c_k_-_1 - c_k$	$x_k = (c_k_-_1 + c_k)/2$	$f_k$

Moments about an arbitrary point A The $r^t^h$ moment of a variable X about any arbitrary point A on the observations $x_1, x_2, x_3, ...., x_n$ is defined as:

For ungrouped data $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}(x_i - A)^r$ For grouped data $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}f_i(x_i - A)^r$ where $n = \sum_{i=1}^{k}f_i$

Moment about any arbitrary point in Python – Consider the given data points. Following are the time (in hours) spent by 20 different persons at GeeksforGeeks portal every week.

15, 25, 18, 36, 40, 28, 30, 32, 23, 22, 21, 27, 31, 20, 14, 10, 33, 11, 7, 13

Python3

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23, 22, 
        21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
 
# Arbitrary point 
A = 22
 
# Moment for r = 1
moment = (sum([(item-A) for item in time]))/len(time)

Raw Moments –

The $r^t^h$ moment around origin A = 0 known as raw moment and is defined as:

For ungrouped data, $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}x_i^r$ For grouped data, $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}f_i x_i^r$ where, $n = \sum_{i=1}^{k}f_i$

Notes:

-> We can find first raw moment ( $\mu^'_1$ ) just by replacing r with 1 and second raw moment ( $\mu^'_2$ ) just by replacing r with 2 and so on. -> When r = 0 the moment $\mu^'_0 = 1$ for both grouped and ungrouped data.

Raw moment in Python –

Python3

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23,
       22, 21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
 
 
# Moment for r = 1
moment = sum(time)/len(time)

Central Moments –

The moments of a variable X about the arithmetic mean ( $\overline{x}$ ) are known as central moments and defined as:

For ungrouped data, $\mu_r = \frac{1}{n}\sum_{i=1}^{n}(x_i-\overline{x})^r$ For grouped data, $\mu_r = \frac{1}{n}\sum_{i=1}^{n}f_i (x_i-\overline{x})^r$ where $n = \sum_{i=1}^{k}f_i$ and $\overline{x} = \sum_{i=1}^{k}f_ix_i$

Notes:

-> We can find first raw moment ( $\mu_1$ ) just by replacing r with 1 and second raw moment ( $\mu_2$ ) just by replacing r with 2 and so on. -> When r = 0 the moment $\mu_0 = 1$ , and when r = 1 the moment $\mu_1 = 0$ for both grouped and ungrouped data.

Python3

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23, 22,
       21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
 
# Mean 
A = sum(time)/len(time)
 
# Moment for r = 1
moment = (sum([(item-A) for item in time]))/len(time)