ML | Raw and Central Moments

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 , 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 which occur with frequencies respectively and .

Moments about an arbitrary point A The moment of a variable X about any arbitrary point A on the observations is defined as:

For ungrouped data For grouped data where 

  Moment about any arbitrary point in Python – Consider the given data points. Following are the time (in hours) spent by 20 different persons at w3wiki portal every week.

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

The moment around origin A = 0 known as raw moment and is defined as:

For ungrouped data, For grouped data, where, 

Notes: 

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

  Raw moment in Python – 

The moments of a variable X about the arithmetic mean () are known as central moments and defined as:

For ungrouped data,For grouped data,where and 

Notes: 

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

  Relationship between Raw and Central moments –