Unit 2 : 2.1 : Measures of Central Tendency: Averages

 

Measures of Central Tendency (Averages)©

    Statistical data is huge. From this a single figure is calculated, which will represent the data. It will natural to expect that, the representative figure i.e. measure should be more or less close to the most of the members of the data. Hence it lies more or less at the center of the data.  It is therefore called as ‘Measure of Central Tendency’.

 Definition

A single figure which represents whole data and lies at center of data la called as ‘Measure of Central Tendency’ or 'Average'.

Types

There are three main and in all five types of averages.

i) Arithmetic Mean (A.M.) (ii) Geometric Mean (G.M.) iii) Harmonic Man (H.M.)

iv) Median

v) Mode

The first three are called as’ Mathematical averages’ & last two are called as ‘Positional averages’.

Requirements (Requisites/Characteristics) of an ideal (good) Measure of Central Tendency (Average)

i) It should be rigidly defined i.e. It should have a definite value.

ii) It should be easy to understand and easy to calculate.

iii) It should be based on all items.

iv) It should not be affected by extreme values.

v) It should be suitable for further calculations.

vi) It should have sampling stability

 i) Arithmetic Mean (A.M.)

It is also popularly known as "Average"

 Def - Arithmetic Mean of a set of observations is defined as, the sum of all observations divided by the total number of observations and is denoted by X-bar. Thus if X1, X2,..,Xn are n  observations then their a.m. is, 

If we are given a discrete frequency distribution with values X1, X2,..,Xn and frequencies f1, f2,..,fn then the a.m. is, 

If we are given a continuous (grouped )frequency distribution with classes having midpoints m1, m2,..,mn and frequencies f1, f2,..,fn then the a.m. is,

Effect of change of origin on A.M.:

 If we change the origin to some convenient value A  i.e., if we write di-xi-A, then

multiplying both the sides by fi

fidi = fi (xi-A) =fixi - fi. A 

taking summation and dividing by N on both sides, we get


Thus a mean is affected by change of origin.

Effect of change of origin & change of scale on A.M.:

If in addition to the change of origin, we change the scale i.e. if we write di'=xi-A/h, 

then xi =A+h di', where A is some arbitrary point & h is the common magnitude of the class interval.

Then multiplying both the sides by fi,  fixi-fi A +h* fidi'

 taking summation and dividing by N on both sides, we get

Σ fixi/N=Σ fi*A/N +h/N Σ fidi'

Thus, a mean is affected by change of origin as well as change of scale.

 Properties of A.M.:

 i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. i.e. if fi/xi, i = 1,2,-, n is the frequency distribution, then Σ fi (xi-X bar) = 0, X-bar being the mean of the distribution.


                                                 
                                                




                                                          
                         


                    


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