Measures of Central Tendency

Generally, the term average is related to a measure of central tendency.

It is nothing but a value to represent a series of data/observations.


There are 3 measures of central tendency :

1.Mean

2.Median

3.Mode

_________________________________________________________________________

Before we get into a detailed description, lets know about the relationship among these three measures.There are 2 cases where the relationship can be explained:
     
Case 1. When the distribution of data is symmetrical. In this case all the three measures are equal. In other words ;

                                            MEAN = MEDIAN = MODE

Case 2. When the distribution of data is asymmetrical. In this case the measures aren't equal. So, an equation is given ;

                                          MODE = 3MEDIAN - 2MEAN  

And this equation is known as the empirical relationship between the mean, the median and the mode.

________________________________________________________________________

Now let's go into a detailed description about the measures.

Mean is the result you get by adding two or more amounts together and dividing the total by the number of amounts .

It is denoted by  .

$$
{\displaystyle {\bar {x}}}

          There are 5 main types of mean :

          1.Arithmetic mean : It is the quantity obtained by summing two or more numbers or variables and then dividing by the number of numbers or variables. It is very much similar to the normal mean .

Formulae :

For individual series :

              = ∑x / N  

For discrete series :

              = ∑fx / N

For continuous series :

           = ∑fm / N    

                     where ;m is the mid value.

           
           2.Combined mean : It is the arithmetic mean of the means of two or more series.

Formula :

            c =  [(1)(N1) + (2)(N2 )] / N1 + N2

           

            3.Harmonic mean : It is the reciprocal of arithmetic average of the reciprocal of values of items in the variable.

Formulae :           

For individual series :

            = N / ∑(1/x)

For discrete series :

         = N / ∑f(1/x)

For continuous series :

           = N / ∑f(1/m)

                            where ;m is the mid value.

          
            4.Geometric mean : It is defined as the Nth root of the product of N number of observations in a series of data.

Formula :           

          GM =  N√x1 * x2 * x3 * ..........xn
                   where; n is the number of observations.

         

            5.Weighted mean: When calculating the arithmetic mean, the importance of all the items are considered to be equal. However, there may be situations in which all the items under considerations are not of equal importance. So we use the weighted mean formula to calculate the mean along with the importance or weightage of each observation in the data.

Formula:
           = ∑Wx / ∑W

Now lets look into the merits and the demerits of the mean ;

Merits :
1.It is simple and easy to apply.
2.It is based on all the observations in the data.

Demerits :
1.It is affected by extreme observations.
2.If any one of the variables are missing, mean cannot be computed.

_________________________________________________________________________

Median is the middle value of the series, when the data is arranged in an ascending or a descending order. It is denoted by M.
Before you apply the formula, make sure that the data is arranged in an ascending order.

Formulae:
For individual series:
         M = [(N+1)/2]th item
For discrete series:
         M = Size of [(N+1)/2]th item

For continuous series:

        We find the median class first .
        Median class = Class that has the item of the size (N/2)th item
        M = L1 + {[ (N/2) - cf ] / f } * i
        where; L1 = the lower limit of the median class 
                    cf  = the cumulative frequency of class preceding median class
                      f  = Frequency of median class.
                      i  = the difference of upper limit and the lower limit.

Now lets look into the merits and the demerits of the median ;

Merits :
1.It can be computed in case of frequency distribution with open ended classes.
2.It can be determined graphically.

Demerits :
1.It is not based on all the observations of the data.
2.It is affected by the fluctuation of sampling.

_________________________________________________________________________

Mode is the value which occurs the most frequently in a series. It is denoted by Z.

Formulae:
For individual series :
     The terms are arranged in any order. Ascending or Descending. If each term of the               series is occurring once, then there is no mode, otherwise the value that occurs                     maximum times is the Mode.
For discrete series :
     Here, the variable(x) which has the highest frequency will be the mode.
For continuous series :
     Here, we use a rigid formula;
     Z = L1 + ( f1 - f0 ) / [ 2(f1) - f0 - f2 ] * i
        where ; L1 = the lower limit of the modal class
                      f1 = the highest frequency 
                      f0 = the preceding frequency of f1
                      f2 = the successive frequency of f1
                       i  = the difference of upper limit and the lower limit.

Now lets look into the merits and the demerits of the mode ;

Merits :

1.It is not affected by extreme values.

   It can be obtained even if the extreme values are not known.

2.Mode can be located on the graph also.

Demerits :

1.It is not based upon all the observation. 
2.It is affected to a greater extent fluctuations of sampling. 

_________________________________________________________________________
If you wish to know the basics of statistics, use the following link :

https://basicmathematix.blogspot.in/2018/04/basics-of-statistics.html