Measures of central tendency

    It is a single value within the range of data which represents a group of individuals value in a simple and concise manner. So we get quick understanding of the general size of the individuals in the group. Since the values lies within the range of the data.

Some definitions for Measures of central tendency:

    " An average may be thought of as a measure of central value " - John I. Griffin

    " The inherit inability of human to group in its entirely a large body of numerical data compels us to seek relatively few constants that will adequately show the data " - R. A. Fisher

    " Averages are statistical constants which enable us to comprehend in a single effort the significance of the whole " - A. L. Bowley

Properties of central tendency:

• It should be rigidly defined.
• It should be simple and easy to calculate.
• It depends on all the observations.
• It should be suitable for further mathematical treatment.
• It should be easily located from the graph.
• It should not be much affected by the extreme observations.

Mathematical averages:

1.  Arithmetic mean
2. Geometric mean
3. Harmonic mean

Positional averages: 

1. Median
2. Mode
3. Quartiles
4. Quintiles
5. Octiles
6. Deciles
7. Percentiles.

Commercial averages: 

1. Moving average
2. Progressive average
3. Composite average.

The five measures of central tendency that uses very commonly. 

1. Arithmetic mean 
2. Median
3. Mode
4. Geometric mean
5. Harmonic mean 

    Here the clear explanation,

    

Arithmetic Mean: It is set of observations is their sum divided by the number of observations.

                                                      

                   Where i = 1, 2, 3, ...... n

Merits for mean:

• It is rigidly defined.
• It is easy to calculate.
• It is based upon all observations.
• It is amenable to algebraic treatment. 
• Of all averages, mean is affected least by fluctuations of sampling. This property is sometimes described by saying that mean is a stable average

    Demerits for mean:

• It cannot be determined by inspection nor it can be located graphically.
• Mean cannot be used if we are dealing with qualitative characteristics which cannot be measured quantitatively.
• Mean cannot be obtained if a single observation is missing or lost or illegible unless we drop it out and compute the mean of the remaining values.
• Mean cannot be calculated if the extreme class is open.
• In extremely asymmetrical distribution, usually mean is not a suitable measure of location.

Problems for mean:

1. Calculate the mean of the following data 50, 76, 44, 48, 57, 59, 63, 45, 48, 30.

2. Find the mean of the following data. 

    Given data, 

Median: Median of a distribution is the value of the variable which divides it into two equal parts. It is the value which exceeds and is exceeded by the same number of observations. The median is a positional average.

                        

    

   Merits of median: 

• It is rigidly defined.
• It is simple and easy to calculate.
• It can be located from the graph.

    Demerits of median:

• It does not depend on all the observations.
• It is not suitable for further mathematical treatment.
• In some situations, median is affected by extreme observations.

Problems for median:

1. Find the median for following frequency distribution.

     Given data,

                                            

      The cumulative frequency just greater than N/2 is 47.5, The value of x corresponding to 47.5 is 5.

2. Find the median for following distribution

                   

    Given data,

             

            The c.f just greater than N/2 is 19 is 33 and the corresponding class is 500 - 600.

                

                                                               = 506.66

Mode: Mode is the value which occurs most frequently in a set of observations and around which the other items of the set cluster densely. In the other words, mode is the value of the variable which is predominantly in the series.

      

    

Merits of mode:

• It is rigidly defined.
• It is simple and easy to calculate.
• It is based on all the observations.
• It can be located from graph.
• It is not affected by the extreme observations.

Demerits of mode:

• It is not rigidly defined.
• It does not depend on all the observations.
• It is not suitable for further mathematical treatment.

Problems for mode:

1. Find the mode for following

                                               

    Given data, 

    

    Here the maximum frequency is 14, thus the mode is 8.

2. Find the mode for the following distribution.

         Given data,

 

                                   

            

                                                                         = 46. 67

Geometric mean: It is a set of n observations is the nth root of their product.

                

    Merits for GM:

• It is  rigidly defined.
• It depends on all the observations.
• It is not affected by the extreme observations.

    Demerits of GM:

• The calculation of GM is not simple and easy.
• It cannot be located from the graph

Problem of geometric mean:

1. Find the GM of 

 

                    

   

               

Harmonic mean: Harmonic mean of a number of observations none of which is zero, is the reciprocal of the AM of the reciprocals of the given values.

            

    Merits of HM:

• It is rigidly defined.
• It depends on all the observations.
• It is not affected by the extreme observations.

Demerits of HM:

• It is not simple and easy.
• It cannot be located from the graph.

Problem for the HM:

1. Calculate the HM for the following continuous frequency distribution.

    

Quantiles: The values of the variable x corresponding to (N +1/4)th, (N + 1/2)th and 3(N+1)/4th items of an ordered discrete series are the values of Q1, Q2, and Q3 respectively. The position of the required item can easily be adjudged with the help of cumulative frequencies.

Deciles: Similar to quartiles, the values of the variable x corresponding to the i(N+1)/10th item for i = 1,2,.......9 of an ordered discrete series is Di. The position of i(N+1)/10th item can be located with the help of cumulative frequencies.

Percentiles: Just like deciles, the value of the variable x corresponding to i(N+1)/100th item for i=1,2,....,99 of an ordered series is Pi.i(N+1)/100th item in the series can easily be placed with the help of cumulative frequencies.