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                                  MEASURES OF CENTRAL TENDENCY AVERAGES

Mass data are colleted, classified, tabulated systematically. The data so presented is analyzed further to bring its size to a single representative figure. Descriptive statistics which describes the presented data in a single number. It is concerned with the analysis of frequency distribution or other from of presentation mathematically by which a few constants or representative numbers are arrive.

CONCEPT OF CENTRAL TENDENCY:

            One of the main characteristics of numerical data is central tendency. It is found that the observations tend to cluster around a point. This point is called the central value of the data. The tendency of the observations to concentrate around a central point is known as central tendency.

           The statistical measures which tell us the position of central value or central point to describe the central tendency of the entire mass of data is known as measure of central tendency or Average of first order 

Meaning of Statistical Averages:-

According to CLARK AND SCHAKADE “Average is an attempt to find one single figure to describe whole group of figures”

Objects of Statistical average:

An average is of great significance in all the fields of human knowledge, because it depicts the characteristics of  the whole group of data understudy.

      Following are the objectives of objectives of computing the statistical averages

1. To give or present the complex data in a simple manner and concise form.
2. To facilitate the data for comparative study of two different series.
3. To study the mass data from the sample
4. To establish relationship between the two series
5. To provide basis for decision making

To calculate the representative single value from the given data.

Requisites of a good and ideal Average.

Any statistical average to be good and ideal average must possess some of the characteristics as it is a single value representing a single value representing a group of values. Following are the requisite properties of a good and ideal average.

1.      It should be easily understood .

2.      It should be simple in calculation.

3.      It should be based on all the observations.

4.      It should not be unduly affected by the extreme values.

5.      It should be rigidly defined.

6.      It should be capable of further algebraic treatment.

7.      It should have sampling stability.

            Thus a statistical average should have all the above requisites to be an ideal and good average.

 Limitations of Averages:

Although an average is useful in studying the complex data and is very widely used in almost all the spheres of human activity, it is not without limitation that restrict scope and applicability. Following are the limitations of statistical averages.

1.      The extreme values, if any, will affect the averageble figure disproportionately.

2.      The composition of the data cannot be viewed with the help of the average.

3.      The average does not represent always the characteristics of individual items.

4.       The average gives only a representative figure of the mass, but fails to depict the entire picture of the data.

5.      An average may give us a value that does not exist in the data.

6.      some times an average might give very absurd results.

            In spite of the limitations, the statistical averages still are useful measures which play an important role in analyzing the mass data.

 TYPES OF STATISTICAL AVERAGES:

Broadly speaking, there are five types of statistical averages which are commonly used in practice. They are.

1.      Arithmetic Mean.

2.      Geometric Mean.

3.      Harmonic Mean.

4.      Median.

5.      Mode.

 Arithmetic Mean

            Arithmetic Mean is the most widely used measurement which represents  the entire data. Generally it is termed as an average to a layman. It is the quantity obtained by dividing the sum of the values of the items in a variable by their number.

            Arithmetic Mean may be two types

1.      Simple Arithmetic Mean.

            2. Weighted Arithmetic Mean.

MERITS OF ARITHMETIC MEAN	DEMERITS OF ARITHMETIC MEAN
It is easy to understand and easy to calculate. It is based on all the observations. It is capable of further algebraic treatment. It is rigidly defined and determinate . It is least affected by fluctuations of sampling. AM is as stable as possible.	The mean is unduly affected by the extreme items. It is un realistic. It may lead to a false conclusion. It cannot be calculated in case of open – end classes. It may not be represented in the actual data.

MEDIAN

Median may be defined as the value of that item which divides the series into two equal parts, one – half containing values greater than it and the other half containing values less that it. Therefore the series has to be arranged in ascending or descending order, before finding the median.

            The Median is a positional average and the term position refers to the place of a value in a series.

            The definitions of Median given by different authors are as follows.

 “ The Median is that value of the variable which divides the group into two equal parts, one part containing all  values greater and the other all values less than median. – By L.R. Conner.

Merits of Median

1. It  is easy to understand easy to compute.
2. It is quite rigidly defined.
3. It is eliminates the effect of extreme items.
4. It is amenable to further algebraic process.
5. Median can be calculated even from qualitative phenomena
6. Its value generally lies in the distribution.

Demerits

1. Typical representative of the observations cannot be computed if the distribution of items is irregular
2. It ignores the extreme items.
3. It is only a locative average, but not computed average.
4. It is more effected by fluctuations of sampling than in Mean.
5. Median is not amenable to further algebraic manipulation.

   QUARTILES

               The quartiles are also positional averages like the median. As the median value divides the entire distribution into two equal parts, the quartile divide the entire distribution into four equal parts.
               A measure while divides an array into four equal parts is known as quartile. Basically there are three such points – Q₁, Q₂ & Q₃ – termed as three quartiles. The first quartile (Q₁) or lower quartile has 25% of the items of the distribution below it and 75% of the items are greater than it.
               Incidentally Q₂ the second quartile, coincide with the median, has 50% of the observations above it and 50% of the observations below it.
               The upper quartile or third quartile (Q₃) has 75% of the items of the distribution below it and 25% of the items are above it.
               These quartiles are very helpful in understanding the formation of a series. They tell us how various items are spread round the median. Their special utility Rs. in a study of the dispersion of items.
               The working principle for  computing the quartile is basically the same as that of computing the median.
   Calculation of Quartiles
                                                   Q₁ = The size of (n + 1)^th item
                                                                                 4
                                                   Q₃ = The size of 3(n +1)^th item
                                                                                4
   MODE
   Mode is the most common item of a series. Mode is the values which occurs the greatest numbers of frequency in a series.Mode is defined as the value of the variable which occurs most frequently in a distribution.
   
   
   The chief feature of mode is that it is the size of that item which has the maximum frequency and is also affected by the frequencies of the neighboring items. 
   
   

   Inspection Method:-

               When there is a regularity and homogeneity  in the series, then there is a single mode which can be located at a glance by looking into the frequency column for having maximum frequency.
    GROUPING METHOD:
               When  there are irregularities in the frequency distribution or two or more frequencies are equal then it is not obvious that which one is the maximum frequency. In such cases, we use the method of grouping to decide which one may be considered as maximum frequency. That is we try to find out single mode by using grouping method. This method involves the following steps.
   a.       Prepare grouping table
   b.      Prepare analysis table
   c.       Find mode

   Merits of Mode

               Mode as  a measure of  central tendency has some merits.
   1. It is simple to understand and it is easy to calculate.
   2. Generally it is not affected by end values.
   3. It can be determined graphically & it can be found out by inspection.
   4. It is usually an actual value as it occurs most frequently in the series.
   5. It is the most representative average.
   6. Its value can be determined in an open end class interval without ascertaining the class limit.

   Demerits of mode

   1. It is not suitable for further mathematical treatment.
   2. It may not give weight to extreme items.
   3. In a bimodal distribution there are two modal classes and it is difficult to determine the value of the mode. and it is difficult to determine the value of the mode.
   4. It is difficult to compute, when are both positive and negative items in a series.
   5. It is not based on all the observations of a given series.
   6. It will not give the aggregate value as in average.