Measures of Central tendency

According to Prof Bowley averages are statistical constants which provide an idea about the concentration of values in the central part of the distribution. The five common measures of central tendency are :

1. Mean [Airthmetic mean]
2. Median
3. Mode
4. harmonic mean
5. Geometric mean

According to Prof. Yule following are the characterstics required to measure central tendency:

1. Central tendency should be rigidly defined
2. It should be comprehensive and easy to determine
3. Should be based on all observations
4. It should not be much affected by the fluctuation of sampling
5. It should not be affected by extreme values


Let us talk about these distributions in detail:

1. Arithmetic mean: Arithmetic mean of a set of observations can be calculated as their sum divided by the number of observations

In case f_i is the frequency of variable x_i then

distribution becomes

'

Properties of Arithmetic Mean

1. Algebric sum of deviations of a set of values from their arithmetic mean is zero
2. The sum of squares of the deviation of a set of values is minimum when it is taken about mean


Applications of Mean


Arithmetic means have many merits. it is rigidly defined,easy to understand and calculate,based upon all observations. It is open to mathematical treatment. Further among all the available averages it is least affected by fluctuation in sampling. That is why it is also called a stable average.

However it has some demerits. It cannot be determined just by inspection. Neither we can locate it graphically. Arithmetic mean cannot be used in case we are dealing with qualitative data. In case we have data dealing with intelligence,happiness, honesty we cannot use mean to measure center of tendency. Further we cannot use arithmetic mean even if single observation from data is missing or lost or wrong. Another demerits is that arithmetic mean is highly affected by the extreme values in case there are extreme values it will give distorted picture of distribution and may not represent the distributions. Arithmetic mean may lead to wrong conclusions even if details of data from which it is computed is not given. Just check out the following example:

Let exam marks for 2 students be as follows


Shyam

name	year 1	year 2	year 3
Rama	50	60	70
70	60	50

In both case of Rama and Shyam average marks(arithmetic mean) are 60 however from data we can analyze Rama is improving and Shyam is decreasing.

In case we have extremely asymmetrical (skewed distributions arithmetic mean cannot be suitable measure of location.