Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Measure of Central Tendency

Given a series of observations measured on a variable, there is a general tendency among the values to cluster around a central value, such clustering is called Central Tendency. Therefore any measure of the central tendency gives us an idea about the concentration of the values or observations of distribution in its central part.

 \to Measure of central tendency are also called measures of location of average.

Different measures of central tendency or averages are:

(i) Arithmetic Mean or Simple Mean

(ii) Median

(iii) Mode

(iv) Geometric Mean

(v) Harmonic Mean

(i) Arithmetic Mean or Simple Mean

In case of group or continuous frequency distribution  x_1,\  x_2,\ ...x_n are taken as the mid values of the class intervals.

Then Arithmetic mean or mean is:

 A.M=\overline{x}=\sum_{i=1}^{n}\ \dfrac{f_ix_i}{\sum_{i=1}^{n}}=\dfrac{1}{N}\sum_{i=1}^{n} f_ix_i ,

 N=\sum_{i=1}^{n}f_i

Weighted Mean -:

If  'w_i ' is the weight attached to the  i^{th} item that is  x_i\ (i=1,2,...,n) then the weighted mean is given by:

 \overline{x}_w=\dfrac{\sum_{i=1}^{n}\ w_ix_i}{\sum_{i=1}^{n}\ w_i} .

Merits of Arithmetic Mean

It is rigidly defined, easy to calculate, and easy to understand based on all observations, suitable for further mathematical treatment, and if all averages affected least by the fluctuation of sampling.

Demerits of Arithmetic Mean

Very much affected by extreme value cannot be used for open-end classes. Arithmetic Mean is not suitable for qualitative characteristics. It can’t be obtained if a single observation missing.

(ii) Median -:

Median of a distribution is the value of the variable which divides the distribution into two equal parts i.e. it is the value such that the number of observation above it is equal to the number of observation below it. Thus median is a positional average.

In the case of continuous frequency distribution, the cumulative frequency just greater than or equal to  \dfrac{N}{2} is called the median class and the median is obtained by the formula

Median  =M_d=l+\dfrac{h}{f}\left( \dfrac{N}{2}-c \right)

Where  l = lower limit of the median class

 h = magnitude/height of the median class

 f = frequency of the median class

 c = cumulative frequency preciding the median class

 N = total number of frequency

Merits of Median:

It is rigidly defined, easy to understand, and easy to calculate, not affected by extreme values, calculated for distribution with open-end classes, suitable for qualitative data.

Demerits of Median -:

In case of even number of observation median can’t be calculated exactly affected much by the fluctuate of sampling.

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