Measure of Central Tendency: The commonly used measures of central tendency (or averages) are : (i) Arithmetic mean (AM) or simply mean (ii) Median

Arithmetic mean of individual observations or un-grouped data

Definition: If x_1, x_2, x_3, ..., x_n are n values of a variable X , then the arithmetic mean or simply the mean of the values is denoted by \overline{X} and is defined as

\overline{X} = \frac{x_1+x_2+x_3+... + x_n}{n} = \frac{1}{n} \sum \limits_{i=1}^{n} x_i

Here the symbol \sum \limits_{i=1}^{n} x_i denotes the sum of x_1+x_2+x_3+... + x_n

In other words, the arithmetic mean of a set of observations is equal to their sum divided by the total number of observations.

Properties of Arithmetic Mean

Property 1: If \overline{X} is the mean of n observations, x_1, x_2, x_3, ..., x_n , then \sum \limits_{i=1}^{n} ( x_i - \overline{X}) = 0 . i.e. the algebraic sum of deviations from mean is zero.

Property 2: If \overline{X} is the mean of n observations, x_1, x_2, x_3, ..., x_n then the mean of the observations x_1+a, x_2+a, ... , x_n+a i.e if each observation is increased by a , then the mean is also increased by a .

Property 3: If \overline{X} is the mean of x_1, x_2, x_2, ..., x_n then the mean of ax_1, ax_2, ax_3, ..., ax_n is a \overline{X} , where a is any number different from zero. i.e.  is each observation is multiplied by a non zero number a , then the mean is also multiplied by a .

Property 4: If \overline{X} is the mean of n observations, x_1, x_2, x_3, ..., x_n then the mean of the observations \frac{x_1}{a}, \frac{x_2}{a}, \frac{x_3}{a}, ..., \frac{x_n}{a} is \frac{\overline{X}}{a} where a is any number different from zero. i.e.  is each observation is divided by a non zero number a , then the mean is also divided by a .

Property 5: If \overline{X} is the mean of n observations, x_1, x_2, x_3, ..., x_n then the mean of the observations x_1-a, x_2-a, x_3-a, ..., x_n-a is \overline{X}- a , where a is any real number.

 

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 of the variable such that the number of observations above it is equal to the number of observations below it.

Median of an ungrouped data (or individual observations): If the values x_i in the raw data are arranged in order of increasing or decreasing magnitude, then the middle, most value in the arrangement is called the median.

First Arrange the observations (values of the variate) in ascending or descending order
of magnitude.

Then Determine the total number of observations, say, n

If  n is odd, then

Median = Value \ of \Big( \frac{n+1}{2} \Big)^{th} observation

If n is even, then

Median = \frac{Value \ of \Big( \frac{n}{2} \Big)^{th} \ observation + Value \ of \Big( \frac{n}{2} + 1 \Big)^{th} \ observation}{2}

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