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

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

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

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