Even though mean refers to one of the measures of average in statistics, it is the most commonly used measure of central tendency, because it is based on all the data values and least affected by fluctuations in sampling.
Mean is of three types based on the method of computation:

1)
Arithmetic Mean

2)
Geometric Mean

3)
Harmonic Mean

## Definition of Mean in Statistics

Mean is a central measure which is used in describing a data distribution. Indeed a data set is described by mean along with a measure of dispersion like range or variance.

Mean in statistics is also called Expectation or expected value when computed for random variables associated with probability experiments. In such cases the mean is essential to compute the probabilities of the variable assuming a specific discrete value or a range of values in an interval.

## Arithmetic Mean

The arithmetic mean of a data set is often referred to as mean of the data set. It is the mathematical average of the data values.

The arithmetic mean of $n$ observations $x_{1},\ x_{2}$, .......$x_{n}$ is given by

$\bar{x}$ = $\frac{x_{1}+x_{2}+......x_{n}}{n}$ or using sigma notation = $\frac{\sum_{i=1}^{n}x_{i}}{n}$
When the data is represented by a frequency distribution where $f1,\ f2,......fn$ are the corresponding frequencies of data values $x_{1},\ x_{2},\ x_{n}$, then the arithmetic mean is computed using the formula
$\bar{x}$ = $\frac{\sum_{i=1}^{n}x_{i}f_{i}}{N}$ where $N$ = $f_{1} + f_{2} + .........+ f_{n}$.
While $\bar{x}$ denotes the mean of sample data the letter µ is used to represent the population mean.

The value of arithmetic mean is also used in other computations of statistic measures like variance.

Even though the arithmetic mean is easy to compute and based on all data values, it's value is influenced by outliners or extreme values in the data set and hence lead to misinterpretations.

Example: The AM $\bar{(x)}$ of the given five numbers $10, 12, 8, 14, 2$ is

$\frac{10+12+8+14+2}{5}$ = $9.2$

## Geometric Mean

Geometric Mean of n data values is the nth root of their product.

The formula for finding the geometric mean of $n$ values $x_{1},\ x_{2},\ .......x_{n}$ is given by

$GM$ = $(x_{1}\ x_{2}......x_{n})^{\frac{1}{n}}$ which can be written using Pi notation as $(\Pi_{i=1}^{n} x_{i})^{\frac{1}{n}}$
If $f_{1},\ f_{2},....f_{n}$ are the corresponding frequencies associated with the above data values the $GM$ is given by,

$GM$ = $(x_{1}^{f_{1}}x_{2}^{f_{2}}....x_{n}^{f_{n}})^{\frac{1}{N}}$

where $N$ = $f_{1} + f_{2} + .........+ fn$.

Example: The GM of the given five numbers $10, 12, 8, 14, 2$ is

$(10\times 12\times 8\times 14\times 2)^{\frac{1}{5}}$ = $7.689$

## Harmonic Mean

Harmonic mean of $n$ observations $x_{1}, x_{2}, .......x_{n}$ is defined by

$HM$ = $n(\Pi_{i=1}^{n}\ \frac{1}{x_{i}})^{-1}$ = $\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+......+\frac{1}{x_{n}}}$
Harmonic mean formula for a frequency distribution is as follows:

$HM$ = $\frac{N}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+......\frac{f_{n}}{x_{n}}}$

where $N$ = $f_{1} + f_{2} + .........+ f_{n}$.

Example: The HM of the given five numbers $10, 12, 8, 14, 2$ is

$\frac{5}{\frac{1}{10}+\frac{1}{12}+\frac{1}{8}+\frac{1}{14}+\frac{1}{2}}$ = $\frac{5}{0.8797}$ = $5.68335$