## Arithmetics Mean

The arithmetic mean (or simple mean) of a set of observations is obtained by dividing the sum of the values of observations by the number of observations.

**(i) Arithmetic Mean for unclassified (Ungrouped or Raw) Data**

**(ii) Arithmetic Mean for Discrete Frequency Distribution**

Variate (x) |
x_{1} |
x_{2} |
… | x_{n} |

Frequency (f) |
f_{1} |
f_{2} |
… | f_{n} |

.\bar{x} = {x_1f_1+x_2f_2+...+x_nf_n\over f_1+f_2+...+f_n}

or, \bar{x} ={ \sum_{i=1}^{n}{x_if_i} \over \sum_{i=1}^{n}{f_i} }

(iii) **Arithmetic Mean for Grouped Frequency Distribution**

Class mark (y) |
y_{1} |
y_{2} |
… | y_{n} |

Frequency (f) |
f_{1} |
f_{2} |
… | f_{n} |

**(a) From Direct Method:** \bar{x} ={ \sum_{i=1}^{n}{x_if_i} \over \sum_{i=1}^{n}{f_i} }

**(b) Shortcut or Deviation:** \bar{x} = A + { \sum_{i=1}^{n}{d_if_i} \over \sum_{i=1}^{n}{f_i} }×h

Where

- A = Assumed mean
- d = deviation
- h = class width
- x = class mark or mid-value

**(c) Step-deviation: **\bar{x} = A + { \sum_{i=1}^{n}{u_if_i} \over \sum_{i=1}^{n}{f_i} }×h

Where

- A = Assumed mean
- u = x - A\over h

**(d) Combined Mean: **If \bar{x}_1 and \bar{x}_2 are means of n_{1} and n_{2} observation respectively then,

Combined mean \bar{X} = n_1 × \bar{x}_1 + n_2 × \bar{x}_2\over n_1 + n_2