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
\bar{x} = {x_1+x_2+...+x_n\over n}={ \sum_{i=1}^{n}{x_i} \over n}(ii) Arithmetic Mean for Discrete Frequency Distribution
| Variate (x) | x1 | x2 | … | xn |
| Frequency (f) | f1 | f2 | … | fn |
.\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) | y1 | y2 | … | yn |
| Frequency (f) | f1 | f2 | … | fn |
(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 n1 and n2 observation respectively then,
Combined mean \bar{X} = n_1 × \bar{x}_1 + n_2 × \bar{x}_2\over n_1 + n_2
