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
Properties of Arithmetic Mean
- Mean id dependent of change of origin and change of scale.
- Algebraic sum of the deviations of set of values from their arithmetic mean is zero.
- The sum of the square of the deviations of a set of values is minimum when taken from mean.
Geometric Mean
(i) If x1, x2 …., xn be n positive observations, then their geometric mean is defined as
G = n\sqrt{x_1x_2...x_n}
(ii) Let f1, f2 , …. , fn be the corresponding frequencies of positive observations x1, x2, … xn, then geometric mean is defined as
G = ({x_1}^{f_1}{x_2}^{f_2}...{x_n}^{f_n})^{1\over N}
Harmonic Mean (HM)
(i) If x1, x2 …., xn be n positive observations, then their Harmonic mean is defined as
HM = n\over{{1\over x_1}+{1\over x_2}+...+{1\over x_n}}
(ii) Let f1, f2 , …. , fn be the corresponding frequencies of positive observations x1, x2, … xn, then harmonic mean is defined as
HM = f_1+f_2+...+f_n\over{{f_1\over x_1}+{f_2\over x_2}+...+{f_3\over x_n}}
Median
The median of a distribution is the value of the middle observation when the observations are arranged in ascending or descending order.
(i) Median for Simple Distribution or Raw Data
Firstly, arrange the data in ascending or descending order and then find the number of observations n,
(a) If n is odd, median ({n+1\over2})th term is the median.
(b) If n is even, {n\over2}th\ term +\ ({n\over2 }+1)th\ term\over 2
(ii) Median for Unclassified frequency distribution
(i) Firstly, find N/2, where N = ∑f
(ii) Find the cumulative frequency which is equal to or just greater than N/2
(iii) (a) If N is odd, median ({N+1\over2})th term is the median.
(b) If N is even, {N\over2}th\ term +\ ({N\over2 }+1)th\ term\over 2
