Table of Contents

Toggle## 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

**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 x_{1}, x_{2} …., x_{n} be n positive observations, then their geometric mean is defined as

G = n\sqrt{x_1x_2...x_n}

(ii) Let f_{1}, f_{2} , …. , f_{n} be the corresponding frequencies of positive observations x_{1}, x_{2}, … x_{n}, 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 x_{1}, x_{2} …., x_{n} 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 f_{1}, f_{2} , …. , f_{n} be the corresponding frequencies of positive observations x_{1}, x_{2}, … x_{n}, 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