Let `A` and `B` be two non empty sets and `R : A--> B` be a relation such that `R : {(a, b) I (a, b) in R,
a in A` and `b in B`}.

(i) ` 'b' ` is called image of ` 'a' ` under `R.`
(ii) ` 'a' ` is called pre-image of `'b'` under `R`.
(iii) Domain of `R` : Collection of all elements of `A` which has a image in `B` or Set of all first entries
in `A xx B`.
(iv) Range of `R` : Collection of all elements of `B` which has a pre-image in `A` or Set of all second
entries in `A xx B`.



Note:
( 1) It is not necessary that each and every element of set `A` has a image in Set Band each and every
element of set `B` has a preimage in Set `A`
(2) Elements of set `A` having image in `B` is not necessary unique.
(3) Basically relation is the number of subsets of `A xx B`

number of relations `=` no. of ways of selecting a non zero subset of `A xx B`

`=text()^(mn)C_0 + text()^(mn)C_1 + ....................+text()^(mn)C_(mn)`

`=2^(mn)`

The variance is a measure based on the deviations of individual scores from the mean. As noted in the
definition of the mean, however, simply summing the deviations will result in a value of `0`. The get around
this problem the variance is based on squared deviations of scores about the mean. When the deviations
are squared, the rank order and relative distance of scores in the distribution is preserved while negative
values are e liminated. Then to control for the number of subjects in the distribution, the sum of the
squared deviations, `sum (X - bar (X))^2` , is divided by `N` (population). The average of the sum of the squared
deviations is called the variance.

(a) Variance of individual observations:

If `x_1, x_2, ...... , x_n` are `n` values of a variable `X`, then
var `(X) = 1/n | sum_(i=1)^(n) (x_i - bar(X))^2 | = 1/n sum_(i=1)^n x_(i)^(2) - (1/n sum_(i=1)^(n) x_i)^2 `

`=` Mean of squares - Squares of Mean



(b) Variance of discrete frequency distribution:

If `x_1, x_2, ...... , x_n` are `n` values of a variable `X` and corresponding frequencies of them are `f_1, f_2 ...... f_n`

Var`(X) =1/N | sum_(i=1)^(n) f_i (x_i - bar(X))^2 | = 1/N sum_(i=1)^(n) f_ix_(i)^(2) - ( 1/N sum _(i=1)^(n) f_ix_i)^2` `( sum_(i=1)^(n) f_i =N )`




(c) Variance of a grouped or continuous frequency distribution:

Var`(X) =h^2 [ 1/N sum f_(i) u_(i)^(2) - ( 1/N sum f_(i) u_(i) )^2 ]` `u_i = (x_i - bar (X))/h` where `h =` Class width



Properties :
(1) lf `x_1, x_2, x_3 .........., x_n` be `n` values of a variable `X`. lf these values are changed to `x_1 + a, x_2 + a, .... x_n + a`,
where `a in R`, then the variance remains unchanged.
(2) If `x_1, x_2, ...... , x_n` values of a variable `X` and let `'a'` be a non-zero real number. Then, the variance of the
observation `ax_1, ax_2, ..... ,ax_n` is `a^2 Var(X)`.

The variance is a measure based on the deviations of individual scores from the mean. As noted in the
definition of the mean, however, simply summing the deviations will result in a value of `0`. The get around
this problem the variance is based on squared deviations of scores about the mean. When the deviations
are squared, the rank order and relative distance of scores in the distribution is preserved while negative
values are eliminated. Then to control for the number of subjects in the distribution, the sum of the
squared deviations, `sum( X - bar (X))^2` is divided by `N` (population). The average of the sum of the squared
deviations is called the variance.

(a) Variance of individual observations:


If `x_1 x_2, ...... , x_n` are `n` values of a variable `X`, then

`Var (X) = 1/n | sum_(i=1)^(n) (x_i - bar(X))^2 | =1/n sum_(i=1)^(n) x_(i)^2 - (1/n sum _(i=1) ^(n) x_i)^2`

`=` Mean of squares - Squares of Mean

(b) Variance of discrete frequency distribution:


If `x_1 ,x_2, ...... , x_n` are `n` values of a variable `X` and corresponding frequencies of them are `f_1, f_2, ...... f_n`
`Var (X) = 1/N |sum_(i=1)^(n) f_i ( x_i - bar(X))^2 | =1/N sum_(i=1)^(n) f_ix_(i)^(n) - (1/N sum_(i=1)^(n) f_i x_i ) ^2` `( sum_(i=1)^(n) f_i =N)`


(c) Variance of a grouped or continuous frequency distribution:

`Var (X) =h^2 [ 1/N sum f_i u_(i)^2 - (1/N sum f_i u_i)^2 ]` `u_i =( x_i - bar(X) )/h` where `h =` Class width




Properties :
(1) lf `x_1, x_2, x_3 .........., x_n` be `n` values of a variable `X`. lf these values are changed to `x_1 + a, x_2 + a, .... x_n + a`,
where `a in R`, then the variance remains unchanged.
(2) If `x_1, x_2, ...... , x_n` values of a variable `X` and let `'a'` be a non-zero real number. Then, the variance of the
observation `ax_1, ax_2, ..... ,ax_n` is `a^2 Var(X)`.

The standard deviation (`s` or `sigma`) is defined as the positive square root of the variance. The variance is a
measure in squared units and has little meaning with respect to the data. Thus, the standard deviation is
a measure of variability expressed in the same units as the data. The standard deviation is very much like
a mean or an "average" of these deviations.


Combined Standard Deviation :
If there are two sets of observations containing `n_1` & `n_2` items with respective mean`bar( x_1_` & `bar(x_2 )`and
standards deviations `sigma_1` & `sigma_ 2`, then the mean `bar(x)` and the standard deviations of `n_1 + n_2` observations,
taken together, are

`bar(x) = ( n_1 bar(x_1) + n_2 bar(x_2)) /(n_1+n_2)`


`sigma^2 = 1/(n_1 +n_2) [n_1 (sigma_(1)^(2)+ d_(1)^(2)) +n_2 (sigma_(2)^(2) + d_(2)^(2) ) ]`

where `d_1= bar(x)- bar(x_1) , d_2 = bar(x) - bar(x_2)`