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 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)`.