In this article learn the normal distribution variance. The normal distribution is a completely continues distribution with zero cumulative in all orders on two. The normal distribution have bell shaped to density function in the associated probability of graph at the mean, and also called as the bell curve

`F(x) = (1/ ( sqrt( 2 pi sigma^2) )) e^ ( - ( x - lambda )^2 / ( 2 sigma^2 ) )`

Here `lambda` and a is mean and variance. T he mean value equal to zero and variance equal to 1 means the distribution called standard normal distribution .In the below details of normal distribution variance.

Sample Variance of Normal Distribution Variance:

The variance of normal distribution is used to one or more descriptors and it is one instant of distribution. The sample variance can be used in construct of estimate in this variance and it is very simplest case of estimated .The variance describing theoretical probability of distribution.

Background of variance in Normal Distribution variance:

The variance has random variable. The random variable is mean of the squared devotion of variable and it's expected of that value.

The Definition of normal distribution variance:

The variance has continuous and discrete case for defined the probability density function and mass function. The distribution variance of random variable denoted by x .The x have mean value of E(x), the variance x is as follows,

X= (x-`lambda`)2.

Var (X) = E[(x-`lambda` )2].

The continuous case of variance:

The random variable x is probability density f(x) function in continues.

Var(X) =`int` (x-`lambda` )2f(x) dx.

Here `lambda` = `int` x f(x) dx.

Where integral definite x range from X.

The discrete case of variance:

The random variable x is probability mass function x1->p1…..xn->pn in discrete case.

Var(X)= `sum_(i=1)^n` `Pi` (xi -`lambda`) 2.

here`lambda `= `sum_(i=1)^n` xi `Pi` .

The square root deviation of X ranges from mean of own it.

The Properties of Normal Distribution Variance:

The variance has non-negative value, because the square is + or 0. The constant of random variable has zero of the variance, and it variable in the data set is zero. And the entries have same value. The following rules are maintain in the that properties,

To change in a location parameter means variance is invariant.
The variance is unchanged means the all values added into constant of the variables.
The all values are scaled with variables in a constant and the variances are scaled in the square of that constant. Those are all properties expressed the following formula:
Var(aX+b) = Var(aX) = a2 var(X)

The Example of Normal distribution variance:

In fair dice a six-sided can be modeled by a discrete random variable in outcomes 1 through 6, each of equal probability 1/6. The expected value is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. Hence the variance computed to be:

`sum_(i=1)^6``1/6` (i-3.5)2 =`1/6` 17.50=2.92

Formulas in Mean Variance Theory

Mean:

It is a mathematical term in the chapter statistics in which it is defined as the total average of a group of numbers. It is found or calculated by adding all the numbers in a given set and divide the sum by number of given values.

Formula:

The formula for mean is derived as,

Mean = `(a_1+a_2+............a_n)/n`

Where,

`a_1+ a_2+..........a_n` = sum of numbers

n is the total numbers given.

This is the formula for mean.

Variance:

It is also one of the mathematical terms of statistics. It is defined as the descriptor or a parameter of the theoretical probability distribution.

Formula for variance:

The term variance is found out by taking square root of standard deviation. Hence it depends on the term standard deviation.

Standard deviation = `sqrt((sum d^2)/n)`

Here, d is calculated by (x -`barx`)

`barx` is the mean value and x is the given values.

n is the number of given values.

These are involved in mean variance theory.