Variance is one of the best measures of dispersion which measure the difference of all observation from the center value of the observations.

**Population variance and standard deviation**

*its estimate (sample variance) by*

^{2 }and*s*For

^{2}.*N*population values X1,X2,…,XN

*having the population mean μ, the population variance is defined as,*

So, we can define the population standard deviation as

Thus, the standard deviation is the positive square root of the mean square deviations of the observations from their arithmetic mean. More simply, standard deviation is the positive square root of σ* ^{2}*.

**Sample variance**

*and a set of sample observations will yield a*

^{2}*s*. If x1,x2,…,xn is a set of sample observations of size n, then the

^{2}*s*is define as,

^{2}

**Properties**

**Effect of changes in origin:** Variance and standard deviation have certain appealing properties. Let each of the numbers x1,x2,…,xn increases or decreases by a constant c. Let y be the transformed variable defined as,

where, c is a constant.

Finally we get that any linear change in the variable x does not have any effect on its σ* ^{2}*. So, σ

*is independent of change of origin.*

^{2}

**Effect of changes in the scale:** When each observation of the variable is multiplied or divided by a certain constant c then there occur changes in the σ* ^{2}*.

So, we can say that changes in scale affects and it depends on scale.

**Uses of variance and standard deviation**

Credit: Data Science Central By: Muhammad Touhidul Islam