population variance formula

In the variance of a random variable or distribution is the expectation, or mean, of the deviation squared of that variable from its expected value or mean. Thus the variance is a measure of the amount of variation within the values of that variable, taking account of all possible values and their probabilities or weightings.

Population Variance formula explanation:


The population variance formula for discover the variance for the given population problem

Population Variance formula:

Here, N is the size of the population.

So X-an unbiased estimate of µ. The variance of the population

Where µ is the population mean. n values x1, ..., xn from the population, This is merely a individual case of the universal definition of variance introduced above, but controlled to finite populations.

In many functional situations, the true variance of a population is not known a priori and must be computed someway. When making with countless populations, this is generally impossible.

Most of the time it is not possible to obtain data for the entire population. For example, it is impossible to measure the weight of each male in a particular area to determine the average weight and variance for males of a particular area. In such cases, results for the population have to be estimated using samples.

Population variance formula for Example problems:


Example:

The hourly wages earned by a sample of five students are:

7, 5, 11, 8, 6.

Let a sample consist of n independent readings x1, x2...xn, drawn from a population which is not necessarily Gaussian. We know that the mean µ of our sample is given by

µ  =?X/N

µ = 37/5

µ = 7.40

For this problem the population variance

Formula = ?2 = ?(X-?)2/N

(X- µ)= 7-7.4=.4

(X- µ)= 5-7.4=-2.4

(X- µ)= 11-7.4=3.6

(X- µ)= 8-7.4=0.6

(X- µ)= 6-7.4=-1.4

We take squared on each (X- µ)

=.4 =0.16

=-2.4 =5.76

=3.6=12.96

=0.6=0.36

=-1.4=1.96

?2 = ?(X-?)2/N

?2=21.2/5

Population  variance   ?2=4.21