(1)If data for all of the population under investigation is known, then the mean and variance for this population can be calculated as follows:
Population Mean: (1)
Population Variance:
(2)
Here, 
is the size of the population.
The population standard deviation is the positive square root of the population variance.
Most of the time it is not possible to obtain data for the entire population. For example, it is impossible to measure the height of every male in a country to determine the average height and variance for males of a particular country. In such cases, results for the population have to be estimated using samples. This process is known as statistical inference. Mean and variance for a sample are calculated using the following relations:
Sample Mean:
(3)
Sample Variance:
(4)
Here, 
is the sample size.
The sample standard deviation is the positive square root of the sample variance.
The sample mean and variance of a random sample can be used as estimators of the population mean and variance respectively. The sample mean and variance may be referred to as statistics. A statistic is any function of observations in a random sample.
You may have noticed that the denominator in the calculation of sample
variance, unlike the denominator in the calculation of population variance,
is 
and not 
. The reason for this difference is
explained in "Unbiased
and Biased Estimators."
See Also:
