Let μ be the population mean of a random variable X. Suppose we have a simple random sample from the population, Xi, i=1,...,N. The best estimate of μ is the sample mean, .
After we collect a random sample from the population, denoted xi, i=1,...,N, we can calculate an observed value for the sample mean, . If we collect a different random sample, we will calculate a different value for the sample mean. Every random sample gives us a different sample mean. Each sample mean, however, provides an estimate of the population mean.
Sampling distribution
The sample mean is a statistic, and is a random variable.
If the population has a normal distribution with mean μ and standard deviation σ, then the sampling distribution of is normal with a mean of μ and a standard deviation of
.