The variance of the sampling distribution of the sample means drawn from a population having a finite variance σ2 of is equal to σ2/n. It is because the sample mean calculated from a sample will have a lower variance than the population data. For example, the population data for returns are -11 percent, -10 percent, -8 percent, -6 percent, - 4 percent, 0 percent, 4 percent, 6 percent, 8 percent, and 12 percent. The return data varies from -11 percent to 12 percent. A range of 23 percent points. But if take the sample size of 5 and then take the sample mean then that sample mean will have the much lesser range and variance as compared to the original data. The worst case of returns will be all five negative returns and then the sample mean will be -7.8 percent (=(-11-10-8-6-4)/5 = -39/5). The best possible sample mean value will be 6 percent (=(0+4+6+8+12)/5 = 30/5). So, the range will be from -7.8 percent to 6 percent which is much lesser than the range of the underlying population returns.
The standard deviation of the sampling distribution of the sample mean is called as the standard error of the sample mean. It is the positive square root of the variance of the sample mean.
σX ̅ = σ/√n when we know the population standard deviation σ
sX ̅ = s/√n when we do not know the population standard deviation and s the sample standard deviation.
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