Some basic properties of the sampling distribution of can be expressed in terms of the population mean and variance when the observations form a random sample. Let

In a random sample, the random variables *X*_{1}, …, *X _{n}* are independent, and each has the distribution of the population. Consequently, each observation has mean

Next,

and *n* is a constant. Using the additivity properties of expectation and variance discussed in Appendix A3.4, we obtain

Furthermore, taking the square root yields

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