APPENDIX A4
The Expected Value and Standard Deviation of 
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 X1, …, Xn are independent, and each has the distribution of the population. Consequently, each observation has mean μ and variance σ2, or
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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