Standard deviation of the sampling distribution of x̄

The standard deviation of the sampling distribution of x̄ is equal to the standard deviation of the population divided by the square root of the sample size.

$$ \sigma_{\overline{x}} = \frac{\sigma }{\sqrt{n} } $$

σ is the standard deviation of the population and n is the sample size. 

Notice that unlike the mean of the sampling distribution of x̄ which is equal to the mean of the population, the standard deviation, σ, of x̄ is not equal to the standard deviation, σ , of the population distribution unless of course n = 1. 

The formula above can only be used if the sample size or n is small compared to the population size. It has been agreed that the sample size is small compared to the population size when the sample size is less or equal to 5% of the population size. 

In other words, when the following is true:

$$ \frac{n } {N} \leq 0.05 $$

In most practical applications, the sample site will be small compared to the population. Therefore, most likely, you will use the formula above to calculate σ

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