Summary of Central Limit Specials

Means/Averages

Standard Error of Means: the standard deviation for the Averages Distribution: \frac{\sigma_x}{\sqrt{n}}

\overline{X}\:\sim\:N\left(\mu_x,\frac{\sigma_x}{\sqrt{n}}\right)

z=\frac{\overline{x}-\mu_x}{\sigma_x/\sqrt{n}}

Sums

Standard Error of Sums: the standard deviation for the Sums Distribution: \sigma_x\cdot\sqrt{n}

\Sigma x is one sum

Mean of \Sigma x=n\cdot\mu_x

\Sigma X\:\sim\:N(n\cdot\mu_x,\sigma_x\cdot\sqrt n)

z=\frac{\Sigma x-n\cdot\mu_x}{\sqrt{n}\cdot\sigma_x}

Where n is the number of items used in the sum or mean, not the sample or population, and N means the Normal Distribution

To use these in the calculators, make sure to replace the mean and standard deviation with the calculation from above, do not round during your work.

For Example, if you have a mean of 37, and a standard deviation of 4.

If you want to find the Distribution for the Means working with 7 items, you would use N( 37 , \frac{4}{\sqrt{7}}).

If you want to find the Distribution for the Sums as above, you would use N( 7*37 , 4*\sqrt{7})