### 7.1 The Central Limit Theorem for Sample Means (Averages)

Central limit theorem for sample means:
$\overline{X}$ ~ *N*$\left({\mu}_{x}\text{,}\frac{\sigma x}{\sqrt{n}}\right)$

Mean $\overline{X}$: *μ _{x}*

Central limit theorem for sample means *z*-score and standard error of the mean: $z=\frac{\overline{x}-{\mu}_{x}}{\left(\frac{{\sigma}_{x}}{\sqrt{n}}\right)}$

Standard error of the mean (standard deviation ($\overline{X}$)): $\frac{{\sigma}_{x}}{\sqrt{n}}$

### 7.2 The Central Limit Theorem for Sums (Optional)

Central limit theorem for sums: *∑X* ~ *N*[(*n*)(*μ _{x}*),($\sqrt{n}$)(

*σ*)]

_{x}Mean for sums (*∑X*): (*n*)(*μ _{x}*)

Central limit theorem for sums *z*-score and standard deviation for sums: $z\phantom{\rule{.30em}{0ex}}\text{for the sample mean=}\frac{\Sigma x\u2013(n)({\mu}_{X})}{(\sqrt{n})({\sigma}_{X})}$

Standard deviation for sums (*∑X*): $(\sqrt{n})$(*σ _{x}*)