6.3 The central limit theorem for sums

Suppose $X$ is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:

• a ${\mu }_X=$ the mean of $X$
• b ${\sigma }_X=$ the standard deviation of $X$

$\Sigma X$ ~ $N(n\cdot {\mu }_X,\sqrt{n}\cdot {\sigma }_X)$

The Central Limit Theorem for Sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution) which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviationequal to the original standard deviation multiplied by the square root of the sample size.

The random variable $\Sigma X$ has the following z-score associated with it:

• a $\mathrm{\Sigma x}$ is one sum.
• b $z=\frac{\Sigma x-n\cdot {\mu }_X}{\sqrt{n}\cdot {\sigma }_X}$

• a $n\cdot {\mu }_X=$ the mean of $\mathrm{\Sigma X}$
• b $\sqrt{n}\cdot {\sigma }_X=$ standard deviation of $\mathrm{\Sigma X}$