Sampling Error

statistics

Relevant in Hypothesis Testing. Because we know that the Sampling distributions of means is a normal distribution, sampling error is when random chance produces an unrepresentative sample from a population.

If $\mu$ in the SAT population was 500, then $\bar{x}$ should be 500 if it was representative. If you took a group of 25 kids who took the SAT and their mean score was 560, that would be pretty far from the mean if ${\sigma }_x=100$.

$$\begin{array}{rl}{\sigma }_x& =100\\ & \\ \mu & =500\\ & \\ {\sigma }_{\bar{x}}& =\frac{100}{\sqrt{25}}\\ & \\ & =100/5\\ & \\ & =20\\ & \\ z& =\frac{560-500}{20}\\ & \\ & =\frac{60}{20}\\ & \\ & =+3\end{array}$$

A z-score of +3 is only ~2% likely, so we’d we’d call that a sampling error. It’s just not close enough to the true population mean.

There are two reasons for this:

1. Sampling Error: You got a weird random chance that put you out on the fringes of probability
2. Sample doesn’t represent this population: Your sample of kids are maybe from a fancy school that tutors specifically towards the SAT.