z-score

These are just how many standard deviations away something is from the mean.

Formula:

$\frac{x-\bar{x}}{s}$ for std deviation of a sample. It’s over $\sigma$ for the population.

If you’re working on a sample mean, you instead divide by the Standard Error of the sample: $\frac{\bar{X}-\mu }{{\sigma }_{\bar{x}}}$ where ${\sigma }_{\bar{x}}=\frac{{\sigma }_x}{\sqrt{N}}$

You can reverse it back into a raw score with $x=(z)(S_x)+\bar{x}$

Note that this is different from Z statistic testing, in that it doesn’t have to do with acceptance/rejection of a hypothesis.

It can be particularly helpful to transition a distribution into a “z-distribution” (which is just turn all the raw scores into their equivalent z-score) so that you can compare different distributions (e.g. ACT scores vs SAT scores) e.g. Sampling distributions of means