6.5: Key Terms

Key Terms	Definition
Normal Distribution	a continuous random variable $(RV)$ with pdf $f(x) =$ $$\frac{1}{\sigma \sqrt{2 \pi}} \mathrm{e}^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}\nonumber$$ where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation; notation: $X \sim N(\mu, \sigma)$. If $\mu = 0$ and $\sigma = 1$, the $RV$, $Z$, is called the standard normal distribution.
Standard Normal Distribution	a continuous random variable $(RV) X \sim N(0, 1)$; when $X$ follows the standard normal distribution, it is often noted as $Z \sim N(0, 1)$.
z-score	the linear transformation of the form $z=\frac{x-\mu}{\sigma}$ or written as $z=\frac{|x-\mu|}{\sigma}$; if this transformation is applied to any normal distribution $X \sim N(\mu, \sigma)$ the result is the standard normal distribution $Z \sim N(0,1)$. If this transformation is applied to any specific value $x$ of the $RV$ with mean $\mu$ and standard deviation $\sigma$, the result is called the z-score of $x$. The z-score allows us to compare data that are normally distributed but scaled differently. A z-score is the number of standard deviations a particular $x$ is away from its mean value.