Normal distribution

In a normal distribution, data clusters around the mean $\mu$ with spread measured by the standard deviation $\sigma$. The empirical rule states that about 68% of data lies within one standard deviation of the mean ($\mu \pm \sigma$), about 95% within two standard deviations ($\mu \pm 2\sigma$), and about 99.7% within three standard deviations ($\mu \pm 3\sigma$). To compare values from different normal distributions, convert a data value $x$ into a standardized score (z-score) using $z=\frac{x-\mu }{\sigma }$. This tells you how many standard deviations $x$ is above or below the mean. To convert a z-score back to a data value, use $x=\mu +z\sigma$.