SL 4.12—Z values, inverse normal to find mean and standard deviation

Z-value (Z-score): A z-value represents the number of standard deviations a particular data point lies away from the mean of its distribution. It places any value from any normal distribution onto a single, universal scale.

The formula for calculating a z-value is:

$z=\frac{x-\mu }{\sigma }$

Where:

• $x$ is the raw data value
• $\mu$ is the mean of the distribution
• $\sigma$ is the standard deviation of the distribution

Standardization is powerful because it lets us compare values from completely different normal distributions on the same scale , the standard normal distribution $Z\sim N(0,1)$, which has mean 0 and standard deviation 1.