Measurements collected to represents height, weight and temperature fall under the classification of Continuous Random Variables. Many such continuous variables have bell shaped distributions and are know as Normal Random variables. Theoretically the distribution of a normal random variable is continuous, bell shaped and symmetric about the mean with median and mean coinciding with mean.
Normal Random Variable
The equation for a normal curve is y = $\frac{1}{\sigma \sqrt{2\pi }}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}$.where μ and σ represent the mean and standard deviation of the random variable which describe a normal distribution completely. The notation used to denote a normal distribution is N(μ, σ2) where σ is the variance of the distribution.

The probabilities of normal random variables are proportional to the area under the normal curve between the two end points that limit the variable. The theoretical normal curve can also be used to study random variable which are only approximately normal.

To enable the study of all normal random variables easy using a single area table, standard normal variables are used. The standard normal distribution has mean μ =0 and standard deviation σ =1.
The formula for standard normal distribution is
y = $\frac{e^{-\frac{z^{2}}{2}}}{\sqrt{2\pi }}$ where z is called the standard normal variable.

The formula used to transform a normal random variable X to a standard normal variable z is
z = $\frac{X-\mu }{\sigma }$ where μ and σ are the mean and standard deviation of the normal random variable X.