The most general normal distribution is

where:

The normal distribution:

1. Is center-symmetric
2. Is normalized (area under curve equals one)
3. Has a most probable value of x_mp=m
4. Has a width indicated by s

The normal distribution is often given in terms of the generalized parameter, z

where:

Combined, these two formulas are equivalent to the general form.

When we calculate the average and standard deviation from our data set, we are estimating the parameters of the parent population

In this case

Areas under the normal distribution tell us the probability of occurrence. Areas are tabulated as a function of z (commonly found in tables). The area is that of the integral equation