# Normal Distribution

A binomial distribution describes a population whose members have only certain, discrete values. This is the case with the number of 13C atoms in cholesterol. A molecule of cholesterol, for example, can have two 13C atoms, but it can not have 2.5 atoms of 13C. A population is continuous if its members may take on any value. The efficiency of extracting cholesterol from a sample, for example, can take on any value between 0% (no cholesterol extracted) and 100% (all cholesterol extracted).

The probability of obtaining a particular result, f(X), from a normal distribution is

where μ is the expected, or true mean for the population and σ2 is the population’s variance.

The area under a normal distribution curve is an important and useful property as it is equal to the probability of finding a member of the population within a particular range of values. Because a normal distribution depends solely on μ and σ2, the probability of finding a member of the population between any two limits is the same for all normally distributed populations. As shown in the figure below, 68.26% of the members of a normal distribution fall within the range μ ± 1σ, and 95.44% of the population’s members have values within the range μ ± 2σ. Only 0.17% of a population’s members have values exceeding the expected mean by more than ± 3σ.