If you could work in only whole numbers of SDs, the number of problems that could be solved would be limited. To calculate the proportions or probabilities of lying within so many SDs of the mean, you need to know what is called the probability density function , p.d.f.

A distribution for a continuous populations is defined by its p.d.f., denoted by f(x) and is such that

P(X ≤ a) = \int_{-∞}^{a} f(x) dx

and

\int_{-∞}^{∞} f(x) = 1

Although the normal distribution is symmetric, not all distributions are.

The famous German mathematician, Carl Friedrich Gauss (1777-1855) developed the normal distribution, sometimes called the Gaussian distribution.

It is given by the formula

f(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^2}

z is called the standard normal variate and represents a normal distribution with mean 0 and SD 1. The graph of the function is shown opposite.

Note that the function f(z) has no value for which it is zero, i.e. it is possible, though very unlikely, to have very large or very small values occurring.

In order to find the probabilities of all possible SDs from the mean you would have to integrate the function between the values, that is, find the area under the graph between certain limits. This is a tedious task and to avoid this, tables of the function are commonly used and these are readily available on some calculators.

We define

So Φ(z) denotes the cumulative probability up to the value of z.

For positive z, the function gives you the probability of being less than z SDs above the mean.

For example, Φ(1.0) = 0.84313, therefore 84.13% of the distribution is less than one SD above the mean.

Tables usually give the area to the left of z and only for values above zero. This is because symmetry enables you to calculate all other values.