Normal Distributions

Probability of an event depends on the range that is specified in the overall distribution. A broad range that includes the peak will be highly likely to include the event. Area under the range to the baseline relates to probability. There are tables of these areas, but we have a simple computer program that performs the calculation. Since the curve for normal distribution is symmetrical, we can work with half of it. The area under the curve equals 1 because there is a 100 per cent chance that an instance that belongs to the set will occur. If carried from the peak to infinity, the area in the calculation becomes .5 (50 per cent) of the total area. The fraction of area for a range over the total area is its probability.

We model a normal distribution with the expression

where

• ƒ(x) = equation of the curve
• x = the intensity of the property, e.g., grade on an exam
• µ = mean value for x
• s = the standard deviation

The coefficient sigma has a large effect on the shape of the distribution. Please experiment with the Java applet below that draws distributions. Note that a small value of sigma gives a sharp distribution while large values give a broad distribution. The areas under the curves should be the same, and this means that the broader distributions have much lower total heights.

Pearson3 distributions that are common in environmental situations and in biological problems are not symmetrical. The calculation of skewness has standard deviation in the denominator, so it makes sense that the curve gets narrower as skewness increases.