Properties of the Normal Distribution

The Normal Distribution

When Sir Francis Galton measured thousands of his fellow countrymen and women at his Anthropometric Laboratory at the 1884 International Health Exhibition in London, the found that many of the human traits and skills he measured where normally distributed. Earlier, Gauss and other mathematicians had worked out the mathematical formula for the normal distribution. Gamblers were especially interested in the mathematics of the normal distribution as a means of increasing their chances of winning. A great body of research shows that many psychologically interesting distributions are normally distributed. So, normally distributed means that a distribution approaches the mathematical characteristics of the normal curve or bell-shaped curve. (See below)

Relation of the Normal Distribution and the Standard Deviation

In any normal distribution there is a relationship between the proportion of cases in between + or - each standard deviation from the mean. (See the graphic, The Normal Curve, to see the normal curve and the relationship between it and the standard deviation.) Here is that relationship in tabular form:

Spread
Proportion of Cases
+ or -1 standard deviation
68.26%
+ or -2 standard deviations
95.44%
+ or -3 standard deviations
99.74%


So, if you can assume a normal distribution, and if you know the standard deviation and the mean, you can get a good idea of the degree of variability within that distribution.


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