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Normal Density Function

From the probabilistic point of view a data set of a certain population is merely a collection of values randomly drawn from a common distribution which is not yet known. Such a distribution is characterized by a probability density function (PDF) which is often described by means of parameters. The parameters are used to represent a family of functions having the same general shape, and the introduction of such parameters gave rise to the discipline of statistical modeling.

A normal density function is determined by two parameters and . The common shape of the normal density function is often described as ``bell-shaped'' curve.

The first parameter is called mean and it determines the center of the density. The second parameter is called standard deviation (SD). The normal density function is unimodal and symmetric around . A small value of leads to a high peak with sharp drop, and a larger value of leads to a flatter shape of function. The actual function is formulated as

Assuming the normal density function for a certain population, we can predict the chance that the observation from the population is between and . Having formed the density function , we can obtain such a probability, denoted by , as the area under the curve over the range between and . The calculation of this probability is what is known as ``integration'' in light of calculus, and can be obtained numerically by

In order to compute the probability
or
in the form above,
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