Normal Distribution

Characteristics of the Normal Distribution

The normal distribution has several properties, including the following:

• The curve is completely determined by the mean (average) and the standard deviation (the spread about the mean, or girth).
• The mean is at the center of this symmetrical curve (which is also the peak or maximum ordinate of the curve).
• The distribution is unimodal; the mean, median, and mode are the same.
• The curve is asymptotic. Values trail off symmetrically from the mean in both directions, indefinitely (the tails never touch the x axis) and form the two tails of the distribution.
• The curve is infinitely divisible.
• The skewness and kurtosis are both zero.

Different normal density curves exist because a normal distribution is determined by the mean and the standard deviation. Figure 2 presents three normal distributions, one with a mean of 100 and a standard deviation of 20, another normal distribution with a mean of 100 and a standard deviation of 40, and a normal distribution with a mean of 80 and a standard deviation of 20. The curves differ with respect to their spread and their height. There can be an unlimited number of normal distributions because there are an infinite number of means and standard deviations, as well as combinations of those means and standard deviations.

Figure 2 Examples of Three Different Normal Distributions

Although sometimes labeled as a “bell curve,” the curve does not always resemble a bell shape (e.g., the distribution with a mean of 100 and standard deviation of 40 is flatter in this instance because of the scale being used for the x and y axes). Also, not all bell-shaped curves are normal distributions. What determines whether a curve is a normal distribution is not dependent on its appearance but on its mathematical function.

The normal distribution is a mathematical curve defined by the probability density function (PDF):

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-{(x-\mu )}^2/2{\sigma }^2},$$

where f(x) = the density of the function or the height of the curve (usually plotted on the $y$ axis) for a particular variable x (usually plotted on the $x$ axis) with a normal distribution

This equation dictates the shape of the normal distribution. Normal distributions with the same mean and standard deviation would be identical in form. The curve is symmetrical about its mean because (x − μ) is squared. Furthermore, because the exponent is negative, the more $x$ deviates from the mean (large positive number or large negative number), $f(x)$ becomes very small, infinitely small but never zero. This explains why the tails of the distribution would never touch the $x$ axis.

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