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— Written by Triangles on November 06, 2015 • updated on November 06, 2015 • ID 21 —

The most famous kind of distribution and its relationship with the real world.

The *normal distribution* of a *continuous* random variable is a particular kind of probability distribution. We know that there are tons of probability distributions out there, and the normal distribution is one of those. Nonetheless it deserves its own name because it has several interesting characteristics.

First of all, the normal distribution can be described mathematically by a scary probability density function (PDF):

§ rho_X(x) = 1/{ sigma sqrt{2pi} } e^ { - {(x-mu)^2} / {2sigma^2} } §

As you may see the PDF is made up with noteworthy statistical components:

- §sigma§ — the standard deviation;
- §sigma^2§ — the variance;
- §mu§ — the mean.

Those parameters change the actual shape of the function plot, which is in general defined as a *bell curve*. Generally §mu§ shifts the bell on the horizontal axis while §sigma^2§ controls its "fatness".

A normal distribution plot is always symmetric about the mean, which is the center of the bell curve and holds an important property:

§"mean" = "median" = "mode"§

In other words: the average of all the numbers in your set (mean), the number in the middle of you sorted set (median) and the number that occurs more often in your set (mode) are the same thing.

Finally a normal distribution has a skewness of 0 (it is always symmetric around the mean) and a kurtosis of 3.

Another important feature of the normal distribution is the so-called *68-95-99.7 rule*. Every normal curve, regardless of its shape or position, obeys the followings:

- about 68% of the area under the curve falls within 1 standard deviation from the mean;
- about 95% of the area under the curve falls within 2 standard deviations from the mean;
- about 99.7% of the area under the curve falls within 3 standard deviations from the mean.

This is actually useful. If you know you are dealing with a normal distribution, you can easily say that any value is *likely* to be within 1 standard deviation, *very likely* to be within 2 standard deviations or *almost certainly* within 3 standard deviations.

Many real-world random variables are distributed normally. For example heights and weights of people, exam scores, IQs and even the risk in the stock market tend to follow a normal distribution (even if the latter might not be entirely true).

The normal distribution comes up also with the so-called *Central Limit theorem*. It basically states that if you repeatedly take samples from a population and graph all the *means* of those samples, the distribution of those means will approach a *normal* look as the sample size gets bigger. The distribution will be normal regardless the initial distribution of your population, be it regular or a total random mess.

Stat Trek - *What is the Normal Distribution?* (link)

Nist/Sematech e-Handbook of Statistical Methods - *Normal Distribution* (link)

Math Is Fun - *Normal Distribution* (link)

RapidTables - *Normal Distribution* (link)

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