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The Normal distribution

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.

Main features of the normal distribution

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:

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".

Normal distribution plot
1. Two types of normal distribution plots.

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.

Normal distribution and the standard deviation

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.
Normal distribution with standard deviations
2. A normal distribution plot with standard deviation points.

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.

Why the normal distribution is so important

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)