The explanation of Normal Distribution Rule

Tags: maths technical

Empirical Law:

While studying Statistics, I came across the Normal Distribution Curve and learned that it has a very interesting property called the Empirical Law :

/For symmetric, uni-modal, bell-shaped curves, around 68% of the values fall around 1 SD of the mean. 95% of the values fall around 2 SD of the mean, and 99.7% of the values fall around 3SD of the mean./

Image credits: Dan Kernler/Wikipedia

Its graph, called the bell curve is represented by the mathematical function:

\[ f(n) = \frac{1}{\sqrt{2\pi}}e^{{-n^2}/{2}} \]

The following is probably a statistical way of showing why the graph follows this property. But before delving further, we also need to understand the Central Limit Theorem:

Central Limit Theorem:

As an experiment is repeated a large number of times, the probabilities of the average results will converge to the normal bell shaped distribution.

Explanation:

Let an experiment be repeated a large number of times, each time with the outcome \(X\_{1}\) , \(X\_{2}\), upto \(X_{n}\).

Since the experiment is the same, the average value and the variance will also remain the same. Let these be called as \(m\) and \(v\) resp.

Therefore, the average of these averages can be written as:

\[ \overline{X} = \frac{X\_{1} + X\_{2} + \ldots + X\_{n}}{n} \]

Taking the expected value of both sides:

\[ E(\overline{X}) = \frac{E(X\_{1}) + E(X\_{2}) + \ldots + E(X\_{n})}{n} \\ \\ = \frac{nm}{n} \\ \\ = m \]

Coming to Variance, which shows us how much the values are spread out from the mean value:

\[ \overline{X} = \frac{X\_{1} + X\_{2} + \ldots + X\_{n}}{n} \]

Taking the variance of both sides:

\[ Var(\overline{X}) = Var\left(\frac{X\_{1} + X\_{2} + \ldots + X\_{n}}{n}\right) \
\implies Var(\overline{X}) = \frac{Var(X\_{1}) + Var(X\_{2}) + \ldots + Var(X\_{n})}{n^{2}} \
= \frac{nv}{n^{2}} \
= \frac{v}{n} \]

Therefore, as the number of experiments \(n\) increases, the variance keeps on decreasing. And for large enough \(n\), the variance is almost equal to zero. This again reinforces that the average of the whole is equal to the individual average.

The instructor, in a previous lecture, told how this is a very important law – and rightly so. I guess whenever there’s average of averages being calculated the result will follow this law.

One application which comes to my mind is the performance cycle of employees, where this curve is used to fit the employees into their respective categories. It is a large number of people, and the expected value of their average performance is bound to fill this curve – averages in the center and outliers (both good and bad) on the either side of the peak.


Since we now know the mathematical definition of Normal Distribution Curve, we can plot the graph of its function and the area under the graph will show that the empirical law (68%-95%-99.7%) holds well.