The central limit theorem

The central limit theorem is one of the keystones of six sigma. Unfortunately, not each six sigma belts knows it as good as he should.

The central limit theorem says that when independent random variables are put together they sum up towards a bell curve, also known as a normal distribution.

Okay, let's break this down with an example.

One of the most classic representations of a random variable is a dice out of our daily life is a dice that we use in many games. A classic dice has 6 sides and you have a change of 1 to 6 to throw each number. If we use multiple dices we get independent random variables since the outcome from one dice has no effect at all at the outcome of the other dice. If you throw a "6" with your first dice, it does not affect the chance that you throw again a six with your second dice. It is still 1 to 6.

Let's say we have five dices we will throw. This means that the sum of the dices will be between 4 and 24. The central limit theorem says that if we create a graph with on the X-axis possible outcome of a throw and on the Y-axis the amount of times the number has been thrown and we throw enough times the shape of the graph will convert towards a bell curve. The graph below shows this for 4 dices which are thrown 50000 times. This probability distribution is called also the normal distribution. You can try it yourself with our dice simulator.

Bell Curve

Let's translate this example into a real-life example. the strength of your coffee depends on several variables:

the type of coffee, the amount of coffee, the amount of water, the temperature of the water, the filter system of your coffee machine, ... And each of these variables depends again on other variables. So can the amount of water depend on the way you are measuring the amount, the water crane which is used, the person that is filling it, the water pressure, .... and these variables are depending again on other variables and so on.

Adding all these variables up will result in the fact that the strength of your coffee can follow a normal distribution.