Test your Ears: The Normal Distribution
The Normal or Gaussian Distribution comes up for discussion with the A-B-X-test, for the calculation of the reliability of the result.
The Normal Distribution gives the chance for a certain result in experiments which give a random result. It is an important mathematical tool for statistic calculations.
For example: You throw a lot of times with a certain number N of dice. On average about half of the dice, N / 2 will show an even number, but it will be a bit different in each throw. How large is the chance for a certain deviation?
It is the Normal Distribution which gives the answer.
Imagine the following experiment:
You throw 100 times with 10 dice. Note in a table how often each number of even-showing dice appears.
That may look as follows. (also with 10000 throws of 10 dice. The numbers come from a computer simulation of throwing dice)
| Number even | How often (100) | How often (10000) |
| 0 | 0 | 4 |
| 1 | 3 | 86 |
| 2 | 2 | 437 |
| 3 | 14 | 1164 |
| 4 | 20 | 2070 |
| 5 | 27 | 2485 |
| 6 | 16 | 2064 |
| 7 | 12 | 1117 |
| 8 | 5 | 445 |
| 9 | 1 | 120 |
| 10 | 0 | 8 |
Obviously the 5 will appear the most often. If you have a close look you can see that the other values are more or less symmetrical around the 5. 4 and 6 appear about the same number of times, as do 3 and 7, etc. If you put these values in a graph you will find this bell-shaped curve:
The physicist and mathematician Carl Friedrich Gauss has derived a formula for this curve:
Looks more difficult than it is. x is the horizontal axis of the graph. p(x) is the chance (probability) to find the value x.
The numbers sigma (
) and mu (
) determine the shape of the curve. exp
means: raise the number e = 2.71828 to the power of what is between the
brackets. Mu means a shift over the x-axis. With
Mu = 0 the peak of the curve would be on X = 0. In the example above Mu
= 5. The sigma gives the width of the curve at a height
of 0.779 of the height of the peak. Sigma is often called
"Standard Deviation". The smaller sigma is in
relation to the x-range, the shaper the curve will be.The term in front of exp is a scale factor. If you leave it away the peak height will always be 1.
In the dice-throwing experiment the sigma should be one half of the square root of the number of dice in the throw. Obviously the x-axis goes from zero to the number of dice, including. In the example above the sigma is 1.6 on an axis from zero to 10. With 100 dice the sigma would become 5 on an x-axis of zero to 100, so giving a much sharper peak.
The A-B-X-test resembles the throwing dice experiment, at least if you cannot recognize the difference between A and B very well, that is if you guess wrong almost as often as right. The Normal Distribution tells how large the chance is that the result is based on random chance.
In this case the sigma must be half of the number of tests you do, x must be the number of right guesses, and mu half the number of tests. The outcome of the formula is the chance that that result could have been obtained with dice throwing.
The reliability factor now is 1 - that chance.
And there is an extra: If you guess wrong more often than right you will have a negative number. Consequently guessing wrong is also a performance !