Random Number Series

The phrase “random number series” refers to a series of numbers in which 2 rules are obeyed: (A) that there is an equal probability of all possible outcomes prior to the generation of the series, and (B) that knowing one number in a series does not allow you to know any other number. It is impossible to know whether or not (A) is exactly correct, but it is possible to say it is probably approximately correct. Usually one should be 90%, or more, certain. This report will be focusing on how to discover the probable truth value of (A).

Normal Distribution

Suppose I have thrown a die 100 times. I then graph it, the x axis being all possible frequencies of rolling a certain number, so, 0-100, and the y axis being the number of each frequency. If rule (A) is true, the curve that then would be graphed should resemble a bell. This does not solve the problem, of course, as you still have to know what qualifies as a bell.

The reason the graph will resemble a bell is that most frequencies of possible solutions, with 100 sample frequencies, will be close to the expected frequency, 16.6 reccuring¹, with a few farther away, and even fewer farther away than that.

| _ _ = highest point = 16.667
| + + +=curve
| + + |
| + + |= y axis
| + + __=x axis
| + +
+________________________+





That concept is called normal distribution.

Standard Deviation

You have the bell. The mean of the frequencies of possible solutions is the highest point along the edge of the bell, which should be near the expected frequency ².

Stop viewing the bell as a curve, and view it as a plane. Cut it in half, using the mean as the point of division. The line cutting through the bell should be parralel to the y axis and perpendicular to the x axis. See below picture.

This is the property of the standard deviant:

Go the standard deviant along the bells edge, starting at the mean, and going both ways. Use those two newfound points as the points of division for 2 new lines. The area between these two lines should account for about 68% of the bell.

From the 2 new points, go outward another standard deviant and draw two more lines. The area between these two lines should account for about 95% of the bell.

Repeat, and the area should account for about 99% of the bell.

The Formula (Lightning and Thunder, Please)

You find the standard deviant by doing the following³:

X = a value of x in the bell, A = the bell’s mean, N = number of Xes.

For each X, subtract that X from A, then square the result. Add up all of your results, then divide by N-1*. Finally, find the square root of that number. That’s the standard deviant.

The higher the standard deviant, the less likely (A) is true.