There are plenty of sites where you can enter data and receive the standard deviation(s), but is there a place where I can enter '4.2 sigma', or '1.3 sigma' and it will spit out the odds that a scientific result, say, is random chance.....
Edit: Think of news you might hear in the world of physics, wherein someone anniunces, proudly, 'five sigma' results, meaning there is only a one in 3.5 million chance (or one in 1.75 million, depending on if you are using both sides of the distribution) their newfound particle is not really there.... But, most ready-made charts only list odds for 3, 3.5, 4, 4.5, etc. stigmas, yet many experimental results say they are certain to within 4.3 sigma, or such.....
How did statisticians come up with the 68-95-99 rule, exactly? Was it arbitrary?
The 68-95-99 rule is from the integrals: $$ \frac{1}{\sqrt{2\pi}}\int_{-1}^1 e^{-x^2/2} dx = 0.681 \\ \frac{1}{\sqrt{2\pi}}\int_{-2}^2 e^{-x^2/2} dx = 0.954 \\ \frac{1}{\sqrt{2\pi}}\int_{-3}^3 e^{-x^2/2} dx = 0.997 $$ correct to three figures.
If you do not know integration, I guess you should wait a while before you will be able to answer your questions.