Indeed - Edit 1
Before modification by Tor at 20/11/2012 07:17:40 AM
If 62.5% of channelers are stronger than Daigan, and strength in channeling is normally distributed, then Daigan is 0.32 standard deviations below the mean. Plain and simple.
Can you explain this? I'm not quite sure what math is being employed here.
The bell curve gives you the probability that a person has a given strength, so if you take the sum of all the probabilities for all the strengths weaker than a given person, then you find the total probability that someone is weaker than that person.
If you go to Wolfram Alpha and type in "-0.32 sd" and scroll down a little bit, you'll se a plot that illustrates this.
Of course, to begin with, we didn't know how far below the mean Daigan was, only that she was stronger than 37.5% of channelers. So to find out how many standard deviations below the mean she was, I look at a normal distribution with zero mean and standard deviation 1, and keep summing the probabilities, starting from minus infinity, and I keep it up until the sum has reached 0.375, at which point I have reached Daigan's strength (in numbers of standard deviations).
Techincally, what I do is say that the integral of exp(-0.5x^2))/sqrt(2 pi) from minus infinity to y should be equal to 0.375, and then solve the equation to find y.
You can do with Wolfram Alpha as well. Scroll down a little, and somewhere it says y=-0.318639, which means that Daigan is 0.318639 standard deviations below the mean.
Of course, for the real problem the mean isn't zero, and the standard deviation probably isn't 1, but the answer in terms of numbers of standard deviations is valid for all normal distributions.