Theoretical stats question re standard deviation

[BruceZ]

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Thanks for the response Bruce. Can you just explain the formula sqrt(0.25*2^2 + 0.75*0^2 - 0.5^2)?

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0.25*2^2 + 0.75*0^2 is E(x^2), the expected value of the square of the win. 0.5^2 is the square of the expected value of the win.


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If I have a series where the outcome can only be 0 or 1, the standard deviation is going to be sqrt(prob of 0*0^2+prob of 1*1^2-mean^2)?

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Correct.


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Also, what does the formula look like if I have 3 outcomes?

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For any number of outcomes, standard deviation is:

sqrt(prob1*value1^2 + prob2*value2^2 + prob3*value3^2 + ... - mean^2)

BTW, this is equivalent to:

sqrt[prob1*(value1-mean)^2 + prob2*(value2-mean)^2 + prob3*(value3-mean)^2 + ...]

The part inside the square root is the variance. This is the expected value of the squared difference between the values and the mean. That's why the standard deviation is called a ROOT MEAN SQUARE or rms average. It is the root of a mean of squares.

Note that this second form is sometimes preferable when working with real data, because the first form can be susceptible to roundoff errors. That is, you have to be careful that you don't subtract two very large numbers that are almost the same, because the difference might be so small that your calculator can't represent it, and you end up throwing away your whole result.