How do you calculate the standard deviation for Roulette? I've seen quoted in a number of places that the standard deviation for a 1-number bet on a 38 number Roulette wheel (0, 00, 1, 2, .., 36) is 5.76.  I can't seem to find a single source that shows how this calculation is made.
I am a student, this is for further understanding of a homework problem, so please assume I'm clueless and be as detailed as possible.
 A: Your question should make explicit what quantity you want the standard deviation of.
You say "a one number bet" but you don't clarify what the outcome is that you're considering. 
I will assume you mean the following:

Let there be a bet of one unit (say \$1), with a payout of 35-1 -- that is you stake \$1 and you end up with an outcome of either \$0 or \$36 (equivalently, a profit of either \$-1 or \$35). The probability of the outcome \$36 is 1/38. What is the standard deviation of the outcome?

Let's start first with the variance. (The variance is the square of the standard deviation.)
See the definition of variance here, and specifically for a discrete random variable here.
The outcome $0$ has probability 37/38 and the outcome $36$ has probability 1/38.
So the variance is $E(X^2)-E(X)^2 = \frac{36^2\times 37}{38}-(\frac{36\times 37}{38})^2 = 33.20776...\quad$
and the standard deviation is the square root of that.
A: sorry, I don't have enough reputation for a comment, this is a comment to Glen_b's answer.
When $X$ is the outcome,
isn't $E(X) = 0\cdot 37/38 + 36 \cdot 1/38$ ?
It leads to the same variance though.
A: The standard deviation is the square root of the sum of squares of deviations from the mean. So I suggest you finding the mean first, which is in this case the Expected Value. You have two X values, the gain amount and the loss amount.
