Which of the following is the standard deviation of the random variable x, the number that appears on a die, if the probability of any number on a fair, 5-sided die is 1/5? A. 0.2 B. 1.581 C. 2.5 D. 1.412

**Answer:**

By my accounts, it’s none of the above. I come up with a value around 2.68. Here’s my approach:

Each outcome is equally likely, for example P(roll=1) = … = P(roll=5) = 1/5

So, we use the formula for the standard deviation:

Where σ is our standard deviation,xi is our individual outcomes and N is the total number of possible rolls (in this case 5).

First, we need to get x, which is our sample mean (x should have a bar over it, but I cannot figure out how to do so with LaTeX, so just imagine it’s there!) so to do so, we add up each individual outcome times its probability of occurring:

Now, we calculate the deviations from the mean for each possible roll (those rolls being 1,2,3,4, or 5). For the sake of simplicity, I will show one of these calculations, but it is the same for the others.

Now, you would repeat this same process for when the roll equals 2, 3, 4, and 5; then sum all these up and that should give you a standard deviation. Given the symmetry of this problem and the fact that it’s a fair die, you only need to calculate the deviations for when the roll is 1 and 2 and multiply each result by 2 and sum them to get the same result as calculating it for all 5 possible outcomes.

So, when we sum them all up, we get something around 2.683 (with some other stuff after it that doesn’t matter given the lack of decimal places in the possible answer). Now, I know that isn’t a possible answer- so either I made a mistake (which is EXTREMELY possible seeing as how it’s 2 am), or there is a mistake somewhere in the problem. I would try to review this with your instructor and see what they have to say.