What's an intuitive description of the meaning of standard deviation in a discrete uniform distribution?

Question

Just starting out with distributions, so I'm looking for an every day explanation to help me understand.

I've read that for a discrete uniform distribution, the standard deviation is a measure of the spread one can expect from the mean on a given trial.

The SD formula given in the book I'm reading results in an SD for a single die of 1.71, which I am taking as correct.

But the average difference from the mean of 3.50 when you roll a single die is 1.5

So in plain English, I would 'expect' the difference from the mean to be 1.5 on any given trial. So what, in plain English, does the standard deviation actually tell you for a discrete uniform distribution?

Answer 1

The short answer is "Not a whole lot"...standard deviations are useful when they have meaningful probability statements (e.g., like with the Normal distribution).

In your cases, all SD really telling you is that typical fluctuations seem to be on the same order of magnitude as your mean, so you cannot treat it as "essentially constant" but the mean has some use as a central estimate (not really the case when SD>>>>mean).

Answer 2

It's not the average difference, exactly. The squaring creates a weighting effect: large fluctuations contribute disproportionately more to the standard deviation than small ones. For your die roll, you get rather large deviations from the mean when you hit 1 or 6, which happens 1/3 of the time. The standard deviation being larger than the "median deviation" (that is, the 1.5 you calculated) reflects this.

To play with this, you might try working with a variable which is $0$ with probability $1-1/p$ and $p$ with probability $1/p$, and increase $p$. You'll find that the mean and the "median deviation" stays the same, while the standard deviation increases.

The standard deviation is a lot more useful in working with things like Chebyshev's inequality, rather than as a number out of context.