# How to interpret the standard deviation of a discrete random variable

How would you interpret the standard deviation of a discrete random variable and state it simply for someone with limited statistical background? For example,

Assume I am betting on an event with a 21% chance of occurring. If I bet it will occur and am correct, I win \$50. If I lose, I pay \$20. Let $$X=\text{profit from bet}$$.

$$P(x=\50) =0.21$$

$$P(x=-\20)=0.79$$

I logically understand the expected value $$\mu$$, which in this case $$\mu\approx-\5$$. If I was to explain what this value meant to someone, I would say:

If you make this bet many times under the same conditions, your long term outcome will be an average loss of $5 per bet. The standard deviation ($$\sigma$$) is equal to approximately \$29. But I struggle for some reason with how to explain this conceptually. Does this refer to the average deviation of each individual bet, just like $$\mu$$? Would it be valid to say:

If you make this bet many times under the same conditions, your long term outcome will be an average loss of $5 per bet, plus/minus \$29.

• Standard deviation is not the average amount by which an observation differs from the mean. That’s called mean absolute deviation. $MAD=\mathbb{E}[\vert X-\mu_X\vert]\le \sqrt{\mathbb{E}[(X-\mu_X)^2]}=sd$ (usually strict inequality, $<$). // Getting to your question, how would you explain standard deviation in the case of a continuous or even Gaussian random variable?
– Dave
Oct 1 '20 at 9:56
• Your final conclusion is definitely wrong. The uncertainty in the average long term outcome depends on how many bets you made and, for a sufficiently large number of bets, is arbitrarily small. You appear to confuse the standard deviation with the standard error. Good intro stats books do an excellent job of explaining this to people with limited (and no) statistical backgrounds. I am fond of Freedman, Pisani, & Purves, but many others of similar expository quality have appeared in the last generation, too. Consider consulting one of them.
– whuber
Oct 1 '20 at 13:21
• I have consulted a textbook and multiple websites and have yet to find a clear explanation, which is why I've posted my question. Oct 1 '20 at 17:08

Would it be valid to say: "If you make this bet many times under the same conditions, your long term outcome will be an average loss of $5 per bet, plus/minus \$29."