There's a room full of CEOs. Each would like to know whether they are paid above average, and how much above average. But no CEO wants to tell their own salary. How can this be done?
I'm indebted to the BBC show "More or Less" for this trick.
The first CEO picks a random number -- any number -- and adds his salary to it. He whispers this number to the second CEO.
The second CEO adds their salary to this number and whispers the new number to the third CEO.
The third CEO adds their salary to this number and whispers the new number to the fourth CEO, and so on.
The last CEO adds their salary to the number and whispers that final number to the first CEO. The first CEO then subtracts the random number, divides by the number of CEOs in the room, and voila! there's the average.
To get the standard deviation, the first CEO picks two random numbers, and adds his salary to one and the square of his salary to the other. At the end, once the random numbers are subtracted off the total, we have the sum of the X's, the sum of the X-squared values, and n, which is all you need to compute the standard deviation using the computational version of the formula.