How is Sigma Computed

I will try to provide an intuitive explanation of Sigma by showing how it would be determined for the spinner below.

Technically the definition is THE SQUARE ROOT OF THE AVERAGE OF THE SQUARES OF THE DIFFERENCES BETWEEN ALL THE POINTS OF THE DISTRIBUTION AND THE AVERAGE POINT. Could this be why no one understands it?

Finding SIGMA for the spinner can be done in a few steps using Calculus , but if you know calculus you probably already know how Sigma is defined, so I will describe a more experiential approach instead that you may not have seen.

First, remember that the average outcome of the spinner was 0.5. We will need this for the calculations ahead. Here goes.

See? That wasn’t so hard. You’ll just have to trust me on the actual number above unless you want to spend a lot of time with a spinner.

Oops sorry, we’re not done. What we found was the Variance or Sigma Squared, not Sigma itself.

Before continuing in our quest for Sigma of the spinner, here is a geometrical interpretation of what we have accomplished so far. Define points in the XY plane with both X and Y values equal to the average, and all other numbers that come out of the spinner, as shown below.

For each number that comes out of the spinner, squares are drawn with one vertex at the average, and the opposite vertex at the corresponding point. Then the Variance is the average area of all the squares after hundreds of thousands of spins. The Variance is a Steam Era Measure of uncertainty in its own right. It has the useful property that the Variance of a sum of uncertain numbers is the sum of the Variances of each of the uncertain numbers.

Unfortunately the Variance is in squared units. That is, the Variance of giraffe necks would be expressed in square feet, and as for that naked phone call: “Boss, I expect profit to average $500,000 with a Variance of 2 billion five hundred thousand square dollars.”

That’s where the square root comes in. Sigma is the square root of Variance and would be expressed in feet and dollars for giraffe necks and profit respectively.

So (drum roll please) Sigma for the spinner is the square root of 1/12th… ta dah!

Oh did I mention that SIGMA's only redeeming feature (the confidence interval stuff) only applies to Normal Distributions anyway? So after all that work, it turns out that with the exception of an historical footnote, Sigma for the spinner is a bit like that ham dish you worked on all Sunday only to discover that you had re-created Spam.

Historical footnote

When early computer programmers wanted to generate normal random numbers, they would add up twelve uniform random numbers (spinners) and subtract the number 6. This was enough spins to give a good approximation of a normal, and further, they got a normal variable with a variance of twelve times 1/12 = 1, and a mean of 0. This is known as a standard normal.