Without evidence of benefit, an intervention should not be presumed to be beneficial or safe.

- Rogue Medic

What Math Do We Need To Know?

An interesting and very short – less than 3 minutes – presentation from a professor of mathematics about what we should be teaching in math class.

I can’t leave you hanging like that without telling you a little bit about what 2 standard deviations means. But first you should read what he said in, If Arthur Benjamin got an extra minute on stage ….

Now to what 2 standard deviations means. This is easiest if you look at a bell curve. Bell curves only represent normal distribution. You’ve seen this curve before.

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for 68.27 % of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45%; three standard deviations (light, medium, and dark blue) account for 99.73%; and four standard deviations account for 99.994%. The two points of the curve which are one standard deviation from the mean are also the inflection points.[1]

Here is another way of looking at the same information, just rotated to the side


The red circles with tails are the Greek letter Sigma, which is the symbol for standard deviation. The data points (dots) on the left are far too few to produce the smooth curve that appears on the right (which is just the bell curve from above with different colors and flipped on its side, so that the standard deviation bars approximately match). No the numbers on the left are not grades of a class from paramedic school before grading on a curve.

Why is it important to understand the concept of 2 standard deviations?

Anything outside of 2 standard deviations is unlikely. Anything outside of 3 standard deviations is rare.

In the video, he mentions the financial mess. One of the big problems was that bankers were making sub-prime loans that initially were just a tiny part of their business. As long as they remained a tiny part of the business, that was not a problem. Grouped in with an overwhelming number of safer loans, they would not present a huge problem, even if they all failed at once.

Mortgage writers were apparently ignoring the increasing proportion of loans that were sub-prime. They went from between 2 and 3 standard deviations to around 1 standard deviation. now it did not even take a coordinated failure of these loans to cause a huge problem, but the mortgage writers kept acting as if things had not changed. They were making their consistent commission. They were selling the risk to people with even less understanding of statistics. On the other hand, they were selling a risky asset, rather than keeping it as an investment. So, maybe they were not so unaware.

When the problem loans were only a few percent of business, the risk was small. You can see from the bell curve, that only a little of the shaded area is there. As the percent of problem loans increased, the chance of many failing at the same time increased. It was not a question of if, but when.

Things changed. Not a little bit, but a lot. The change was ignored.

We do the same thing in EMS in many ways.

We claim that some risk is small, insignificant.

We do this for many things.

We ignore the accumulation of rare, or just unlikely, risks.

We act surprised when the inevitable happens.

We blame someone else.

Footnotes:

^ 1 Standard Deviation
All of the graphs are from Wikipedia.
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