Playing At Dice - What That "Weekend Exercise" Was All About
By Skanderbeg Posted in Archived — Comments (28) / Email this page » / Leave a comment »'Morning, class!!
Prof. Skanderbeg was pleased to see that some of the readership actually took up the challenge of doing the homework assignment - and the exercise was valuable, since (judging by the comments) it did seem to be getting the point across.
Just to explain what that was all about, I thought I'd put some comments in a separate post.
There are basically three items to explain.
The short take is that using a pair of dice and summing up the results hands you about the simplest possible physical "system" that actually presents the real behavior of a "natural system" that is behaving statistically (rather than deterministically) - that is, it gives you a statistically-correct probability density function.
If you have one die, the probability density function is boring and trivial - each value has an equal 1/6th probability. But with two dice and the addition of the results, you get a very good and properly-behaved statistical system. That is, if you histogram-out the 36 possible additive combinations, there is a stable statistical mean value (7.0), and the distribution of other probabilities (out to 2 and 12) follows a Gaussian distribution.... along with proper standard deviation behavior and so forth. So it's about the cheapest and easiest laboratory for examining the behavior of statistical systems (both natural and man-made).
If a system is behaving "statistically," it will show a stable central mean, a Gaussian statistical distribution, and proper allocation of standard deviations. The two-dice system does this.
As "nilram" noted....
I wrote a quick perl program to simulate a dice roll and plugged the output into gnuplot. The average was 6.917, sd=2.37As long as you have access to a good pseudo-random number generator, you can code this up. (It's more spectacular to flummox people by doing the by-hand version with real dice, of course, but we can save that for when we're doing Congressional testimony.) From what I can tell, nilram's experiment involved 1000 rolls of the dice. I had done this some time ago with my own code and had rolled only 100, but produced the same results (though I rounded it off a bit more strongly, producing a mean of 6.9 and a standard deviation of 2.3). I'm too lazy this morning to actually go directly compute the standard deviation from the system histogram, but given that we got the same results from the same approach, I'd say that these are the real results for this statistical system.
The "bottom line" here is that a statistical system must be described in terms of not just its mean (average), but also in terms of its standard deviations. This two-dice system is very well-behaved and comprehensible.... but note that the standard deviation is rather large. That's the rub about dealing with "climate" and such - if you go look at any measured temperature data, the standard deviations (of say the high temperature for a particular place on a particular calendar date over a period of say a century) are huge - one standard deviation is at least multiple degrees, and frequently reaches double digits. This calls into question the sanity of obsessing over various forms of "data processing" that boil down to blind panic over apparent variations of fractions of a degree.
Sequential Plotting - The "Trend Chart"
So we have shown that we have a nice, simple statistical system to study and benchmark against - the humble pair-of-dice. Nothing strange about it.
Or is there? This gets us into the slightly wacky world of statistical process control (SPC). This is important stuff, and we should give much credit to Walter Shewhart and his protege W. Edwards Deming for their insights in this field. The name "Deming" might ring a bell, even though (sadly) he is no longer with us; he is the famous "quality guru" who finally got his due in the 1980s. He was the first to realize the implications of SPC for manufacturing; he got brushed off in the U.S. (especially by Detroit), but got a ready audience in Japan, where his ideas were adopted enthusiastically by the nascent Japanese auto industry.... and the rest is history.
The short take (too short) is that in a manufacturing process, the first goal is to create a stable process - that is, one that is (of course) meeting the basic targets but which has been "stabilized" - that is, under monitoring, it is behaving statistically, with a stable central statistical mean, a Gaussian profile to the bulk data, and reasonable standard deviations. The first task in stabilization is to get the mean to the right place, and then to get the magnitude of the standard deviations down to an acceptable level - without, of course, dorking up the stability along the way. When that has been achieved, the process is healthy and ready for routine manufacturing use.
But here is where the Deming-based "fun" really starts. You need to have ways of deciding if the process is indeed stabilized - and, perhaps more importantly in the long run, if a "stabilized" process is actually remain stable (statistically), or if it is actually systematically drifting away from the desired stability - and heading toward moving away from acceptability.
This is where SPC comes into play. The simplest approach is to identify a few things that you can actually measure well and measure sequentially - the thickness of a thin film layer, the width of a metal rod, whatever. As "stuff" comes along, you sequentially measure a list of items, and plot the trend that they show.
When you make plots of this sort, you can start to see trends - and this is where Deming's brilliant insight came into play. A well-behaved, well-centered statistical system will, when examined "sequentially," appear to show trends that look like "drift." This is how nature behaves, but if people don't know that, they tend to panic. They see an apparent trend, think that the manufacturing process is drifting, and make changes to the process to correct the "drift." Now they really have changed the process, but they've basically dorked it and it then does become broken - to the great cost of the manufacturing, who suddenly can't make things - as the engineers involved run around in circles chasing the "trend" that really isn't there.
That's the challenge of SPC - the need to be able to determine if some apparent "trend" represents a real, systematic drift of the process - or if it's just a statistical fluctuation.
Does that challenge sound at all familiar in the "climate change" context?
As "bennjneb" asked,
What point are you trying to make by recording a bunch of dice rolls?Prof. Skanderbeg gave out the homework assignment of basically creating a "trend chart" (or control chart) for the simple system of the two dice. If you do this, you will actually see sections that will seem to show a "trend" of some sort - an apparently significant trend of the numbers rising or falling. If you haven't tried this yourself, it's a highly recommended learning exercise - literally one of the most enlightening (about nature) that one can do.
So as "rdbwiggins" replied,
Given the degree of extrapolation required to determine the average temperature of the earth at any given point in time, random rolls of the dice would be about as accurate at predicting future climate change as the current computer models.Yes, that's more or less the point. If the system is behaving statistically, it will show apparent sequential trends that in reality are mirages. The dice experiment demonstrates that - and if you look at statistical and sequential temperature data, you see the exact same behavior!
Now, "Neil Stevens" said something really interesting:
Red Paint causes global warming!I'm assuming this is actually a report of results. But it basically gets the point. If you take some red dice and also some green dice and roll them independently, they can both have the exact same bulk statistical behavior - yet they can show completely different apparent trends!
I'm serious! In clinical trials, red dice show HIGHER average climate temperatures than any other color!
This is why we must be careful with any "data" regarding climate and/or temperature and/or whatever. We are looking at a system that is behaving statistically - not deterministically. Trying to impose determinism onto a statistically-behaving system leads to conclusions that are comical - except when they cause you to "intervene" and take the manufactured-product yield to zero.... or to destroy the global economy and global society.
(N.b. I'm aware of Nassim Taleb's book "Fooled by Randomness" - in fact, a copy of it is in my "travel reading stack" and will come to the top on some plane at some point. His "The Black Swan" seems to be showing up everywhere - I have that as well, but haven't read it yet either. The touchstone on this for Prof. Skanderbeg was Henry Neave's "The Deming Dimension," which looked at SPC in manufacturing and then went beyond to apply it to other things. Later in his long and productive life, Dr. Deming got interested in notions of how SPC principles were catastrophically not being understood in a variety of other settings. I really wish he were alive today to comment on SPC and "climate change." I honestly believe that he'd produce an analysis much like this one.)