One of the analogies sometimes used to explain the impact of global warming on the weather is that we’re “loading the dice.” Perhaps a better desrciption is that we’re changing the dice.

An ordinary die has six faces, with a single spot on one face, two spots on another, etc. etc. up to six spots. When you roll the die, you get an essentially random result between 1 and 6. It’s not uncommon in many games (craps, for instance) to roll two dice and add their numbers to compute the result. We could even roll three, or four, or as many as the game requires (Yahtzee, anyone?).

In any case, let’s call the result you get weather. The way the faces of the dice are numbered, with the six faces having the numbers 1 through 6, let’s call that climate.

Climate (the labels on the dice) determines what you expect to get. It also determines how much you can expect your result to vary. This fits the definition of actual climate, which is the mean and variability of weather over long time spans over large areas. Although climate determines the average result and how much variation we can expect, it doesn’t by itself determine the actual result. That’s a random result of rolling the dice … or if you prefer, a random result of natural variations in weather.

Suppose we roll three standard dice. The result will be some number between 3 and 18, and here’s the probability of each possibility:

The mean value is 10.5 with a standard deviation of 2.958. Now let’s change just one of the dice so that instead of having faces with 1, 2, 3, 4, 5, and 6 spots, its faces have 1, 3, 4, 5, 7, and 8 spots. Now the result of rolling all three will be some number between 3 and 20. Here’s the probability with the changed dice (in red) compared to that with the standard dice (in black):

The mean result with the changed dice is 11.667, its standard deviation is 3.375. Notice that we haven’t just increased the mean, we’ve also increased the expected variation.

Note also that this change doesn’t alter the chance of getting the lowest possible result. The chance that all three dice will turn up “1” is still 1/216. But we’ve reduced the probability in the middle of the distribution, and for most of the low numbers.

It’s on the high end that we see the greatest change, relatively speaking. The chance of rolling an “18” is way more than it was before. We even have a chance (albeit a small one) of results so high they used to be impossible — we could get a 19 or a 20.

When we change either the mean value or the variance of a distribution, then relatively speaking the most profound changes in the probability are likely to occur in the tails of the distribution, i.e., for the extreme events. Let’s take a look at how this might affect a different probability function, the normal distribution (the familiar “bell curve”).

Here’s the “standard normal” distribution (in black) compared to the same distribution shifted to a higher mean value (0.3) in red:

Here’s the ratio of the probability for a given result with the shifted distribution, to that with the standard distribution:

Note that for almost all results above zero the probability is higher but for all results below zero the probability is lower. For really extreme results the ratio becomes sizeable. The chance of a one-in-a-million result (roughly 4.75 standard deviations) is about 4 times greater.

While increasing the mean definitely increases the likelihood of high extremes and lowers that of low extremes, the greatest danger of more extreme events happens when we increase the variance of our normal distribution. Here’s the standard normal again (in black) compared to a normal distribution with increased mean value of 0.3, and only slightly increased standard deviation (from 1 to 1.1):

Here’s the probability ratio between the two distributions:

Note that for extreme highs, the probability is much increased. The chance of a 4-sigma even is about 10 times greater. We can perhaps get a clearer picture by plotting the same thing with a logarithmic y-axis:

This makes clear an interesting phenomenon: we have also increased the probability of extreme low events! The chance of a one-in-a-million low is twice as great as before, while the chance of a one-in-a-million high is just about 20 times greater.

And this illustrates one of the greatest potential dangers of global warming. If we increase the mean temperature (and we already have), of course we increase the likelihood of extreme heat waves (and we already have). But if, in addition, global warming increases the variance of regional temperatures, then we increase the likelihood of extreme heat waves by a lot. A helluva lot. The effect was profound when we only increased the standard deviation by a factor of 1.1 — what if it increases by a factor of 1.2 or even more? The increased likelihood of extreme heat would be astounding. What’s more, we would also increase the likelihood of extreme cold spells!

In fact this applies to all weather phenomena, not just temperature. Climate change is likely to change the mean value of each, and increased variance will dramatically increase the likelihood of extremes, bringing more heat waves and cold spells, more flood and drought, etc. We may, in fact, already be witnessing exactly this phenomenon. Welcome to the rest of our lives.


27 responses to “Craps

  1. Is there a reason to expect that global warming will change the variance of climate?

    • DO: Is there a reason to expect that global warming will change the variance of climate?

      BPL: Yes. More energy in the system, so more chaotic behavior. Do a between-means t test for the standard deviation of temperature anomaly 1850-1930 and 1931-2011 to see what I mean.

    • MightyDrunken

      James Hansen et al, in their paper, look at the variation and variance of the Earth’s climate over the past few decades. Fig 4 in the paper gives some evidence that the distrubution of temperatures have shifted upwards, while the coldest possible temperatures have not warmed as much widening the possible temperature range.

      P.S. I hope my url worked.

      • This is a more “formal” link for it

        Click to access 1204.1286.pdf

      • MightyDrunken

        Well my URL http skills failed.
        My post should have began,

        “James Hansen et al, in their paper “Perceptions of Climate Change:
        The New Climate Dice”, looked at the variation and variance of the Earth’s climate over the past few decades.”

        vv Thanks Flakmeister for giving the full report, I guess mine was the earlier version..

    • The notion (coincidentally) receives some empirical support here, if you read the whole story:

    • Brian Dodge

      A physical mechanism underlying increased climate variance is the decrease in temperature difference between the tropics and the poles. This causes the meanders in the jet stream to grow in extent, reaching farther north in ridges and farther south in troughs. They also move west to east more slowly, allowing more persistent weather patterns time to grow to extremes. Expansion of the Hadley circulation pushes the average position of the jet stream towards the poles, so I would think that Winnipeg or Copenhagen would get the effect of a larger, slower moving, record producing warm ridge more often than Atlanta or Rome would get a record snowfall. More heat waves on the north edge of the northern hemisphere jet stream would enhance polar amplification by earlier snow melt – once the snow pack is melted by an early spring heat wave, it won’t recover before the next winter.
      As for the water cycle, more moisture from more heat makes for bigger rainstorms, which dump a lot more water when they’re slower moving. Plus, the water vapor is concentrated near the surface because of the lapse rate, water vapor is less dense than N2/O2, and the latent heat concentrated near the surface but released by convection means more intense storm cells when they are eventually triggered. When you get a year’s worth of rainfall in the spring, and more runs off than soaks in, it won’t do much to alleviate the drought come summer time. The lack of soil moisture will intensify heat waves, which will intensify the adverse effects of drought – if there’s enough soil moisture to establish a crop in the spring, the growing foliage decreases albedo, but evapotranspiration carries away the energy captured as latent heat instead of temperature rises. If the soil moisture isn’t replenished by rain, evapotranspiration slows, the crop doesn’t set seed, eventually dies, and evapotranspiration stops, and all the energy appears as sensible 42 degrees C in the shade heat – think France 2003, Russia 2010, US 2012.

  2. Is there a more elegant way than monte carlo to determine the standard deviation for a number of X-sided dice?

    [Response: Since the dice are independent, the mean of the sum is the sum of the means, and the variance of the sum is the sum of the variances ( being the square root of the variance).]

  3. Bravo, Tamino, that is an absolutely brilliant article! I especially liked how you perceived the increased probability and degree of “extreme cold.” Your the first I’ve seen mention it. Regarding your last sentence, I am absolutely convinced that we are “witnessing exactly this phenomenon”… and in spades.
    Have you ever considered doing (or have done) any kind of analysis using something similar to the non-linear mathematics of “monster waves?” Might the “changes” actually be occurring on a trajectory described by a step-function and we may be “in” the 2nd, perhaps 3rd, such step? Thanks again, you do great works, I thoroughly enjoy reading “most” of your posts. :)

  4. Interesting and very clear. Thank you. But is there any evidence that there is a higher incidence (I realize the effect is slight) of extreme cold spells due to climate change, either in observations or in models? In other words, is this (slightly more cold extremes) a physical effect or an artefact from the statistical model you have chosen?

  5. Tony Abbott, leader of the Austalian federal Opposition, and almost certainly the next prime minister of the country, is on record as believing that “climate change is crap”. I don’t think that one could write a more succinct explanation of why climate change is actually “craps” with altered dice, but the sad reality is that one can lead an ass to water, but not force it to drink.

    I honestly think that some people could be transported a hundred years forward in time to see the results of their ideological refusal to accept what climate change means for humanity, and they would still hold to their guns. Frustratingly there are too many of these people in government and Big Business.

    • And herewith is the dragon’s head of ideology over reality.

      “I honestly think that some people could be transported a hundred years forward in time to see the results of their ideological refusal to accept what climate change means for humanity, and they would still hold to their guns.”

      Brave new world, indeed.

  6. Another fine post, making what is non-intuitive in statistics clearly understandable (for this layman at least).

  7. Andy S,

    The IPCC report on climate extremes came out earlier this year.

    Includes observations, and some distribution graphs very like those above.

  8. Tsk, not ‘climate extremes’. Extreme events.

  9. Sounds to me as worthy for publication. Especially the bit about increased variability and extreme cold spells. Any plans? Or is this actually common knowledge in the literature already?

  10. Hi, I’m a redditor that was just recently suggested to ask you a particular question because you’d be well suited to answer.

    My question is related to global warming and how there’s a new breed of “well 3 years don’t make a trend”, “well 4 years don’t make a trend”, “well n years don’t make a trend” talk that is coming from self professed “scientists”.

    My question was simply the following: given our long term knowledge of climate data, overal temperatures and known variations(? – do we even have variation data?) and our short term record of weather data, are we able to attribute a statistical odd to an n-string of record breaking years (record temperatures, record ice extent etc).

    (e.g. the odds that 5 consecutive hottest years is x% that it is merely a coincidence)?

    Thanks for your time.

    (sorry, I wasn’t able to find a way to contact you directly)

    [Response: Yes, it can be computed. If the data are not autocorrelated the calculation is simple — if not, it becomes quite complicated and is probably best estimated by Monte Carlo simulations. There’s also the question what “null hypothesis” you assume — there’s one probability in an unchanging climate but quite another in a warming climate. See this for one (in my opinion unsatisfactory) example.]

  11. Something is amiss – 1,3,4,5,7,8 would not give a range of 3 thru 20 for three rolls.

    [Response: Note that only one of the dice has been changed to that pattern, the other two are unchanged.]

  12. Pekka Kostamo

    “Weather” is not random, although it is still convenient to model it as such. Every weather event has a physical process behind it, The processes are, of course, complex and for that reason not very easily modeled with our current understanding and computing power.

    Increasing CO2, the several feedbacks and time constants result in warming, evidenced by higher measured average temperature. The processes producing our daily weather are modified as more energy is circulating in the system, ever so slightly for a tenth of a degree. For a full degree the change is more easily seen. When the global mean will be +18 degC instead of the current +14 degC, the processes and their weather products will be very much different. Unfortunately our understanding so far is quite limited, although not entirely missing.

    As an example on how the process runs consider the following. Higher average results in less temp differential between the North polar and equatorial regions. This produces a slower circumpolar jet stream with lesser diameter. (So far theory and measurements apparently exist).

    A slower jet stream has less turbulence and fewer eddies, and its meanderings are slower but (perhaps) more extensive as well. This impacts the generation of blocking high pressure areas, their life cycle, strength and shape are modified. A different jet stream and modified blocking highs in turn change the tracks of depressions that often determine our daily weather. These developments are not well understood yet, but are amenable for study.

    Slowing down of changes in “prevalent weather” type is apparently one result. So when it is hot and sunny, it stays hot and sunny for a longer period. Likewise when the depressions roll in, they continue to do so for a longer time. My intuitive comment would be that what used to be 2 – 4 weeks durations are now frequently 5 – 8 weeks. This slowing down is then manifested as a “hot summer” or “cold winter”, or whatever, as the sequence duration approaches the commonly used season descriptors “winter”, “summer” and so on. Temperature records are being broken in North America now. They were broken in summer 2011 in Europe – although less famously.

    Currently Northern Europe (where I live) experiences a cool summer (not record cold). The Siberian/Russian seasonal blocking high has taken a peculiar form. The Atlantic depressions roll in at their normal rate (one every 2-4 days), but they stop and decay over Northern Europe instead of moving on to the Arctic Ocean as would be more common. That pattern has been rather stable for 2 months now, and apparently will continue into August. The result is our rather stable current weather of variable cloudiness, almost daily showers, daytime max temp of 15 – 21 degC and night min temp +10 -15 degC. This has happened before, of course, but maybe not for all the summer.

    The above attempts to give just one idea on how the process works down from a higher global average temperature to the level of our changed daily weather. There are many other sub-processes that produce regional/local weather and they will change as well. Nothing will stay the same with more energy retained and circulating in the global system, and probably the changes will be quite remarkable. “Climate” is much more than the global average temperature. The climate scientists are busy working to understand these impacts.

    (Incidentally, the +16 degC future is now a lost chance. Better focus on a more realistic +18 degC world. It is hard enough to reach, political and economic realities considered.)

  13. Better focus on a more realistic +18 degC world.

    That world will not have a functioning global human civilisation.

    It will also likely not have a global human population greater than about a quarter of what it is today.

    • Your assertion is likely correct, and I am glad I’ll be dead and gone. Makes my blood run cold…pun NOT intended.

  14. If one just changes the values on a six-sided dice, one has to chose which values are exchanged. So is there a reason just to chose the 2 and the 6 to be excluded?
    I would suggest to change not the values, but the dice, or better the number of surfaces, by chosing 8-sided dice.

    • RFP:

      His example changed one of the dice to give it a slightly higher mean and variance. He used three dice in total to give a distribution that was both easy to understand and illustrative of the concept.

      He could have used an 8-sided dice for one (or all) of them and shown a similar (if rather more dramatic) effect, I expect he wanted to illustrate how even a smaller effect changes the odds of extreme events dramatically.

  15. Horatio Algeranon

    “Carbon Di(ce)oxide”
    — by Horatio Algeranon

    Normal dice
    Are quite precise.

    But carbon dice
    Are not so nice.

    Are you feelin’ lucky
    With Dice of Bucky?

  16. (Sorry, I meant to put this here, not under next post)
    A timely post–a record low for July 21 was set at BWI Airport outside of Baltimore, MD–69F, breaking the old record of 73F set in 1984. I’m sure this will be cause for denialists to claim global warming is all a hoax.

  17. “I’m sure this will be cause for denialists to claim global warming is all a hoax.”

    Though they don’t actually need a ’cause’ per se, historically the biggest incitements to doing so are pending climate change legislation and pending climate Framework conferences.