Five Sigma threshold crossed for Global Warming
What the heck is “Five Sigma?”
The online publication Earther recently reported the strengthening of the statistical case for global warming based on satellite data. If you find Earther’s Evidence for Global Warming Passes Physics’ Gold Standard Threshold readily comprehensible, then you need not read most of this post. (The “Gold Standard” is five sigma.)
If you have been pretty satisfied with the Earther article, I invite you to skip way down to the section “Three further notes of clarification [etc.]“
If Earther’s piece is not easily comprehensible, that’s probably because writer Daniel Kahn did not explain just what “five sigma” means statistically. My hunch is that 30% of my readers will understand and remember it, another 45% understood it in the past but have forgotten a lot of it, and the remaining 25% may never have had it presented to them. If you belong to the latter group, don’t blame yourself—blame the scattershot American education system.
Simply said, “five sigma” comes down to we have zillions of measurements and 99.99994% of them confirm that global warming is real. There is an item of faith here: you have to trust that NASA has amassed enough data over a long enough period of time to meet the requirements of a statistical analysis. My “zillions of measurements” encapsulates that faith to my satisfaction. Very scientifically of course. (Well . . . The Shorter Oxford English Dictionary defines zillion as “a very large but indefinite number.”)
Does understanding “five sigma” matter?
It matters if you’d like to have a scientifically bulletproof case for the existence of global warming. Don’t you want bulletproofness?
At this point, those who think the case for global warming is bulletproof enough, damn the details, can skip down seventeen paragraphs to the “Three further notes [etc.]” section which could be useful in fleshing out the argument. (TWO is where you find the stunning Ted Cruz reference!!)
Otherwise, you can slog through the next seventeen paragraphs that explain more fully what “five sigma” tells us. Math-averse folks may find it tedious, but believe me I’m not getting any deeper into it than arithmetic. Addition, subtraction, multiplication, division.
“Sigma” ( σ ) is just shorthand for standard deviation, borrowed from the Greek alphabet (for conciseness, scientists have borrowed from Greek for various quantities, e.g. pi, lambda, alpha, rho, mu, etc. It’s simpler to put “σ” into an equation rather than “standard deviation.”
You can search the web for “standard deviation” and get a more precise explanation than I’m giving below, but I’m trying to make it as simple as possible to save time, and get to what standard deviation is good for.
Standard deviation (aka sigma) is the average of the difference between individual data points in sample and the mean (average)* of those all points.
To take the simplest possible example: say you take temperatures on two successive days in January at 6 am and you get minus one and plus one (-1 and +1.) (I’m using degrees Celsius.) The mean is the sum of those two numbers—zero in this case—divided by the number of measurements (two), giving you a mean of 0/2 = 0.
That doesn’t tell you a whole lot. The mean is obvious. But how far those measurements deviate from the mean tells you something you might want to know—such as how much variability there is in a set of data points. If you had successive measurements of -10 and +10, the mean would still be zero, but there’s a striking difference in the meaning. Does the temperature bounce wildly around the mean, or does it stick pretty close?
Here, the average difference in the first case is pretty obvious, because both measurements differ from the mean (zero) by the same amount—one (1). That’s the standard deviation = sigma = σ. Almost as obvious is the second case where the standard deviation is 10.
The practical meaning of the standard deviation
Let’s suppose you take the temperature 7 days in a row and came up with something like this: -17, -3, 8, -12, -4, 0, 22 (the last = 72 degrees F.). The mean is still zero, but . . . you’ve been aware that from morning to morning there are pretty wide temperature swings, although you don’t have a strong grasp of how wide these swings are around the mean. Can you express it in one number? Ta da! enter the standard deviation, aka σ, aka sigma: 12 degrees** Celsius, or 22 degrees Fahrenheit.
Sounds like a lot. Based on this, you wouldn’t be greatly surprised if on any day the temperature is 22 degrees or more different from the day before! Big enough to remind you of the notorious Polar Vortex, and make you speculate that maybe these wide temperature swings are signs of something bigger than day-to-day variability.
Needless to say, my contrived example of seven measurements is so small it doesn’t represent reality very well, say of the entire month of January. More data points should smooth things out, but I didn’t want to concoct 31 fictional temperatures and run all the calculations and fuss around with Celsius and Fahrenheit etc. and . . .you wouldn’t want me to.
Those simple examples illustrate the principle of mean and standard deviation. In reality, you want as big a sample as reasonably possible. (Sample size, and the methods of collecting the sample, come into play when evaluating the accuracy of opinion polls, which do not always tell you what you think they tell you.)
What makes “five sigma” so compelling?
Here’s where the magic of statistics comes in. The (blue) graph below of a “normal distribution” of data conveys the principle pretty clearly. The vertical axis is the number of instances of the values on the horizontal axis—a frequency distribution. Imagine you take 30 years worth of daily 6 am January temperatures (930 measurements), and the mean (sum of all temperatures divided by 930) is zero. Then zero will be at the center of your horizontal axis, and the highest point on the vertical axis will be the number of days with a temperature of zero, or very close to it. (I’ve fudged a little here; see my footnote***.) As you move away from zero in either direction, there will be fewer and fewer numbers of days with those temperatures as you slide down towards the tails.
(Caution: the symmetry of the curve shown—an idealized “Bell curve”—is more perfect than you usually find with real-world data. But even where the curve appears lopsided, the standard deviations have the same meaning. Search on “long-tailed distribution.”)
The vertical lines in the curve mark where the standard deviations fall. The center line is the mean. On either side, the first line out shows one standard deviation; the next line out shows two standard deviations, the third line three standard deviations, and so forth. At the scale of this example, the fourth and fifth standard sigmas are not even marked, since they fall so far out on the tails of the distribution, that the curve vanishes. (Nevertheless, with a huge quantity of data, there will often be measurements under the curve even out on the tails.)
As it turns out, if you take the mean and the standard deviation of a whole ginormous pile of measurements, you will find that roughly 68.5% of your measurements will fall within one standard deviation (one sigma) on either side of the mean, 95.4% within two sigma, 99.7% within three sigma . . . and out to 99.99994% of measurements falling within five sigma.
That is, a five sigma result determines there’s a .9999994 probability that a phenomenon under study is real. (Certainty is 1.0.) Something you can bet the farm on. Unfortunately, it’s something that know-nothings such as Donald Trump are betting civilization against.
Higgs boson analogy
“Five sigma” is such a big deal that it has led most physicists to agree that the Higgs boson—the so-called “God particle”—was discovered by the Large Hadron Collider (LHC). The LHC, you may recall, is the enormous machine in Switzerland where countless protons are smashed into each other at nearly the speed of light. The smashes produce showers of particles of various characteristics that are analyzed mathematically, most of which belong to previously known particles. Then along came a new particle, identified as very probably (five σ) as the Higgs. Named after the theoretical physicist Peter Higgs, who postulated its existence back in the 1960s, the Higgs (more technically, the Higgs field) is believed to give mass to all the stuff in the universe, from Rice Krispies to red supergiant stars. (Photons, traveling at the speed of light, do elude the Higgs field, and they have no mass.)`
Pretty darned fundamental. In the LHC, there was a “bump” in the distribution of measurements that tested positive for the Higgs within five sigma of the center of the bump. (For more on the Higgs evidence, see this in Physics Central.)
Therefore, IF the five-sigma test is good enough for most physicists to agree that the “God particle” was found by the LHC, THEN the five-sigma test should be good enough to establish the reality of global warming. So goes the case that Daniel Kahn makes in Earther.
Three further notes of clarification on Brian Kahn’s article in Earther
ONE: Keep in mind that—this is the tricky part that Brian Kahn doesn’t fully explain—what’s under analysis is not just a series of temperatures with their means and standard deviations (as in our examples above), but a series of differences between the long-term temperature means (referred to as 10-year “smoothings”) and new trends. The five-sigma refers to the difference of the differences—not only that there is a trend (temperature goes up), but also how consistent the trend is (.9999994 of measurements either up or down from the trend line fall within five sigmas of the upward-sloping trend line.) The up-and-down movement of the measurements around the trend line result in the “sawteeth” appearance of the typical temperature graph (see graph of temperature vs CO2 levels a few paragraphs down).
TWO: Kahn refers to Ted Cruz’s**** bogus arguments against global warming based on a distorted interpretation of satellite data. Cruz has been right about one thing: upper atmosphere (stratospheric) warming has been less pronounced than surface data, and satellite data measure heat from above the stratosphere. But that is exactly what one would expect from the Greenhouse Effect that traps heat in the troposphere (lower atmosphere) in which we on the surface live, love, laugh, and curse a dysfunctional U.S. Congress. Heat is transmitted from the troposphere up into the stratosphere very slowly. It has been slow enough that the five-sigma result has been a long time coming.
THREE: the caption of the graph of surface temperatures in Kahn’s article refers to “naturally forced climate.” Natural forcing is what influences climate in the absence of greenhouse gases produced by human activity. Natural forcing is what has raised long-term temperatures above those in the last major Ice Age about 18,000 years ago. The most consistent short-term forcing comes from the most prevalent greenhouse gas, water vapor. Water vapor participates in a positive feedback effect, wherein the more water vapor the more warming, the more warming, the more water vapor (from evaporation), and so forth. The natural increase in water vapor and its consequent warming is a part of what is referred to as “noise” in the data that NASA’s analysis has to account for. Kahn refers to the “noise” in the Earther article beginning with “Gavin Schmidt, the director . . . ”
Also contributing to the “noise” are things such as sunspots, volcanic eruptions, and big ocean circulation phenomena such as the North Atlantic Current of which the Gulf Stream is a part, and the well-publicized El Niño and La Niña in the Pacific.
Climate is complicated, which is what allows the conspicuousness of contrarians such as David Happer, the physicist Trump has invited to join his Climate-Change-Is-a-Hoax panel. (Happer also believes there is no such thing as the HIV virus. )
Another natural forcing factor is solar radiation received by Earth. Over long time spans this has significant effects (see Milankovitch cycles), but for a discussion of why this has little to do with current warming, see this from the Union of Concerned Scientists.
FINALLY: Global Warming is real (and so is the climate change driven by warming). BUT . . .
The foregoing arguments point us to the more narrow interpretation of warming we call Anthropogenic Global Warming (AGW), the term we use to assign blame for warming to humans. That’s because we are responsible for the steep increase in greenhouse gases, principally carbon dioxide. The increase has accelerated what might otherwise be a merely natural warming (search the Holocene).
THAT brings us to the skeptics’ fallback position (sigh), that even if global warming is real, human activity has little to do with it. Apologists for the fossil fuel industry (whose own scientists actually know better) and stooges such as Scott Pruitt repeat this mantra ad nauseam. Thus the scientific distinction made between warming due to natural forcing and warming due to artificial forcing. There’s such a close correlation between the increase of atmospheric carbon dioxide due mostly to burning fossil fuels and temperature increases, that only the most dug-in deniers any longer bother to question the extreme likelihood of a causal relationship, given all the other physical science pointing to the same conclusion.
This graph represents a correlation between CO2 increases and temperature increases 1964 through 2008. The trend going back into the 19th Century (not shown) displays an upward trend that has been steadily rising, but has accelerated during the last five decades. That is probably due to a shift in the carbon cycle involving feedbacks added to the unleashing of more greenhouse gases by us, discussed in my earlier post Whose Hoax? The Carbon Cycle and Climate Change Denial
============= footnotes follow ===============
* “average” usually refers to the sum of the data points divided by the number of data points, i.e., the mean. Another kind of “average” that’s thrown around loosely is the median, which is where there are as many points above as below. Both mean and median are statistical “measures of center.” Often, the median is a better indicator of a real-world situation. For example, see my post Hollowing out of the middle class – a second look) for real-world implications of the median vs the mean to evaluate wealth disparities.
** The quick way to compute this (if you have a calculator handy) is to take the squares of the differences on each day (since the square of a negative number is always a positive, it simplifies the procedure), add the squares up, divide by the number of measurements, then take the square root of the answer to get the standard deviation. You can see it works for something as simple as our example of -1 and +1. Adding the squares of those gives you 1 + 1 = 2, divide by the number of measurements (2) and you get 1, of which the square root is 1. Easy-peasy! Not quite as easy if your mean was not 0, but, say, 3—you have to take the difference between the measurement and the mean, and then square that. E.g. if the mean is 3 and your reading is 9 you would square (9-3 = 6) and get 36. Three (3) would be the center of the graph of the distribution.
*** Warning (SORRY!): In our example of 930 January days, it could be that the temperature was NEVER EXACTLY ZERO —and how closely your data match a normal distribution depends on how fine-grained your temperature readings are. Say you’re using integers (a coarse measure), and you have no readings of zero, but have fifty days of -1 and fifty days of +1, while lesser frequencies on either side of them keep falling away in a normal-distribution sort of way. It would appear a little weird with a gap in the center, even while the other measurements fit. But it’s only an appearance—a tiny fraction of anomalies does not negate the normal distribution. (I’m pretty confident 930 temperature measurements will give you a normal distribution with few gaps.)
If you’re acquainted with frequency histograms, you will recognize the curve as a smoothing out of the height of bars in a histogram—it’s the same sort of approach used to derive how the integral calculus gives you the area under the curve of a function.
At right is a histogram over which a normal curve is fitted even though the center bar—the mean— is lower than its shoulders. The sample here is small, but if you were to take measures of temperatures for 930 days you’d have a lot more, and narrower, bars for the histogram. The range of temperatures on the horizontal axis might be -25 to +25, in which case you’d have 51 bars.Therefore, if there were no exact zero measurements, there would be a very narrow vertical gap in the exact center, but you’d still probably have a histogram you could fit a normal curve to. Statisticians have a formula to compute what they call “goodness of fit” which in principle you should get for your temperature graph before you actually fit the curve, but I’ve forgotten how to do it and I guess you don’t care.
For more info on the histogram above, see sample histogram with normal distribution
**** I have heard several times that Ted Cruz was a “champion debater” in college, which suggests an intelligence at least one standard deviation above the mean. This begs the question of how much he is just faking (consciously lying) about the satellite data and warming, versus how much he’s actually deluded himself that he’s talking about reality. My guess is he’s faking on behalf of his friends in the corporate world. This suggests that Lying Donald Trump has also been right about one thing: “Lying Ted” is a good nickname for the senator.