
Calspace Courses
Climate Change · Part One
Climate Change · Part Two
Climate Change 2 Syllabus
1.0  The Ice Ages: An Introduction
2.0  Discovery of the Ice Ages
3.0  Ice Age Climate Cycles
4.0  Climate Through the Last 1000 Years
5.0 Determining Past Climates
· 5.1  Reconstructing Climate Change
· 5.2  Stories Told by Trees and Corals
· 5.3  Warming Since A.D. 1850
· 5.4  The Statistics of Change
6.0  Causes of MillennialScale Change
7.0  Climate and CO_{2} in the Atmosphere
8.0  Recent Global Warming
9.0  Climate Change in the Political Realm
10.0  The Link to the Ozone Problem
11.0  Future Energy Use
12.0  Outlook for the Future
Introduction to Astronomy
Life in the Universe
Glossary: Climate Change
Glossary: Astronomy
Glossary: Life in Universe


The Statistics of Change
An example of a perfect Gaussian distribution of data, like global temperatures, where X represents the deviation from the “mean” value of the data set and F(X) represents the actual values of a data set. Most values fall near the mean, that is, near X = 0, while less data points would be found at value at X = 3σ (that is, 3 standard deviations away from the mean). 
Statistics are a familiar tool in making judgments about what is really happening. If a basketball player makes, on average, 70 percent of his freethrows, and he only got 3 out of 10 in a game, he had a bad day. If he got 6 out of 10 it was close enough the average that we are not surprised. The same is true for 8 out of 10. We expect that from him, with a little luck. Statistics compares expectations with what is actually observed. It helps us decide whether something strange is going on, or whether things are normal, and follow established patterns. We can use statistics to discuss the question of whether climate has changed recently or not.
The course of recent temperature history within the fluctuations of the last ten centuries leaves little doubt that climate did change. There is nothing very subtle about it. Only addicts of wishful thinking will close their eyes to what is evident. So why do we need statistics? We need it to define the meaning of "obvious." Remember, scientists are always in doubt, as a matter of principle. We should be able to offer statistics that describe the odds that the statement "the climate has changed" is false, even if we are quite sure that it is correct. How likely is it that the last two decades were as warm as they were simply by chance, as a matter of natural fluctuations?
First we note that the last two decades are the warmest on record for the last 1000 years. Let's say, just to be sure to avoid error, there are only two other such warm decades. We then have four extremely warm decades, two of which are the last ones. Can we say that global warming has arrived, as expected? The probability of finding a warm decade in a certain position along the one hundred decades that make up the last 1000 years is four in one hundred. The probability of finding another warm decade following the one we have previously fixed for the 1980s is three in ninetynine (derived from the number of decades left over divided by the number of slots left over). The product between the two is twelve in ten thousand. We could also fix the 1990’s and have one other decade preceding: another twelve in ten thousand.
By this simple argument, the probability of the 1980’s and 1990’s being an extremely warm decades, by chance alone, if only four such decades are available in the last millennium, is twentyfour out of ten thousand, or 0.0024, or 2.4 per thousand.
If the two last decades are in fact the two warmest (not just two among the four warmest), the probability of finding them by chance at the very end of the record is 0.0002, or 0.2 per thousand, twelve times smaller again.
Alternatively to the simple combination approach, we can also use Gaussian statistics (named after the German mathematician Karl Friedrich Gauss, 17771855, who invented error analysis, among other things). In this
type of calculation, we work not just with the rank of the decades in terms of warmth ("the two warmest") but with how warm they actually are ("very warm"). The distance from the expected mean is given by an artificial parameter called "standard deviations," which are symbolized by the Greek letter sigma (σ), where roughly two thirds of the numbers are within one standard deviation of the mean, at least in a wellbehaved Gaussian distribution.
Our two decades are at three standard deviations from the mean, or beyond. The probability of finding such an unusual value at a given place in the series is 3 in a thousand or less. The probability of finding another one in the position next to it (on a prescribed side) also is less than 3 in a thousand. Thus, the probability of finding the pair in a certain position is the twice the product, which comes to less than 18 per million, a number ten times smaller than if we use the argument building on the position of the "two warmest decades" without specifying how warm.
In conclusion, we can safely say that the chance that we are wrong when saying that "unnatural" climate warming has arrived is somewhere between less than three in a thousand and less than one in ten thousand, with a best guess of less than one in a thousand.

