The Method to Our 1-in-27 Million Madness
By Climate Central
By now, you've probably heard that 2014 was the warmest year on record. You may have also heard that 13 of the 15 hottest years occurred in the past 15 years, which left to chance, is the longest of long shots. How long? Like 1-in-27 million. We calculated those odds — and illustrated them with a nifty animation — and a lot of folks were interested in how we arrived at that number. Behold the methodology:
Here's how we set out to calculate the probability that 13 of the 15 hottest years would occur in the past 15 years without the influence of climate change. It's a little geeky but bear with us. Our statistical approach was to consider all the possible combinations you can get by reordering the 135-year climate record, and then calculate how many of those combinations have 13 of the hottest 15 within a 15-year period.
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Under naive assumptions of independence, one could randomly reorder the 135-year global temperature record as single years to represent the absence of a long-term climate trend. In that case, there are 105 ways to choose 13 of the hottest 15 years and 7,140 ways to choose two more years from the remaining 120. That gives you 749,700 ways of choosing 15 years from the 135-year record that contain exactly 13 of the hottest 15 years. That seems like a lot. But of all the ways to randomly choose 15 years from 135 (3.075637e+19 or approximately 31,000,000,000,000,000,000), that's only 1 in 4.102491e+13 or approximately 1 in 41,000,000,000,000 or 1-in-41 trillion.
A less naive approach would allow for short-term climatic variability, which may extend beyond a single year (but doesn't represent a long-term climate change trend). To capture we randomly reordered the period of record using 2-year blocks. By randomly choosing a start year for the block assignments, there are two sets of year pairings. The top 15 are found in the last 20 years of the series. Looking at those 10 pairs, the two possibilities are set1 = {2, 2, 2, 1, 2, 2, 2, 0, 2, 0} and set2 = {1, 2, 2, 1, 2, 2, 2, 1, 1, 1}. There are 472 ways to choose an appropriate subset from set1 and only 10 ways to choose them from set2. With equal chances of which set is chosen, the average is 241 ways to choose 8 two-year blocks that contain at least 13 of the hottest 15 years. There are 6,522,361,560 ways to choose 8 blocks from 67 (you can fit 67 2-year blocks into a 135-year period of record). So a conservative estimate of this probability is 1 in 27,063,741 or about 1-in-27 million.
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