Last edited Fri Jul 5, 2024, 12:12 AM - Edit history (4)

The graph illustrates that a small shift to the right in the average shifts the whole bell curve to the right, and, in this illustration makes hot weather (orange) much more common and extreme hot weather (red) from almost zero probability to considerable probability

if, for example, the average daily high in July in some locale is 85 degrees with a 6 degree standard deviation, and normally distributed: [1]

then the number of days when the high is 103 or above (3 standard deviations above the mean) is 0.1350% of July days.

In Excel, the formula for finding the area under the normal distribution from 103 to infinity with an average of 85 and standard deviation of 6 is:
=1-NORM.DIST(103,85,6,TRUE)
which gives an answer of 0.001350 which is 0.1350% [2]

OK, so no biggie. So what?

Now lets say that due to climate change so far, the average July daily high temperature has shifted by just 2% to the right, from 85 to 86.7

then the number of days when the high is 103 or above changes to 0.3297% of July days.

That's a 2.44 fold increase (144% increase) in the number of 103+ degree days for just a 2% increase in the average.

Shift the average 4% to the right, from 85 to 88.434, and you get 0.7598% of July days, a 5.63 fold increase (463% increase) in the number of 103+ degree days.

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Repeating the above exercise, but with a less extreme example, finding the number of days with a high of 97 or above (2 standard deviations above the mean)

with an average July daily high of 85 degrees, the number of July days when the high is 97 degrees or above is 2.275% of July days.

Now lets say that due to climate change so far, the average July daily high temperature has shifted by just 2% to the right, from 85 to 86.7 degrees

then the number of July days when the high is 97 or above changes to 4.302% of July days.

That's a 1.89 fold increase (89% increase) in the number of 97+ degree days for just a 2% increase in the average.

Shift the average 4% to the right, from 85 to 88.434, and you get 7.669% of July days, a 3.37 fold increase (237% increase) in the number of 97+ degree days.

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The same principle applies to other climate change events, e.g. the severity of storms or what have you -- a small shift in the average results in a huge increase in the number of extreme events.


======= FOOTNOTES ===========

[1] Temperatures are normally distributed according to a Google search.

Picking New York City for example (with an 84 degree average daily high in July)
https://weatherspark.com/m/23912/7/Average-Weather-in-July-in-New-York-City-New-York-United-States

The 10 and 90 percentile bands are shown on the graph, and they are (reading from the graph and using the peak July date) are 77 to 92, which is a band width of 15.

That occurs when X is -1.281552 to +1.281552 standard deviations
https://en.wikipedia.org/wiki/Standard_deviation
See the big table about 2/3 of the way down the article that has this row:
1.281552sigma   80%   20%

Meaning that between minus and plus 1.281552sigma, 80% are within that confidence interval and 20% are outside of it.
(The greek symbol sigma is the symbol for standard deviation. DU replaces it with a "?" so I show "sigma" in the above)

So for a 10 to 90 bandwidth of 15, the standard deviation is 5.85228
(15/2 = 7.5,   7.5/1.281552 = 5.85228)

So, rounding, I used 6 as my standard deviation in the above example.

[2] The TRUE means its the cumulative normal distribution as opposed to the probability density function
The area under the normal distribution curve from minus infinity to 103 (and having an average of 85 and standard deviation of 6) is NORM.DIST(103,85,6,TRUE)
The area from 103 to infinity is
1-NORM.DIST(103,85,6,TRUE)