Extreme heat is arguably the easiest event for scientists to model.

Temperature is one-dimensional and more or less follows a normal distribution for a given region. As climate change continues, temperatures increase (shifting the bell curve to the right) and become more variable (flattening the bell curve).

The end result is a significant increase in extremely hot weather.

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As any statistics student knows, 99.7% of the Earth’s surface should have temperatures within three standard deviations of the mean (this is just an interval, with length dependent on how flat the bell curve is) at any given time.

So if we still had the same climate we did between 1951 and 1980, temperatures more than three standard deviations above the mean would cover 0.15% of the Earth’s surface.

However, in the past few years, temperatures three standard deviations above average have covered more like 10% of the Earth’s surface.

Even some individual heat waves – like the ones in Texas and Russia over the past few years – have covered so much of the Earth’s surface on their own that they blow the 0.15% statistic right out of the water.

Under the “old” climate, they almost certainly wouldn’t have happened. You can only explain them by shifting the bell curve to the right and flattening it.

For this reason, we can say that these heat waves were caused by global warming.

Here’s a graph of the bell curves we’re talking about, in this case for the months of June, July, and August. The red, yellow and green lines are the old climate; the blue and purple lines are the new climate.

Look at the area under the curve to the right of x = 3: it’s almost nothing beneath the old climate, but quite significant beneath the new climate.