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Comment by _alternator_ | Hacker News Reader

Fair, I had to click into the full size (I pinch zoomed on my phone). But now the fact that 1988/1989 are outliers raise a few more questions about the variance calculation. In particular, the title suggests that those weren't included in the variance calculation.

A 3.5-sigma event is a 1-in-1000 chance. Yet we have 2 of them in our data set that goes back only 40ish years? Something is suspect. Maybe I'm not accounting for the likelihood of a random walk excursion probability, but if a 3.5 sigma event isn't a 1-in-1000 event I have difficulty interpreting.

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I think a 3.5-sigma event being a 1/1000 probability is only under the assumption of random data - a constant climate would deliver more-or-less random data around some mean.

The appearance of two 3.5-sigma events suggests that the data is no longer being randomly generated around that mean, but is instead drifting away from that mean.

More concretely, if all the data were within 2-sigma, then you could invoke the null-hypothesis and refute climate change. Beyond 2-sigma (and even worse two 3.5-sigma events) refuting climate change becomes (much) less reasonable.

> The appearance of two 3.5-sigma events suggests that the data is no longer being randomly generated around that mean, but is instead drifting away from that mean.

Not if the two 3.5-sigma events are in opposite directions, as in this case.

> ... opposite directions ...

Yes, I'd forgotten that detail. Still, though, chaotic bifurcation could be present with Il Niño and La Niña driving high & low extremes. Same mean, maybe, but with a bimodal distribution?

The event in this case is a daily observation, so in the last 45 years there have been 16k events. Under Gaussian assumption, 3.5 sigma outliers in positive direction are 1 in ~4,000 events, in either direction 1 in ~2,000, so approx. 8 days would be outliers in the recorded period. Judging by the looks, it's pretty close. And the data is absolutely not Gaussian: at best it's the result of a Gaussian process, as the daily recordings are not independent, but dependent on the previous day.

Why are you assuming gaussianity?

The tails could be heavy, it's true. This could make my usual 1-in-1000 heuristic overly conservative. Let's go with that, heavy tails. The data shows we are at something between a 1-in40 and 1-in-20 deviation. Normally (pun intended) that's ~2-sigma deviation. So... I guess it doesn't really seem like an epoch defining El Niño.

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