Figure 2: Side-by-side comparison of (a) the magnitudes reported for mean-temperature warming [1], [2] versus (b) the magnitudes observed in extreme-heat-day frequency [3].

Figure 2 lays the disparity out side by side. The bars in panel (a) show what climate headlines lead with: roughly 1.24–1.55 °C of mean warming. The bars in panel (b) show what climate impacts actually look like on the ground: extreme-heat-day frequencies that have already multiplied by factors of two, four, or even more. The same warming has two very different stories, depending on whether you read the mean or the tail.

Earlier work by Fischer and Knutti, published in Nature Climate Change in 2015, came to a similar conclusion using a different approach. They estimated that under the warming of around 0.85 °C at the time, roughly 75% of moderate hot extremes over land worldwide were already attributable to human-caused warming, and that this fraction grows non-linearly with further warming [4]. The headline number was the mean; the story was in the attribution of the extremes.

Recent work goes further still. Across global land areas, state-of-the-art climate models underestimate the strongest observed positive trends in the widening of the upper tail of regional temperature distributions by a factor of about four [5]. In plain language, the models policymakers rely on are, in the regions where extremes are growing fastest, too conservative about how much the tails are widening. The extremes are running ahead of both the average and the projections.

Why ‘Once in a Hundred Years’ No Longer Means What It Used To

So far I have used the normal distribution as a teaching tool, because it makes the geometry obvious. But the real story is worse than the bell curve allows. Climate variables of interest (e.g., daily temperature maxima, multi-day heatwave intensities, extreme precipitation) are increasingly understood to behave in ways that depart from neat normality. Their tails are heavier and their behaviour is non-stationary, which means the historical distribution we are comparing against is itself changing.

Fischer, Sippel, and Knutti showed that, in high-emission scenarios, week-long heat extremes that break standing records by three or more standard deviations are projected to be 2–7 times more probable in 2021–2050 than in the last three decades, and 3–21 times more probable in 2051–2080 [6]. The crucial point is statistical. These record-shattering events are intensifying, but their occurrence probability depends on the rate of warming, not just the level. Pathways matter. The faster the mean moves, the more often the tails throw out something previously unprecedented.

This is why the phrase “once in a hundred years” is doing genuine communicative damage. It was always a statement about long-run probabilities under a stationary climate. Under a non-stationary climate, that phrase suggests a rarity that no longer exists. The honest version is something closer to “one in a hundred years, if the climate had stopped changing in 1990”. Almost nobody hears the conditional.

Why It Matters How We Talk About the Numbers

None of this is to suggest climate scientists are confused about their own data. They are not. The IPCC’s Sixth Assessment Synthesis Report devotes substantial attention to changes in extremes, variability, and tail risk [7]. The problem lies at the interface between science and the public—in news headlines, in government press releases, and in the targets that frame political debate. The “1.5 °C” in the Paris Agreement is genuinely useful as a coordination device for negotiators; it is genuinely misleading as a description of what is happening to the people living through that warming.

As a statistics student, I think we can do better. When we report a mean, we should report what is happening to the variance and the tails alongside it. When we say the world has warmed by 1.55 °C, we should also say that the chance of crossing a historical 99th-percentile heat threshold has roughly quadrupled. When we talk about “average sea level rise”, we should specify what is happening to storm-surge maxima. The bell curve is no longer a safe metaphor; the public deserves to know that.

Better statistical communication will not by itself slow emissions. But the way risks are framed shapes the policies a society is willing to accept. A population told that the world has warmed by 1.5 ºC may shrug. A population shown that the same warming has multiplied the frequency of lethal heat events several-fold is a population that takes the question seriously. We have spent decades training the public to focus on a single, reassuringly small number. The next decade of climate communication, and arguably the next decade of climate policy, will depend on whether we can train them, and ourselves, to look at the whole distribution.

Reading the Full Distribution

If there is one habit I would urge fellow undergraduates to take away from this piece, it is a small but stubborn statistical instinct. Whenever you see a headline reporting a climate average, ask the second question: what is happening to the tail? What is the 99th percentile doing? What is the variance? How much heavier are the extremes than they used to be? Those questions are not pedantic, they are the difference between a number that sounds manageable and a story that sounds urgent.

The mean has never lied to us. It has simply told us only one moment of a far more complicated truth. The science is in the tails. The danger is in the tails. The honest conversation about climate change has to be there too.

Acknowledgements

Written for SUSTAIN 200 and submitted to UoA Scientific as part of the SUSTAIN special edition. Both figures were created by the author using Python and Matplotlib. The numerical values plotted in Figure 2 are taken directly from the cited peer-reviewed sources; Figure 1 is a teaching schematic derived analytically from a standard normal distribution and is labelled as such. The author thanks the UoA Scientific editorial team for feedback on earlier drafts.