Day-to-day temperature at a given place and season has always fluctuated around a seasonal average — historically this scatter is well described by a normal (bell-curve) distribution of the daily anomaly x (°C from the local baseline mean):
PDF(x) = 1/(σ√2π) · exp(−(x−μ)² / (2σ²))
P(x > threshold) = 1 − Φ((threshold − μ) / σ) [Φ = standard normal CDF]
Climate warming does two things to this curve, both visible in the bars below: it shifts the mean μ to the right by the global warming ΔT̄, and — because a warmer, more energetic atmosphere holds and releases moisture and heat less evenly — it can also widen the spread σ. Hansen, Sato & Ruedy (2012) showed observed summer-temperature distributions doing exactly this since the 1980s. A shift that looks small in the mean produces a much larger change in how often the tail is crossed, because P(x > threshold) depends on the exponential tail of the curve, not on the mean directly — a rare 3σ event under the old climate can become a routine 1σ event under the new one.
- ΔT̄ slider — shifts μ, the whole curve, to the right.
- Variability slider — widens or narrows σ, fattening or thinning both tails.
- Threshold slider — sets what counts as an "extreme heat day", in standard deviations of the 1951–1980 baseline climate (σ = 1 °C here).
- Falling markers — each one is a simulated day, sampled from the current distribution and landing on its bin; red markers cross the extreme threshold. The running tally converges on the analytic P(extreme) shown above.
This is the same "loaded weather dice" argument used to explain why heatwaves that were once one-in-a-generation events are now showing up every few years: the dice are not just moved, they are also reshaped.