Rarity is relative
An extreme value is rare only relative to a reference distribution.
1 of 8
When Rare Events
Stop Being Rare
Follow one 41.2°C day to see how probability, variability, feedbacks, and thresholds shape the meaning of an extreme event.
Enter the Story ↓Scene 01
A hot day becomes an extreme event
At 3:40 PM, fictional Meridian City records 41.2°C. It is a large value, but a large number alone does not yet define rarity.
41.2°C
Daily maximum temperature
What makes this an extreme event?
Scene 02
“Rare” always refers to a background
Rarity is a comparison. Relative to Meridian City’s reference distribution of daily maximum temperature, 41.2°C lies far into the upper tail.
Select “Reveal the tail probability” to isolate the rare-event region.
Scene 03
A fixed threshold can become easier to cross
Keep the extreme threshold fixed at 40°C and move the center of the background distribution only modestly. The probability beyond the threshold grows much more visibly.
Background AThe threshold lies deep in the tail.Background BThe same threshold is crossed more often.
The center moved modestly. The far-tail exceedance probability changed much more.Estimated exceedance area4.9%The threshold has not moved.
Scene 04
The average is not enough
Two backgrounds can have approximately the same average while their day-to-day spread differs. The more variable system produces more threshold crossings.
similar averagedifferent variabilitydifferent number of extremes
System A · lower variability
same averageextreme threshold
0 threshold crossingsSystem B · higher variability
same averageextreme threshold
4 threshold crossingsSimilar averagebutdifferent spread→different number of extremes
Scene 05
Some extremes amplify themselves
When soil is moist, part of the incoming energy supports evaporation. Under dry conditions, less energy follows that pathway and more heats the surface and near-surface air.
incoming energy↓
surface stateMoist soil
evaporationmore
surface heatingless
Evaporative coolingpart of the energy is buffered
Scene 06
A smooth change can produce an abrupt consequence
Physical intensity can increase smoothly while impacts change sharply. Day A and Day B are close in temperature, but they fall on opposite sides of an operating threshold.
cooling demand exceeds capacitynighttime safety thresholdvegetation stress rises sharply
Scene 07
One extreme event can have several layers of explanation
The 41.2°C day does not need one exclusive cause. Statistical background, a dynamical trigger, local amplification, and an impact threshold can all contribute to the explanation.
01
Background probabilityHow unusual is 41.2°C relative to the reference distribution?
→02
Immediate dynamical triggerWhat weather pattern organized the event?
→03
Local amplificationDid dry soil or another local condition intensify it?
→04
Impact thresholdWhich systems were pushed beyond their operating range?
→MERIDIAN CITY41.2°Cone event
Scene 08
The same event can have a different statistical meaning
Return to exactly the same 41.2°C event. Its physical value has not changed. Its rarity depends on the background against which it is judged.
Background A41.2°C
commonunusualrarevery rare
Deep in the tailVery unusual relative to this backgroundBackground B41.2°C
commonunusualrarevery rare
Same reference positionShift the background to compareTail probabilities can change rapidly
A modest shift in center or variability can strongly change threshold exceedance.
Dynamics and feedbacks matter
Processes operating during an event can amplify its physical intensity.
Consequences can be nonlinear
Smooth physical changes can cross thresholds and produce abrupt impacts.
A rare event is not defined by one large number. Its meaning emerges from probability, dynamics, amplification, and thresholds.
Importance Sampling: Rare Events and Tail Probabilities
A direct introduction to rare-event probabilities and the mathematical challenge of sampling the tails.
Watch Video →Near the Edge
Distinguish rare excursions, nonlinear amplification, and transitions between regimes.
Open the Exploration →One Planet, No Controlled Rerun
How can we investigate causes and counterfactuals when the Earth cannot be rerun under controlled conditions?
Continue →