Question: Why a groundhog?
The story structure requires an animal that is not routinely active in winter (foxes, rabbits...) but also isn't hibernating (bears...).
Question: What is the relevance of the groundhog seeing its shadow?
An embellishment. The story works better if someone sees the shadow, rather than simply saying "If there are shadows..."
Question: What is the relevance of shadows?
Shadows are simply a proxy for whether the weather: If it is cloudy, no shadows. There is a lot of parallel folklore, contemporaneous and earlier, that skips the groundhog and its shadow and explicitly talks about cloudiness (the Wikipedia article on Groundhog Day has some examples).
Question: What's the relevance of cloudiness?
Clouds, and the moisture in the air that accompany them, creating an insulating blanket that slows the radiating of heat into space.
Question: But isn't winter cloudiness random?
In some places, yes. But others, no. For example, I grew up in western New York state, downwind of the Great Lakes. When the lakes froze over, they stopped moisture for the clouds and snow and we would have long series of cold nights and cold days. However, when the lakes didn't freeze over, the warmer days would produce modest melting of the snow which in turn would provide localized moisture for the clouds, which in turn would generate more warming. Other areas have similar dynamics.
Question: Why February 2nd?
The reasoning above argue for looking at the trend of cloudiness, not just the situation at a particular time on a specific date. However, that doesn't make for a good story, a good ceremony or good celebration. If you have to choose a date, February 2 is roughly halfway between the Winter and Spring solstices, thus the pattern of winter has likely been established, but spring is still far enough off that a prediction is interesting. Plus February 2 is a Christian feast day, and thus allow bundling the two together.
Question: Many of the current Groundhog Day events produce results that are worse than chance. Does this mean that the folklore was nonsense?
No. Climate conditions have change substantially since the origin of the folklore and some of the crucial underlying conditions may well have changed. Recognize that these were people who didn't have either the data nor the computational resources to make long-range predictions using causation. Instead they had to rely on correlations that they had observed.
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