Example of the claim:
I've been told that if you see cows lying down in the field, that means it's going to rain.
Pretty much "common knowledge" among the PA Dutch but that doesn't mean it's true.
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It is doubtful that there is much correlation. Cows lie down too often for other reasons to be a useful predictor of rain.
Cows are ruminants and lie down to chew their cud. They also lie down for other reasons including sleep. Cows lie down perhaps up to sixteen times a day. At any given time there is a 50% chance a cow will be lying down but they tend to stand up more in hot weather and perhaps tend to lie down more in cold weather.
We humans tend to find significance in coincidences and, if primed by knowledge of this adage, might have stronger recall of occasions when cows were seen lying down prior to rain than for occasions when cows were seen lying down prior to no rain or when cows were seen standing up prior to rain.
Some frivolous media have speculated that air temperatures often drop before rain and that this therefore supports the old adage. It is notable that the study they cite says no such thing and is more focused on heat-stress.