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For any given point on Earth, the length of winter daytimes is just slightly longer than the length of summer nighttimes. So if the longest daytime where you live is 15 hours, the shortest daytime is 9 hours, and if the longest daytime there is 14 hours, the shortest daytime is 10 hours, and so on.
(The correlation between winter days and summer nights would be exactly the same if not for refraction caused by the atmosphere making the sun visible when it's actually a little below the horizon. That's why the length of daytime on the equinoxes is not exactly 12 hours.)
Exception: Daytime at the north pole is about 7½ days longer than nighttime at the north pole, and nighttime at the south pole is about 7½ days longer than daytime at the south pole. This is because Earth travels fastest in its orbit when it's closest to the sun, which happens in early January, and it travels slowest when it's farthest from the sun, which happens in early July.
