We can have the words "March comes in like a lion, goes out like a lamb" right in front of us, in 18-point type, and we still will hem and haw and not feel entirely confident about the veracity of the old saying. Maybe it is because people flip the whole "lion/lamb" bit regularly; maybe it is because they're not sure of the origin, thinking it is probably weather-related but also might have to do with cuddly-wuddly livestock, big cats, the tides, the networks' fall television schedules and/or the fluctuations of the stock market.
Those are all fine guesses, but the answer has more to do with the night sky, and the constellations of Leo and Aries. Temperatures also play a part in this saying's longevity -- the end of March can be more mild-ish than the beginning. Right, is your jaw dropping? We didn't think so, seeing as how the beginning of March is still in winter and the end is not. And here in Southern California, we've got an awful lot of lamb-like days, which is wonderful, but they can all run together after awhile in a beautiful, 72-licious haze. Is it the end of March? The beginning of December? Who knows? We're going outside for a jog.
In LA, you can honor the famous lion/lamb phrase with an informative trip to Griffith Observatory. Spy the stars that inspired this old chestnut, and ponder how the cosmos impact our other well-known sayings (we keep trying to start "Betelgeuse may make a man jig every fourth of May" but it doesn't seem to be catching on like we hoped).