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Believe it or not, this question has received some serious scrutiny from the scientific community over the years. Even the syndicated Straight Dope columnist Cecil Adams and the producers of the television series Mythbusters have conducted their own studies on the debate. The general consensus as to whether you'd get wetter if you run or walk in the rain appears to favor walking. If you don't want to get soaked any more than is strictly necessary during a rainstorm, run very quickly.
There are those who suggest this conclusion is based on some faulty physics. They argue that a person who runs through rain should get wetter than a walker, because he or she would be pelted with rain from above and ahead of them. The runner should get wetter because he is deliberately putting himself in the path of more raindrops than a walker. A steady walker would only receive raindrops falling from above, and therefore he or she should be drier than the runner.
As it happens, the decision whether to run or walk in the rain has more to do with time than volume of rainfall. Ultimately, the runner will be out of the rain in less time than the walker, which means the runner would be exposed to less overall moisture. The walker might benefit slightly from not running into the raindrops ahead of him, but the added time spent in the rain would make him wetter overall.
For many people, the decision to run or walk may be based on logistics more than anything else. If shelter from the rain appears to be close, a person might decide to run in order to reduce exposure time. If shelter is further away, another person might decide to walk, since he is already soaking wet and running would use up too much energy or be dangerous under the conditions. It is not unusual to see different pedestrians move according to their natural preferences.
In short, if your goal is to remain as dry as possible, then you should probably run through a downpour. If you are wearing protective rain gear, then you may want to conserve your energy and walk. If you are already soaked to the bone, it really doesn't matter much if you run or walk in the rain. Finding dry, warm clothes or a friend with a towel would most likely take precedence over scientific inquiry, anyway.