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Let's say that an experiment has to determine the number of cosmic muon at sea level. The appropriate equipment is ready to measure the counting rate.

I can think of two ways

  1. Count for 10 minutes, so as to reduce the relevant error.
  2. Count 10 times(1 minute/measurement)

Which is the best measurement? Are they equal?

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up vote 2 down vote accepted

Well, in essence, they're equivalent. But you're probably comparing apples to oranges here – depending on your detector.

The error in your measurement will almost certainly be determined by statistical noise. Depending on the size of your detector (and its orientation), you might get 10 muons in far less than 10 minutes, meaning you could improve your statistics by using the first method. On the other hand, if you have a small detector, it may take far more than 10 minutes to get 10 muons, so the second method would be preferable.

Cosmic rays arrive basically independently of each other, so you will see a lot of what's called shot noise. This is just statistical noise in the number of arrivals (which follows a Poisson distribution, if you want to look up the statistics). This noise will be your dominant source of error unless your method of keeping time is wildly noisy. If you make $N$ measurements, the statistical uncertainty in your rate will be proportional to $1/\sqrt{N}$. To minimize this uncertainty, you always want to measure as many samples as possible to improve your statistics. Since the underlying rate is actually constant, this is equivalent to measuring for as long as possible.

If, as you seem to suggest, your detector is just the right size so that it really does average 1 muon per minute, then the methods are identical. In particular, the results you get with the two methods will be within the statistical uncertainty of each other.

But generally, you just want to measure for as long as possible to minimize the noise. Deciding how long you have is probably dominated by personal/social factors more than science.

[Depending on how you do things, your second method may also be prone to fencepost error.]

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Good answer. I'd also say this can become a practical question if the experiment is powered by a battery. For example, what if you were counting earthquake on Mars - is it better to count for one long continuous time interval, or make repeated shorter duration measurements over a longer period of time. –  Mark Rovetta May 22 '13 at 16:55
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Thank you very much for your answer! That was exactly what I had in mind. They are practically the same, but I hoped that statistics could say which is the best, regardless of the practical limitations... –  Thanos May 22 '13 at 19:22
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