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5.2 Combining Data with Different Errors

Example 2. It is necessary to use the lifetime of the muon in a calculation. However, in searching through the literature, 7 values are found from different experiments:

2.198 ± 0.001 µs 2.203 ± 0.004 µs 2.202 ± 0.003 µs
2.197 ± 0.005 µs 2.198 ± 0.002 µs 2.1966 ± 0.0020 µs
2.1948 ± 0.0010 µs

What is the best value to use?

One way to solve this problem is to take the measurement with the smallest error; however, there is no reason for ignoring the results of the other measurements. Indeed, even though the other experiments are less precise, they still contain valid information on the lifetime of the muon. To take into account all available information we must take the weighted mean. This then yields then mean value

tau = 2.19696

with an error

sigma (tau) = 0.00061.

Note that this value is smaller than the error on any of the individual measurements. The best value for the lifetime is thus

tau = 2.1970 ± 0.0006 µs.

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