To be precise means to have all measurements close to each other, regardless of whether or not those measurements are correct. Accuracy means that a measurement is correct. To have both accuracy and precision is imperative in science because precision and accuracy together maintain the validity of the results found. This is exemplified by a recent experiment in neutrino physics. When scientists first measured the possibility that neutrinos could travel at a speed faster than that of light they did tests and found it to be a question of mere nanoseconds. In this case especially, sig digs are imperative to precisely calculate the difference in speed and the associated error to achieve the correct results.
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