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How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

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This question refers to significant figures. The reason we follow significant figure rules when performing calculations in science is that a calculated answer (which is an extrapolation of a manipulation of measurements) can only be as precise as the least precise measurement recorded. This means if you record measurements to a certain precision, the calculated answer at the most can only match the recorded level of precision as to balance the level of confidence in accuracy before and after calculations.
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