How include instrument tolerance in uncertainty?

Use the manufacturer’s tolerance as a systematic uncertainty component and combine it with the random uncertainty by adding variances: square the systematic and random uncertainties, add them, then take the square root to get the combined uncertainty.

Think of uncertainty like two kinds of wobble: one wobble you cannot control because it comes from the instrument itself, and one wobble changes when you repeat the measurement. The manufacturer’s tolerance is the instrument wobble that stays the same every time, so treat it as a systematic uncertainty. The random uncertainty comes from repeated measurements and you estimate it with the standard deviation or the standard error. To combine them correctly, square the systematic number and square the random number, add those two squares together, and then take the square root of that sum to get the total uncertainty that includes both effects.

Students often add the systematic and random uncertainties directly instead of adding their squares, which overestimates the true uncertainty; the fix is to add variances not values. Another common mistake is ignoring the manufacturer’s tolerance when it is significant compared with the random error; always compare magnitudes and include the tolerance if it is nontrivial. People sometimes confuse tolerance with a confidence interval; tolerance is a fixed instrument limit and should be treated as a systematic term rather than a statistical one.

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Combining Instrument Tolerance with Random Error

1. Instrument wobble is systematic
2. Random wobble from repeated measurements
3. Estimate random with standard deviation
4. Square systematic and random values
5. Add the squared values
6. Take square root of sum

Common Errors When Combining Uncertainties

1. Adding uncertainties directly instead of variances
2. Ignoring significant manufacturer tolerance
3. Confusing tolerance with a confidence interval

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