Uncertainty in Measurements

Dealing with uncertainty in measurements

Measurements always have uncertainties attached to them because the techniques used for measurement are never infinitely precise. The level of uncertainty is determined by the limitations of the measurement technique that you're using.

Different ways of communicating accuracy

Explicitly

Accuracy is stated explicitly by giving the recorded measurement along with the level of uncertainty (which is some range in which the actual value could be). For example, 57.38 ± 0.03 mm says "the value is probably in between 57.35 and 57.41 mm." You can also express it with percent error where the percent is just multiplied by the measured value in order to get the range. So 57.38 mm ± 10% is "57.38 mm ± (10% * 57.38 mm)

Implicitly

Accuracy is stated implicitly through the number of significant figures. The number of significant figures is just the number of measured digits in a measurement. So for 123mm, the number of significant figures is 3. For 0.025cm it's also 3. For 3,840,000 it's also 3. When multiplying and dividing measurements, the number of sig figs in the result is the same as in the measurement with the fewest amount of sig figs (and it rounded up appropriately). When adding and subtracting measurements, the decimal place stays in the same place and the number of significant figures is determined by the measurement with the largest uncertainty (aka the one with the fewest numbers after the decimal).

Rules of significant figures

Dealing with big numbers

For really large numbers with lots of trailing zeroes, scientific notation is usually used to condense the size of it in writing and to clearly show the number of sig figs.

Other notes

Numbers that are coefficients in equations and whatnot, it's treated as having no uncertainty. You can think of it as having infinite significant figures. This makes sense because these coefficients and exponents are not measurements. They are quantities without a unit that are expressing a relationship.

Difference between accuracy and precision

Accuracy is how close a measurement is to the true value. Precision is how close measurements taken of the same thing are to one another.