Maths and data tools

Accuracy and precision fail differently

Precision is how tightly repeated readings cluster. Poor precision is random error, and it averages down: take more measurements and the mean tightens as the square root of the count. Accuracy is how close the result sits to the true value. Poor accuracy is systematic error, and averaging does nothing at all. A balance reading half a gram heavy reads half a gram heavy however many times you use it.

This is why the sign of an error is worth looking at before you discard it. Readings scattered either side of the accepted value point to noise you can average away. Readings consistently on one side point to a calibration problem, and no amount of repetition will help.

That square root also sets what more data is worth. Halving the standard error of the mean takes four times as many measurements, and quartering it sixteen times as many. Somewhere on that curve the random component drops below the systematic one, and past that point more repetition buys nothing.

Combine uncertainties in quadrature

For independent random uncertainties, the combined uncertainty is the square root of the sum of the squares, not the plain sum. Adding linearly assumes every error conspires in the same direction at the same moment, which overstates the true uncertainty. Add linearly only for correlated or systematic effects, where they genuinely do move together.

One consequence is worth internalising: quadrature is dominated by the largest term. Take uncertainties of 1 and 3 in the same units, combining to about 3.16. Eliminating the smaller one entirely gets you to 3.00, an improvement of 5 percent. Halving the larger one gets you to 1.80, an improvement of 43 percent. Spend the effort on the dominant term.

Absolute or relative depends on the operation

For sums and differences, absolute uncertainties combine in quadrature. For products, quotients and powers it is the relative uncertainties that combine, and a power multiplies the relative uncertainty by its exponent. That last rule bites: a 1 percent uncertainty in the radius of a sphere becomes 3 percent in its volume, because the radius is cubed.

The dangerous case is subtracting two similar numbers, because the absolute uncertainty survives while the result shrinks. Two masses of 15.2 g and 14.8 g, each good to 0.1 g, are individually known to better than 0.7 percent. Their difference is 0.4 g with a combined uncertainty of about 0.14 g, which is 35 percent. Nothing went wrong in the measurement. The precision was destroyed by the subtraction.

Weighing by difference is how this shows up at the bench, and the fix is to make the difference large relative to the reading error rather than to buy a better balance. The same reasoning explains why percent error needs a non-zero accepted value: dividing by something near zero inflates the percentage without adding information.

How many digits you are entitled to

Quote the uncertainty to one significant figure, or two when the leading digit is 1, then round the value to the same decimal place. Reporting 4.28371 ± 0.05 is wrong; 4.28 ± 0.05 is right. The extra digits are not more information, they are a claim the uncertainty figure contradicts.

Significant figures are a crude stand-in for a stated uncertainty. Writing 1.0 rather than 1.00 does convey something about precision, but it cannot express an uncertainty that is not a power of ten. Where it matters, state the uncertainty and say whether it is a standard deviation, a standard error or a confidence interval, since the three differ by a factor of several and readers cannot tell which you meant.

Round once, at the end. Carry a couple of guard digits through intermediate steps and round only the final answer, because rounding at every stage accumulates a bias that has nothing to do with the measurement. For the same reason a conversion should use the exact defined factor where one exists, such as 25.4 mm to the inch or 101,325 Pa to the standard atmosphere, rather than a remembered approximation.