Errors and Uncertainties

Every measurement has an uncertainty. Quoting a result without one is quoting an opinion.

Random error — causes readings to scatter on both sides of the true value, differently each time. Judging when a swinging pendulum passes a mark, or reading a slightly fluctuating meter, both produce random error.

Reducing random error — repeat the measurement and take a mean. The scatter partly cancels, and more repeats give a better mean. Using an instrument of finer resolution also helps.

Systematic error — shifts every reading by the same amount in the same direction. A balance reading 2 g with nothing on it adds 2 g to every mass you measure.

Reducing systematic error — repeating does nothing at all, which is what makes systematic error dangerous. It has to be found and removed: check and correct the zero, calibrate the instrument against a known standard, or change the technique.

Example: a zero error is the obvious case, but parallax is the common one. Reading a scale from an angle shifts every reading the same way. Reading at eye level, or using a mirror behind the pointer, removes it.

Resolution — the smallest change an instrument can show. A metre ruler marked in millimetres has a resolution of 1 mm. Resolution sets the floor on precision but says nothing about accuracy: a badly calibrated instrument can have superb resolution and still be wrong.

Absolute uncertainty — the uncertainty in the same units as the measurement, written with ±. For a single reading from a scale, it is usually taken as half the smallest division.

Percentage uncertainty —

percentage uncertainty = (absolute uncertainty ÷ measurement) × 100

Example: 2.50 m ± 0.05 m is (0.05 ÷ 2.50) × 100 = 2.0%. The same ±0.05 m on a measurement of 0.10 m would be 50% — the same absolute uncertainty, and a completely different quality of measurement. This is why percentage uncertainty is the useful comparison.

Combining uncertainties — two rules cover almost everything at this level:

Adding or subtracting quantities — add the absolute uncertainties.

Multiplying or dividing quantities — add the percentage uncertainties.

Raising to a power — multiply the percentage uncertainty by the power. A quantity squared has twice the percentage uncertainty; cubed, three times.

Example: finding the volume of a sphere from its radius. V depends on r³, so a 2% uncertainty in the radius becomes a 6% uncertainty in the volume. This is why the quantity raised to the highest power usually dominates, and is the first thing worth measuring better.

Which uncertainty to attack — identify the largest percentage uncertainty and improve that measurement. Improving anything else is wasted effort.