Uncertainty and error

Two kinds of error, treated differently

Random errors scatter readings either side of the true value. Electrical noise does it. So does a reaction time that varies from one press of the stopwatch to the next, and so does a draught nudging a balance. Because the direction of the scatter is unpredictable, repeating and averaging helps, and more repeats pull the mean in tighter, though some random scatter survives any number of them.

Systematic errors shift every reading the same way. A ruler whose scale starts at 1 mm does it, and so does an ammeter showing 0.02 A with nothing connected. Averaging cannot touch them. Repeat a hundred times and the readings stay consistently wrong, so the only remedy is to find the cause and remove it. Most often the cause is a zero error, and the fix there is the model for all the rest. Check the instrument reads zero when it should, then subtract the offset from every reading you took.

Precision, accuracy, resolution, repeatability and reproducibility

Precision is how tightly repeated readings cluster, and it says nothing whatever about being right. Accuracy is how close a result sits to the true value. The two are independent, exactly as the two kinds of error predict. A systematic error leaves you precise and wrong. Heavy random scatter leaves you imprecise even when the mean lands on the truth.

Three further terms have specific meanings in mark schemes. Resolution is the smallest change an instrument can display, one millimetre for an ordinary ruler. Repeatability means the same experimenter with the same equipment gets consistent results; reproducibility means a different experimenter, or different equipment, still agrees, which is the stronger claim.

Quantifying uncertainty

The uncertainty is the interval within which the true value can reasonably be expected to lie, written after a ± sign. For a single reading, the usual board convention is half the instrument's resolution, so a millimetre ruler contributes ±0.5 mm at each end. Measuring a length means judging both ends, which gives ±1 mm overall. For a set of repeated readings, use half the range, the largest reading minus the smallest, divided by two.

One uncertainty, three forms. The absolute uncertainty keeps the unit, ±0.5 mm. Divide by the value and you have the fractional uncertainty; multiply that by 100 and you have the percentage.

Combining uncertainties

Calculations mix measured quantities, and three rules combine their uncertainties. Add or subtract quantities, and you add the absolute uncertainties. Multiply or divide, and you add the percentage uncertainties. Raise a quantity to the power n, and its percentage uncertainty is multiplied by n, so a radius known to 2% yields a volume known only to 6%, because volume goes as r³$r^{3}$r^3.

The subtraction rule has a practical consequence. Take x = 50 ± 1$x = 50 \pm 1$x = 50 ± 1 mm and y = 20 ± 1$y = 20 \pm 1$y = 20 ± 1 mm. Their difference is 30 ± 2$30 \pm 2$30 ± 2 mm. The value shrank while the doubt grew, so subtracting two nearly equal quantities can leave a percentage uncertainty large enough to make the result useless. Experiments are designed to avoid taking the difference of two nearly equal measurements.

On a graph, each point's uncertainty becomes an error bar. Draw the best-fit line, then the steepest and the shallowest acceptable lines, the extremes that stay consistent with the error bars. Half the difference between those two gradients is the uncertainty in the gradient, and where the same two lines cut the axis brackets the intercept. Last, let the uncertainty set your significant figures. Quoting 12.4783 when the doubt is ±0.5 claims a precision the experiment never had.