Errors and uncertainties

Errors in measurements fall into two categories: systematic errors and random errors.

Systematic errors

• Offset every reading in the same direction, by a constant amount or constant proportion
• Arise from faults such as zero errors or calibration faults
• Are reduced by checking and recalibrating the instrument, not by averaging or repeating readings

Random errors

• Cause readings to scatter unpredictably, above and below the true value
• Arise from fluctuations in conditions or in observer judgement
• Are reduced by taking repeats and calculating a mean, or by using a finer-resolution instrument

Key Definition Accuracy describes the closeness of a measurement to the true value, while precision describes the closeness of repeated readings to each other.

A precise set of readings is not necessarily accurate, since a systematic error can shift them all together away from the true value.

When a quantity is calculated from several measured values, the uncertainties combine.

For sums and differences, the absolute uncertainties are added.

For products, quotients, and powers, the percentage (or fractional) uncertainties are added, with each percentage uncertainty multiplied by its power.

Sum or difference, y = a ± b → Δy = Δa + Δb Product or quotient, y = ab / c → Δy/y = Δa/a + Δb/b + Δc/c Power, y = aⁿ → Δy/y = n × (Δa/a)

The final answer is quoted as a value plus or minus its absolute uncertainty, with the uncertainty given to one significant figure and matched to the same decimal place as the value.

Identifying which measured quantity carries the largest percentage uncertainty (especially one raised to a power) reveals which step of the experiment most needs improvement.