Measurement Errors and Uncertainty

Significant figures, absolute, relative error and uncertainty propagation in physical measurements.

Conceptual context

Every measuring instrument has limitations. A millimeter ruler cannot guarantee an exact reading; a manual stopwatch includes human reaction time; a balance has a minimum resolution. These limitations are not avoidable defects: they are an intrinsic property of the measurement process itself. Experimental physics does not ask "what is the exact value?" but rather "in what interval does the true value lie, and with what confidence do we know it?". Error analysis answers exactly that question: it converts the uncertainty of a measurement into a quantifiable number, $\delta x$, and allows us to compute how that imprecision transfers to the final result when we operate with several measured quantities.

🟢 Essential level

When you measure a length with a ruler you obtain a number, $x$. But that number is not the true value: it is your best estimate. The maximum expected difference between your reading and the real value is the absolute uncertainty $\delta x$. The complete result of a measurement is always written as $x$ ± $\delta x$, in the units of the measured quantity.

To compare the quality of different measurements — a length and a mass, for example — you need a common scale. That scale is the relative error $e_{rel}$: the fraction that $\delta x$ represents with respect to $x$. The smaller $e_{rel}$ is, the more precise the measurement. Multiplying $e_{rel}$ by one hundred gives the percentage error $e_{\%}$, the standard way to report precision in a lab report. The number of significant figures $n_{cs}$ directly indicates how many digits of the result are reliable.

For example, a typical school ruler has a smallest division of 1 mm, so its usual $\delta x$ is 0.5 mm: half of that smallest division.

🔵 Formal level

Basic formulas

Quantifying uncertainty starts from four fundamental relations.

Relative error:

Relative error of a measurement

Dimensionless, always positive, defined only when $x$ ≠ 0.

Percentage error:

Percentage error

Multiplies $e_{rel}$ by one hundred. Allows comparing the precision of different experiments on an intuitive universal scale: an error of 5% is directly interpretable, whereas $e_{rel}$ = 0.05 requires mental conversion. It is reported alongside the result whenever the difference in precision between experiments is relevant.

Real student questions

Why do uncertainties add in a subtraction instead of subtracting? Because uncertainty represents the worst case. If $x$ can be as large as $x$ + $\delta x$ and y can be as small as y − $\delta y$, the maximum possible difference is ($x$ + $\delta x$) − (y − $\delta y$) is equal to (x − y) + ($\delta x$ + $\delta y$). Errors always accumulate in the worst direction.

When do I use absolute error and when relative error? Use $\delta x$ to report individual measurements and to propagate in sums/differences. Use $e_{rel}$ to compare the quality of different measurements and to propagate in products/quotients.

What happens if $e_{rel}$ > 1? The uncertainty exceeds the measured value. The result has no useful precision: it cannot distinguish the value from zero. You must change instrument or method.

Final synthesis

Measuring is not simply reading a number: it is quantifying knowledge and its limit. $\delta x$ is the quantitative honesty of the measurement, $e_{rel}$ is its relative quality, and error propagation is the arithmetic of that honesty when results are calculated from several imperfect measurements.