absolute and relative measurement error

When measuring, we obtain a value $x$ that usually differs from the true value $x_0$. Absolute error shows by how much the result may differ from the true value, and it has the same units as the measured quantity: $\Delta x = |x - x_0|.$ If the true value is unknown, $\Delta x$ is often taken as the instrument error, for example half the smallest scale division. Relative error is the ratio of the absolute error to the value of the quantity; it is dimensionless and is often expressed as a percentage: $\delta = \frac{\Delta x}{x_0}\cdot 100\% \approx \frac{\Delta x}{x}\cdot 100\%.$ Relative error is convenient for comparing measurements of different quantities and scales: the same absolute error can be negligible for a large value and large for a small one. Therefore, when reporting a result, it is useful to state both the absolute and the relative error.

The measured length of a rod is $x=12.4\,\text{cm}$, and the true value is $x_0=12.5\,\text{cm}$. Find the absolute and relative measurement errors.

A thermometer showed $x=36.8\,^\circ\text{C}$, while the reference value is $x_0=37.0\,^\circ\text{C}$. Find the absolute error in degrees Celsius and the relative error as a percentage. Round the relative error to hundredths.

Compare two recorded measurement results: mass $m=500\pm5\,\text{g}$ and length $l=2.0\pm0.1\,\text{cm}$. For each quantity, calculate the relative error and determine which measurement has the smaller relative error. Why can accuracy not be compared using absolute errors alone?

Take a ruler with a smallest division of $1\,\text{mm}$ and measure two segments: about $2\,\text{cm}$ and about $20\,\text{cm}$. Take the absolute error to be half the smallest division: $\Delta x=0.5\,\text{mm}$. For each segment, calculate the relative error using $\delta=\frac{\Delta x}{x}\cdot100\%$. Record the results in a table and conclude why, for the same absolute error, the relative error is smaller for the larger segment. If no ruler is available, perform the same calculations for a virtual ruler in a safe online measurement simulation.

Show that absolute error is an interval in the units of the quantity, while relative error is a dimensionless fraction of the value and is convenient for comparison.

Diagram: a horizontal scale with the measured value $x$ and the interval $x\pm\Delta x$; next to it are the formulas $\Delta x=|x-x_0|$ and $\delta=\frac{\Delta x}{x}\cdot100\%$. Graph: the dependence of $\delta$ on $x$ for a fixed $\Delta x$ — a hyperbola showing that relative error decreases as $x$ increases.

On a kitchen thermometer the scale division is $1\,\mathrm{^\circ C}$. The water for yeast shows $38.0\,\mathrm{^\circ C}$, so the absolute error taken as half a division is $\Delta t=0.5\,\mathrm{^\circ C}$. The relative error is $\varepsilon=0.5/38.0=0.013=1.3\%$. If the same thermometer reads $24.0\,\mathrm{^\circ C}$, the same $\Delta t=0.5\,\mathrm{^\circ C}$ gives $\varepsilon=0.5/24.0=0.021=2.1\%$.

The scenario uses the absolute error $\Delta t$ obtained from the instrument resolution and the relation $\varepsilon=\Delta t/t$; it shows that the relative error grows when the measured temperature is lower.

A phone stopwatch shows $12.45\,\mathrm{s}$ for a 100 m sprint. The display resolution is $0.01\,\mathrm{s}$, but the start and finish reaction uncertainty is estimated as $\Delta t=0.05\,\mathrm{s}$, so the result is written as $t=(12.45\pm0.05)\,\mathrm{s}$. The relative error is $\varepsilon=0.05/12.45=0.0040=0.40\%$. A difference of $0.05\,\mathrm{s}$ between runners is therefore about $0.4\%$ of the race time.

This uses absolute error as the timing uncertainty and relative error as the ratio $\Delta t/t$; the theory tells how much a fixed $0.05\,\mathrm{s}$ uncertainty matters compared with the total time.

A car speedometer has a scale division of $2\,\mathrm{km/h}$. At a reading of $90\,\mathrm{km/h}$ the absolute error is taken as $\Delta v=1\,\mathrm{km/h}$, so $\varepsilon=1/90=0.011=1.1\%$. At a reading of $30\,\mathrm{km/h}$ the same $\Delta v=1\,\mathrm{km/h}$ gives $\varepsilon=1/30=0.033=3.3\%$.

The example applies the absolute error from the instrument resolution and the relative error $\varepsilon=\Delta v/v$; it shows that the same speed uncertainty is a larger fraction of a lower speed.