Absolute & Relative Errors

In any measurement, we define the true value $x_0$ as the actual, accepted value of the quantity being measured. The measured value $x$ is what our instrument records. In practice, the true value is often taken to be the mean of many repeated measurements. The discrepancy between these two values is the starting point for all error analysis.

The absolute error $\Delta x$ is simply the magnitude of the difference between the measured value and the true value: $$\Delta x = |x - x_0|$$. It has the same units as the measured quantity. When you take multiple measurements $x_1, x_2, \ldots, x_n$, the best estimate of the true value is the arithmetic mean $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$, and the absolute error for each reading is $\Delta x_i = |x_i - \bar{x}|$. The mean absolute error is then $\overline{\Delta x} = \frac{1}{n}\sum_{i=1}^{n}|x_i - \bar{x}|$. This is also called the mean deviation. The final result is reported as $x = \bar{x} \pm \overline{\Delta x}$, meaning the true value almost certainly lies within this range.

The relative error (also called fractional error) is defined as the ratio of the absolute error to the true (or mean) value: $$\varepsilon = \frac{\Delta x}{\bar{x}}$$. Since the units cancel, $\varepsilon$ is dimensionless. It is often expressed as a percentage by multiplying by 100, giving the percentage error: $\varepsilon_{\%} = \frac{\Delta x}{\bar{x}} \times 100\%$. A small relative error means the measurement is highly precise relative to the size of the quantity being measured. This makes relative error the most useful tool for comparing the quality of measurements of different physical quantities.

Consider two measurements: (A) a length of $5.0 \pm 0.1$ m and (B) a length of $0.050 \pm 0.001$ m. The absolute errors are 0.1 m and 0.001 m respectively — measurement B has a smaller absolute error. However, the relative errors are: (A) $0.1/5.0 = 2\%$ and (B) $0.001/0.050 = 2\%$. Both measurements have the same relative precision! Relative error strips away the scale of the measurement and reveals the true quality of the experimental technique.

When a result $Z$ is calculated from measured quantities, errors combine according to specific rules. For addition or subtraction ($Z = A \pm B$), absolute errors add: $\Delta Z = \Delta A + \Delta B$. For multiplication or division ($Z = A \times B$ or $Z = A/B$), relative errors add: $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$. For a power law ($Z = A^n$), the relative error is multiplied by the exponent: $\frac{\Delta Z}{Z} = |n| \frac{\Delta A}{A}$. These rules follow from calculus (total differential) and are essential for any quantitative experimental work.

• \varepsilon_{\%} — Percentage error (dimensionless (percent)): The relative error expressed as a percentage; indicates how large the error is compared to the measured value.
• \overline{\Delta x} — Mean absolute error (same as measured quantity): The average of the absolute deviations of each individual measurement from the mean value.
• \bar{x} — Mean value (same as measured quantity): The arithmetic mean of all repeated measurements; serves as the best estimate of the true value.

Problem. A student measures the diameter of a steel ball five times and records the following values: 4.21 cm, 4.19 cm, 4.23 cm, 4.20 cm, and 4.22 cm. Calculate (a) the mean value, (b) the mean absolute error, (c) the relative error, and (d) the percentage error. Express the final result in the standard form $\bar{x} \pm \overline{\Delta x}$.

Answer. Mean = 4.21 cm; Mean absolute error = 0.01 cm; Relative error ≈ 0.003; Percentage error ≈ 0.3%; Final result: d = 4.21 ± 0.01 cm.

• Random and Systematic Errors — Absolute and relative errors arise from two distinct sources — random fluctuations and systematic biases. Understanding the difference is essential for reducing measurement uncertainty.
• Significant Figures and Precision — The number of significant figures used to express a measurement is directly related to its absolute error; proper use of significant figures ensures that reported precision matches actual measurement quality.
• Propagation of Errors in Calculations — When derived quantities are calculated from measurements, absolute and relative errors must be combined using propagation rules — a direct application of the concepts introduced here.
• Physical Quantities and SI Units — Absolute error is expressed in the units of the measured quantity, so a solid understanding of SI units is necessary to correctly interpret and report absolute errors.