Quality of Measurement

Accuracy is quantified through the absolute error (also called absolute uncertainty): $\Delta x = |x_{\text{measured}} - x_{\text{true}}|$. The percentage error gives a sense of relative accuracy: $\%\,\text{error} = \frac{|x_{\text{measured}} - x_{\text{true}}|}{x_{\text{true}}} \times 100\%$. Low absolute and percentage errors indicate high accuracy. Systematic errors — such as a zero-offset on a voltmeter or a stretched measuring tape — reduce accuracy without necessarily affecting precision.

Precision is evaluated by taking multiple readings and computing their spread. The standard deviation $s$ of a sample of $n$ measurements $\{x_1, x_2, \ldots, x_n\}$ is: $$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$$ where $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$ is the sample mean. A small $s$ means high precision. The uncertainty in the mean (standard error) is $\sigma_{\bar{x}} = s/\sqrt{n}$, which decreases as more readings are taken — one reason physicists repeat measurements.

Reproducibility extends precision to different laboratories, operators, or instruments. If physicist A measures the speed of light as $(2.9979 \pm 0.0002) \times 10^8$ m/s and physicist B, using a completely different method, obtains $(2.9978 \pm 0.0003) \times 10^8$ m/s, their results are reproducible because the intervals overlap. Reproducibility failures often reveal hidden systematic errors or flawed experimental design. It is distinct from repeatability, which only concerns the same operator with the same instrument in a short time frame.

The resolution of an analogue instrument is conventionally taken as half the smallest scale division (the least count). For a digital instrument it is one unit of the last displayed digit. If a thermometer's smallest division is 1 °C, its resolution is 0.5 °C. Resolution sets a hard lower bound on the random uncertainty you can report: it makes no physical sense to claim a reading to greater precision than the instrument's resolution. Resolution is an intrinsic property of the instrument, independent of the quantity being measured.

When a result $R$ depends on independently measured quantities $a$ and $b$, uncertainties propagate. For $R = a + b$ or $R = a - b$: $\Delta R = \Delta a + \Delta b$ (absolute uncertainties add). For $R = ab$ or $R = a/b$: $\frac{\Delta R}{R} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$ (fractional uncertainties add). For a power law $R = a^n$: $\frac{\Delta R}{R} = |n|\frac{\Delta a}{a}$. These rules ensure that the final reported value honestly reflects the accumulated limitations of all measurements involved.

• s — Sample standard deviation (same as measured quantity): Measures the spread of repeated measurements; small s means high precision.
• x_i — Individual measurement (same as measured quantity): The i-th value in a set of n repeated readings.
• \bar{x} — Sample mean (same as measured quantity): Arithmetic average of all n measurements; used as the best estimate of the true value.
• n — Number of measurements (dimensionless): Total count of repeated readings; larger n reduces the standard error in the mean.

Problem. A student measures the period of a pendulum five times and records: 1.84 s, 1.86 s, 1.83 s, 1.87 s, 1.85 s. The accepted value is 1.90 s. Calculate (a) the mean period, (b) the sample standard deviation, (c) the standard error of the mean, (d) the percentage error in the mean. Comment on accuracy and precision.

Answer. Mean period = 1.850 s; s = 0.016 s; standard error = 0.007 s; percentage error = 2.6%. High precision, low accuracy — suggests systematic error.

• Significant Figures and Rounding — Significant figures are the practical tool for communicating the precision of a measurement in written form; mastering them is inseparable from understanding precision and resolution.
• Systematic and Random Errors — Systematic errors reduce accuracy without affecting precision; random errors reduce precision. Identifying which type of error dominates is essential for improving measurement quality.
• Propagation of Uncertainty — Once individual measurement uncertainties are established using precision and resolution, propagation rules determine the uncertainty of any calculated result.
• Scientific Notation and Orders of Magnitude — Expressing measurements in scientific notation makes it straightforward to compare resolutions and uncertainties across vastly different scales.