Measurement, Uncertainty, and Error Analysis

Every measurement has uncertainty. Reporting a result without its uncertainty is incomplete.

Standard notation:

$$x = x_{\text{best}} \pm \delta x$$

Types of error

Absolute vs. relative uncertainty

$$\text{Absolute: } \delta x, \qquad \text{Relative: } \frac{\delta x}{x} \times 100\%$$

Significant figures — The last digit of a measurement reflects the uncertainty. For example, $L = 3.45$ m means the uncertainty is in the hundredths place ($\pm 0.005$ m).

Rules for significant figures

• Addition/subtraction: match the fewest decimal places.
• Multiplication/division: match the fewest significant figures.

Tip: Always report uncertainty to one significant figure (two at most), and round the measurement to match. For example: $g = 9.81 \pm 0.03$ m/s$^{2}$, not $9.8134 \pm 0.0312$.

Common pitfall: A measurement without an uncertainty is only half a result: "$g = 9.7$" is unfalsifiable until you say $\pm$ what. The error bar is not an admission of failure — it is the claim’s precision.

Measurement and Uncertainty

Every measurement has an associated uncertainty. Repeated measurements of the same quantity cluster around a mean $\bar{x}$ with spread described by the standard deviation:

$$\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$$

The standard error of the mean $\sigma_{\bar{x}} = \sigma/\sqrt{N}$ shrinks as more measurements are taken.

A result is reported as $\bar{x} \pm \sigma_{\bar{x}}$ — the uncertainty quantifies our confidence in the measurement.