Measurement, Uncertainty, and Error Analysis

Measurement & Error

Every measurement has uncertainty. Reporting a result without it is incomplete; an error bar is your claim's precision, not a failure.

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

Absolute vs Relative Uncertainty:

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

Significant figures: The last digit reflects the uncertainty ($L = 3.45$ m implies $\pm 0.005$ m).

Sig Figs & Statistics

Rules: Add/subtract matches the fewest decimal places. Multiply/divide matches the fewest significant figures.

Standard deviation measures spread in repeated trials:

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

The standard error of the mean shrinks with more trials:

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{N}}$$

Tip: Report uncertainty to one sig fig (two at most) and match the measurement: $g = 9.81 \pm 0.03$ m/s$^2$.