Measurement Uncertainty

The absolute uncertainty $\Delta x$ is expressed in the same units as the measured quantity $x$. When reading an analogue instrument, the absolute uncertainty is typically half the smallest scale division. For example, a ruler with 1 mm divisions gives $\Delta x = \pm 0.5$ mm. For a digital instrument, take the last displayed digit as the absolute uncertainty, e.g. a voltmeter reading 3.47 V has $\Delta V = \pm 0.01$ V. The result is written $x \pm \Delta x$.

The percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value, multiplied by 100: $$\text{Percentage uncertainty} = \frac{\Delta x}{x} \times 100\%.$$ A short pencil of length 50 mm measured with $\Delta x = 0.5$ mm has a 1 % uncertainty, whereas a 300 mm ruler length measured with the same instrument has only 0.17 % uncertainty. Percentage uncertainty reveals how significant the error is relative to the measurement itself.

When two quantities are added or subtracted to find a result $z = x \pm y$, the absolute uncertainties add: $$\Delta z = \Delta x + \Delta y.$$ This is the most conservative rule — uncertainties never cancel when combined. Example: measuring a rod by finding both ends on a ruler gives a length $L = x_2 - x_1$, so $\Delta L = \Delta x_2 + \Delta x_1$. Even though one reading is subtracted, both uncertainties contribute positively to $\Delta L$.

When quantities are multiplied or divided to give $z = x \cdot y$ or $z = x / y$, the percentage uncertainties add: $$\frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y}.$$ This rule extends to powers: if $z = x^n$, then $\frac{\Delta z}{z} = n \frac{\Delta x}{x}$. For example, if you calculate the area of a square from a measured side $s$, the percentage uncertainty in the area is twice the percentage uncertainty in $s$, because $A = s^2$.

When a measurement is repeated $n$ times, giving values $x_1, x_2, \ldots, x_n$, the best estimate is the mean $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$. The uncertainty can be estimated as half the range of the readings: $\Delta x = \frac{x_{\max} - x_{\min}}{2}$. In more advanced work, the standard error of the mean, $\sigma_{\bar{x}} = \sigma / \sqrt{n}$, is used, where $\sigma$ is the standard deviation of the data set. Both approaches acknowledge that more repeated readings reduce the random component of uncertainty.

An uncertainty should be quoted to one or two significant figures, and the measured value must be rounded to the same decimal place as the uncertainty. Writing $g = 9.8134 \pm 0.3$ m s$^{-2}$ is incorrect because the value is quoted to four decimal places but the uncertainty is only known to one. The correct form is $g = 9.8 \pm 0.3$ m s$^{-2}$. This discipline prevents the false impression of precision that would mislead anyone using the result.

V = 119 \pm 4 \text{ cm}^3

• Systematic and Random Errors — Uncertainty arises from both random scatter and systematic offsets; distinguishing these two types of error is essential for understanding and reducing overall measurement uncertainty.
• Significant Figures — The rules for significant figures determine how a result and its uncertainty are correctly rounded and presented, preventing the false impression of precision.
• Units and Dimensions — Absolute uncertainty must always be expressed in the same physical units as the measured quantity; understanding dimensional analysis ensures uncertainties are handled consistently.
• Graphs and Data Analysis in Experiments — Error bars on graphs are a visual representation of measurement uncertainty, and the range of possible gradients through error bars propagates uncertainty into any calculated physical constant.