1. Types of Error

All experimental measurements are subject to error. It is useful to separate them into two categories.

Random errors arise from unpredictable variations in conditions or in the way an instrument is read. Examples include electronic noise in an electrical instrument, slight fluctuations in the thickness of a wire, and variability in human reaction time when using a stopwatch.

Reducing random errors:

• Take at least three repeat readings and calculate a mean; this also makes anomalies easier to spot and investigate.
• Use data loggers, computers, or cameras to remove human judgement from the timing or reading process and to enable smaller measurement intervals.
• Choose instruments with higher resolution (e.g. a micrometer at 0.01 mm rather than a ruler at 1 mm).

Systematic errors originate in the apparatus itself or in a flaw in the experimental method. They are not reduced by repeating the measurement. Common examples include a balance that does not read zero when empty (a zero error) and reading a scale at an angle rather than at eye level (a parallax error).

Reducing systematic errors:

• Calibrate apparatus by measuring a known value (e.g. place a standard 1 kg mass on a balance); any discrepancy reveals the systematic offset.
• In radioactivity experiments, measure background radiation beforehand and subtract it from every reading.
• Read analogue scales at eye level, perpendicular to the scale, to eliminate parallax. Using a mirror behind the pointer and aligning the pointer with its image ensures a correct viewing angle.

The target patterns below make the distinction visual: notice that systematic error can produce a tight cluster away from the true value, while random error gives a wider spread around the bullseye.

[DIAGRAM: asset_name: 1.2 - Limitation of Physical Measurements - Diagram 1; asset_slug: 1.2 - Limitation of Physical Measurements - Diagram 1; recommended_method: retained_png; description: Two target diagrams side by side. Left target shows shots clustered tightly together but away from the bullseye, labelled "Precise but not accurate — systematic error". Right target shows shots scattered around the bullseye, labelled "Accurate on average but not precise — random error".]

2. Describing the Quality of Measurements

It is important to distinguish five terms precisely. The cleanest way to remember them is to separate ideas about the spread of readings, the conditions of the repeat, and the quality of the instrument/result.

Repeatability is tested within a single laboratory session. Reproducibility is a tougher test because it checks whether the result survives changes of person, equipment, or method. Resolution is a property of the instrument: a ruler marked every millimetre has a resolution of 1 mm, while a micrometer may resolve 0.01 mm. Higher resolution does not guarantee higher accuracy, but it limits how fine a measurement can be.

Note the important distinction: precision concerns the spread of repeated readings (random error), while accuracy concerns proximity to the true value (systematic error). A set of measurements can be precise without being accurate, and vice versa.

3. Quantifying Uncertainty

Uncertainty can be expressed in three equivalent forms.

Absolute uncertainty — the doubt expressed as a fixed quantity in the same unit as the measurement:

Fractional uncertainty — the doubt expressed as a fraction of the measured value:

Percentage uncertainty — the doubt expressed as a percentage of the measured value:

For the example above: $\frac{0.6}{7} \times 100\% = 8.6\%$, so the reading can be written as $7 \pm 0.6$ V, with fractional uncertainty $0.086$ or percentage uncertainty $8.6\%$.

To reduce percentage (and fractional) uncertainty, measure larger quantities. For instance, timing 10 oscillations of a pendulum rather than 1 reduces the percentage uncertainty in the period by a factor of 10.

How to assign an uncertainty

The method depends on the type of measurement.

Single reading from an analogue instrument (e.g. a thermometer): the uncertainty is $\pm$ half the smallest division. If the smallest division is 1 $^\circ$C, the uncertainty is $\pm\,0.5\;^\circ$C.

A measurement involving two readings (e.g. a length measured with a ruler, where both the start and end positions are judged): each end contributes $\pm$ half the smallest division, so the total uncertainty is $\pm$ the smallest division. For a millimetre ruler, this is $\pm\,1$ mm.

Digital instruments and quoted values: the uncertainty is $\pm$ the last significant digit unless stated otherwise. For example, a digital voltmeter reading 3.2 V has an uncertainty of $\pm\,0.1$ V.

Repeated readings: the uncertainty is half the range of the readings.

The result is expressed as: mean $\pm \frac{\text{range}}{2}$.

Significant figures and uncertainty

The number of significant figures quoted for a value should be consistent with its uncertainty. If the uncertainty in a length is $\pm\,0.01$ m, then quoting the length as 1.2345 m implies false precision; it should be reported as 1.23 m. Uncertainties themselves are usually given to one significant figure (or at most two).

4. Combining Uncertainties in Calculations

When measured quantities are combined in a formula, the uncertainties propagate into the final result. There are three standard cases to handle here. (Trigonometric and logarithmic combinations are not required.)

Case 1: Adding or subtracting quantities — add absolute uncertainties

If $C = A + B$ or $C = A - B$, then:

where $\Delta$ denotes the absolute uncertainty.

Worked example. A thermometer with an uncertainty of $\pm\,0.5$ K shows a temperature drop from $298 \pm 0.5$ K to $273 \pm 0.5$ K.

Notice that subtraction can greatly increase the percentage uncertainty: here each reading has $\approx 0.2\%$ uncertainty, but the difference has $\frac{1}{25} \times 100 = 4\%$ uncertainty.

Case 2: Multiplying or dividing quantities — add percentage uncertainties

If $C = A \times B$ or $C = A / B$, then:

Worked example. A force of $91 \pm 3$ N is applied to a mass of $7.0 \pm 0.2$ kg. Find the acceleration and its uncertainty.

Percentage uncertainty in $F$: $\frac{3}{91} \times 100 = 3.3\%$

Percentage uncertainty in $m$: $\frac{0.2}{7.0} \times 100 = 2.9\%$

Total percentage uncertainty in $a$: $3.3 + 2.9 = 6.2\%$

Convert back to absolute: $6.2\%$ of 13 $= 0.8\;\text{m\,s}^{-2}$

Case 3: Raising to a power — multiply the percentage uncertainty by the power

If $C = A^n$, then:

Worked example. The radius of a circle is $5.0 \pm 0.3$ cm. Find the area and its percentage uncertainty.

Percentage uncertainty in $r$: $\frac{0.3}{5.0} \times 100 = 6.0\%$

Since area $\propto r^2$, the power is 2:

Percentage uncertainty in $A$: $6.0 \times 2 = 12\%$

5. Representing and Extracting Uncertainties from Graphs

Error bars

Each data point on a graph can be given an error bar — a vertical line (and/or horizontal line) extending above and below (and/or left and right of) the point by an amount equal to the uncertainty in that quantity.

In the sketch below, notice that each vertical bar shows the uncertainty in a single y-value and that the best-fit line still passes through the range allowed by the data.

[DIAGRAM: asset_name: 1.2 - Limitation of Physical Measurements - Diagram 2; asset_slug: 1.2 - Limitation of Physical Measurements - Diagram 2; recommended_method: retained_png; description: A graph with data points plotted. Each point has a vertical error bar extending symmetrically above and below it. A straight line of best fit passes through all error bars. Labels indicate "error bar = +/- uncertainty in y" on one point.]

A line of best fit should follow the overall trend of the data and pass through as many error bars as is reasonably possible (excluding any anomalous points). If many error bars are missed, either the line choice is poor, there are anomalous points, or the uncertainties may have been underestimated.

Uncertainty in the gradient

To determine the uncertainty in the gradient of a straight-line graph:

1. Draw the line of best fit through the data points.
2. Draw the steepest acceptable line (the line of worst fit with the maximum gradient) that still passes through all error bars.
3. Draw the shallowest acceptable line (minimum gradient) that still passes through all error bars.
4. Calculate the gradient of each line.

An alternative convention uses both the maximum and minimum gradient lines:

Both methods are acceptable; the important thing is to be consistent.

Note that individual data points on a graph may or may not have associated error bars. When error bars are present, the worst-fit lines must pass through all error bars. When error bars are absent, draw the steepest and shallowest lines that could reasonably fit the data — these lines should still pass through or close to as many points as possible. The procedure for calculating the uncertainty in the gradient and intercept is the same in either case.

Uncertainty in the y-intercept

When the steepest and shallowest lines intercept the y-axis at different points, the uncertainty in the y-intercept is found in the same way:

The figure below shows what to compare: notice how the best-fit, steepest acceptable, and shallowest acceptable lines lead to different gradients and y-intercepts.

[DIAGRAM: asset_name: 1.2 - Limitation of Physical Measurements - Diagram 3; asset_slug: 1.2 - Limitation of Physical Measurements - Diagram 3; recommended_method: retained_png; description: A graph showing a line of best fit, a steepest line of worst fit, and a shallowest line of worst fit, all passing through the error bars. The different y-intercepts and gradients are labelled. Arrows indicate the difference between the best-fit gradient and the worst-fit gradient.]

6. Pulling It Together — Significant Figures and Good Practice

The number of significant figures in a final answer should reflect the size of the uncertainty, rather than just the raw number of digits entered into a calculator. In general:

• Quote uncertainties to 1 significant figure (occasionally 2 if the leading digit is 1).
• Quote the final value to the same number of decimal places as the uncertainty.
• Do not copy all the digits from a calculator display.

Identifying and reducing errors — a practical checklist:

This final problem pulls the measurement ideas together in a realistic analysis task.