Uncertainties on Graphs Revision Notes | A Level AQA Physics

1. Why Graphs Carry Uncertainty

Every measurement you make in a physics experiment carries uncertainty. When you plot those measurements on a graph, that uncertainty travels with each point — and if you ignore it, your gradient and intercept become meaningless numbers with no honest indication of how reliable they are.

Think about measuring the extension of a spring with a ruler. Your ruler has a resolution of 1 mm, so each reading carries an uncertainty of perhaps ±0.5 mm. Plot ten such readings and each point on your graph is not a precise dot — it is a small region of plausible values. Error bars make that region visible.

AQA requires you to represent these uncertainties on graphs using error bars, and then use those error bars to find the uncertainties in both the gradient and the intercept of a straight-line graph. These two skills are examined directly and are worth marks in every data-analysis question.

2. Drawing Error Bars on a Graph

To draw an error bar, you need two things: the plotted value and its absolute uncertainty. If you measured a voltage as 3.2 V with an uncertainty of ±0.1 V, you plot the point at 3.2 V and draw a vertical bar from 3.1 V to 3.3 V through that point. The total length of the bar is twice the absolute uncertainty.

Error bars can be vertical (uncertainty in the y-variable), horizontal (uncertainty in the x-variable), or both. In most A Level practicals, the independent variable is set precisely by the experimenter — for example, a mass placed on a hanger — so horizontal error bars are often absent. The dependent variable, which is measured, typically carries the error bar.

• Vertical error bar: uncertainty in the dependent (y-axis) variable
• Horizontal error bar: uncertainty in the independent (x-axis) variable
• Both: when both variables carry significant measurement uncertainty
• No error bar on a point: when the uncertainty is smaller than the plotted symbol, or when the variable was set rather than measured

That last point matters for the specification: individual points on the graph may or may not have associated error bars. If the uncertainty in a particular measurement is negligible compared to the scale of the graph, it is acceptable to omit the error bar for that point. You should be able to justify this decision — for instance, stating that the uncertainty is smaller than the diameter of the plotted point.

3. Lines of Best Fit and Worst Fit

Once your data points and error bars are plotted, you draw a line of best fit — a straight line that passes as close as possible to all the points, with roughly equal numbers of points above and below the line. This gives you your best estimate of the gradient and intercept.

Here is where error bars become powerful. Because each point has a range of plausible values, the true line could be steeper or shallower than your best fit. The line of worst fit is the most extreme straight line that can still pass through every error bar on the graph. It represents the boundary of what the data could plausibly support.

You need to draw two candidate lines of worst fit: one as steep as possible while still clipping every error bar, and one as shallow (or as negatively sloped) as possible while still clipping every error bar. The one that gives the largest difference from the best-fit gradient is the one you use for your uncertainty calculation.

4. Calculating Uncertainty in the Gradient

With both lines drawn, you can now quantify how uncertain your gradient is. Calculate the gradient of the line of best fit using a large triangle drawn on the graph — the larger the triangle, the more accurate your gradient calculation. Then calculate the gradient of the line of worst fit in the same way.

The result is then expressed as: gradient $= m_{\text{best}} \pm \Delta m$, with appropriate units. This tells anyone reading your report the range within which the true gradient plausibly lies, given the precision of your measurements.

Worked example: Calculating Uncertainty in the Gradient

A student investigates how the resistance R of a wire varies with its length L. They plot R (y-axis) against L (x-axis) with error bars on R. From their graph: the line of best fit has a gradient of 2.40 Ω m⁻¹ and the line of worst fit has a gradient of 2.10 Ω m⁻¹. Calculate the gradient and its uncertainty.

1. $m_{\text{best}} = 2.40 \ \Omega \text{ m}^{-1}$, $\quad m_{\text{worst}} = 2.10 \ \Omega \text{ m}^{-1}$ — Read these directly from the graph using large triangles for accuracy. [M1]
2. $m_{\text{best}} - m_{\text{worst}} = 2.40 - 2.10 = 0.30 \ \Omega \text{ m}^{-1}$ — This is the full spread between the two extreme plausible gradients.
3. $\Delta m = \frac{0.30}{2} = 0.15 \ \Omega \text{ m}^{-1}$ — The uncertainty is half the range, since the best-fit line sits in the middle of the plausible range. [M1]
4. $m = (2.40 \pm 0.15) \ \Omega \text{ m}^{-1}$ — Always include units on both the gradient value and the uncertainty. [A1]

Answer: Gradient $= (2.40 \pm 0.15) \ \Omega \text{ m}^{-1}$

5. Calculating Uncertainty in the Intercept

The same pair of lines gives you the uncertainty in the y-intercept. Read off the y-intercept of the line of best fit and the y-intercept of the line of worst fit, then apply the same half-range formula.

A common source of confusion is that the line of worst fit used for the intercept uncertainty may not be the same extreme line you used for the gradient uncertainty. For the gradient, you use whichever candidate worst-fit line gives the largest difference from the best-fit gradient. For the intercept, you independently use whichever candidate worst-fit line gives the largest difference from the best-fit intercept — this could be the steepest or the shallowest line, and it need not match the line chosen for the gradient calculation.

Worked example: Calculating Uncertainty in the Intercept

Using the same resistance-length graph as before: the line of best fit has a y-intercept of 0.20 Ω and the line of worst fit has a y-intercept of 0.44 Ω. Calculate the intercept and its uncertainty.

1. $c_{\text{best}} = 0.20 \ \Omega$, $\quad c_{\text{worst}} = 0.44 \ \Omega$ — Read these from where each line crosses the y-axis. [M1]
2. $|c_{\text{best}} - c_{\text{worst}}| = |0.20 - 0.44| = 0.24 \ \Omega$ — Take the absolute value — the order does not matter since you are finding a range.
3. $\Delta c = \frac{0.24}{2} = 0.12 \ \Omega$ — Same half-range approach as for the gradient. [M1]
4. $c = (0.20 \pm 0.12) \ \Omega$ — The relatively large uncertainty here suggests the intercept is not well-determined by this data set. [A1]

Answer: Intercept $= (0.20 \pm 0.12) \ \Omega$

6. When Error Bars Are Absent or Unequal

The specification explicitly states that individual points on the graph may or may not have associated error bars. There are two legitimate reasons a point might lack an error bar: the uncertainty is genuinely negligible (smaller than the plotted symbol), or the variable at that point was set rather than measured.

Error bars can also vary in size from point to point. If you measure a quantity using a method whose percentage uncertainty depends on the magnitude of the reading — for example, timing short intervals with a stopwatch — then points at low values will have proportionally larger error bars than points at high values. Draw each bar to reflect the actual uncertainty at that point, not a uniform bar copied from one measurement.

When error bars are absent from all points, you cannot draw a formal line of worst fit. In that situation, AQA expects you to acknowledge that the uncertainty in the gradient and intercept cannot be determined from the graph alone, and that additional information about measurement uncertainty would be needed.

8. Exam Technique and Common Mistakes

Graph-based uncertainty questions follow predictable patterns in AQA papers. The marks are almost always split between: drawing the line of worst fit correctly, calculating the gradient of each line using a clearly shown triangle, and applying the half-range formula. Miss any one of these and you drop marks even if your arithmetic is correct.

When drawing your triangle to find a gradient, make it as large as possible — at least half the length of the plotted line. A small triangle amplifies reading errors. Always read coordinates from the graph axes, not from the data table, and show the triangle explicitly on the graph so the examiner can award the method mark.

A frequent error is using the formula $\Delta m = m_{\text{best}} - m_{\text{worst}}$ without halving. This gives the full range, not the uncertainty. The uncertainty is always half the range, because the best-fit line sits at the centre of the plausible interval.

Mastering error bars and lines of worst fit means you can honestly report what your experiment actually tells you — not a false precision, but a justified range. That is the core of what AQA is testing: whether you understand that every measurement has limits, and whether you can quantify those limits rigorously on a graph.