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Graphs are the physicist's most powerful single tool for extracting information from experimental data. A well-drawn graph converts noisy measurements into a clean straight line whose gradient and intercept carry direct physical meaning. But to use a graph quantitatively, you must also handle the uncertainties on the plotted points. OCR examines this explicitly in practical-skills questions, and it is central to the Practical Endorsement assessments required for Physics A.
This lesson covers: drawing error bars, finding the line of best fit, determining the worst acceptable line, and extracting the uncertainties in gradient and intercept from a graph.
In Lesson 7 we propagated uncertainties analytically — multiply out the percentages, add them up, quote the result. That works, but it has two weaknesses:
A graph, by contrast:
For these reasons, every major OCR practical ends with a graph whose gradient carries the final numerical answer.
An error bar is a short vertical (or horizontal) line drawn through a plotted point, extending from (value − uncertainty) to (value + uncertainty). It shows graphically the range within which the true value is believed to lie.
A student measures the extension of a spring for various loads. Force is measured with ±0.1 N; extension with ±0.5 mm.
| F / N | x / mm |
|---|---|
| 1.0 ± 0.1 | 5.2 ± 0.5 |
| 2.0 ± 0.1 | 9.8 ± 0.5 |
| 3.0 ± 0.1 | 15.3 ± 0.5 |
| 4.0 ± 0.1 | 20.5 ± 0.5 |
| 5.0 ± 0.1 | 25.0 ± 0.5 |
On a graph of x (y-axis) vs F (x-axis), each point has a vertical error bar of total length 1.0 mm (from x − 0.5 to x + 0.5). The horizontal error bars (0.2 N total) are small compared to the range of F plotted and might not be drawn, but should be mentioned.
A line of best fit is a single straight line drawn through the plotted points such that it passes through, or as close as possible to, as many points (and their error bars) as possible. Formally, it is the line that minimises the sum of squared vertical deviations — but for hand-drawn graphs, "eyeball" judgement based on error bars is what OCR expects.
Exam Tip: Examiners check the line of best fit with a transparent ruler. If your line clearly misses several error bars, or has most points on one side, you lose marks. Spend the time to draw it carefully.
Once you have the line of best fit, you must also construct the worst acceptable line. This is the line that passes through the extremes of the error bars of the first and last points (or of the outermost pair of points), and represents the steepest (or shallowest) line that is still consistent with the data.
The worst acceptable line is what gives you the uncertainty on the gradient and uncertainty on the intercept.
Depending on how your line of best fit sits, the worst acceptable line can be either:
You then pick whichever candidate is further from the best-fit line (i.e. the worse case) and use it to calculate the uncertainty.
graph LR
subgraph "Data Points with Error Bars"
A["First point<br/>x₁, y₁ ± Δy"]
B["Middle points"]
C["Last point<br/>x₂, y₂ ± Δy"]
end
subgraph "Fitted Lines"
D["Line of best fit<br/>gradient m_best"]
E["Steep worst line<br/>gradient m_max"]
F["Shallow worst line<br/>gradient m_min"]
end
A --> D
B --> D
C --> D
A --> E
C --> E
A --> F
C --> F
Having drawn both lines, you read off:
Then:
Δm = |m_best − m_worst|
This is the absolute uncertainty in the gradient. Percentage uncertainty is:
%m = (Δm / m_best) × 100%
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