Graphical Treatment of Errors

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.

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Why Graphs Beat Formulas for Uncertainty Analysis

In Lesson 7 we propagated uncertainties analytically — multiply out the percentages, add them up, quote the result. That works, but it has two weaknesses:

1. It relies on single measurements at each point, so systematic errors can hide.
2. It gives only one number — no insight into how the data scatter compared to the expected relationship.

A graph, by contrast:

• Makes random scatter visible at a glance.
• Exposes systematic errors as unexpected intercepts or curvature.
• Averages data across many points, reducing the effective uncertainty on the gradient.
• Allows you to extract both a value and an uncertainty from the slope.

For these reasons, every major OCR practical ends with a graph whose gradient carries the final numerical answer.

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Error Bars

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.

Rules for Error Bars

1. Always draw error bars for the axis with the larger percentage uncertainty. For A-Level, you typically draw them only on the y-axis unless the x-axis uncertainty is comparable.
2. The error bar extends above and below the point by the absolute uncertainty in the plotted quantity.
3. The error bar should be centred on the point and drawn to scale — not with arbitrary length.
4. If the uncertainty is too small to see on the chosen scale, this is worth noting: "error bars not drawn because smaller than the plotted point".
5. If the uncertainty varies from point to point (e.g. because the resolution of the instrument is a fixed absolute value but the quantity changes), the bars will have different lengths on different points.

Example — Constructing Error Bars

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.

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The Line of Best Fit

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.

Rules for Drawing the Line of Best Fit

1. Use a transparent ruler so you can see the points through it.
2. Position the line so that:
   • It passes through as many error bars as possible.
   • Points are roughly equally scattered above and below the line.
   • If Hooke's law (or any other theoretical law) predicts the line should pass through the origin, you must consider whether your data actually support that, rather than forcing it through (0, 0).
3. Draw the line lightly first with a sharp pencil, then firm up.
4. Extend the line across the full range of your data — do not stop at the last data point if you need to read off an intercept.
5. Do not join the dots — a physics graph is not a dot-to-dot; the random scatter should not appear in the line.

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.

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The Worst Acceptable Line

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.

Two Candidates

Depending on how your line of best fit sits, the worst acceptable line can be either:

• Steeper than the best fit, passing through (x_min, y_min + Δy) and (x_max, y_max − Δy) — i.e. the top-left corner of the first error bar and the bottom-right corner of the last error bar, or
• Shallower than the best fit, passing through (x_min, y_max + Δy) and (x_max, y_min − Δy) — reversing the corners.

You then pick whichever candidate is further from the best-fit line (i.e. the worse case) and use it to calculate the uncertainty.

A Mermaid Diagram — Best and Worst Fit Lines

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

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Extracting the Gradient and Its Uncertainty

Having drawn both lines, you read off:

• m_best — the gradient of the line of best fit.
• m_worst — the gradient of the worst acceptable line.

Then:

Δm = |m_best − m_worst|

This is the absolute uncertainty in the gradient. Percentage uncertainty is:

%m = (Δm / m_best) × 100%

Rules for Calculating a Gradient

1. Use a large triangle (at least half the axes' range) — small triangles amplify reading errors.
2. Read intercepts on the actual axes where possible — not some arbitrary subdivision.
3. Include units — a gradient has units (unit of y) / (unit of x).
4. Show the calculation: "gradient = (y₂ − y₁) / (x₂ − x₁) = ..."

Worked Example 1 — Hooke's Law