Graphical Treatment of Errors

Spec mapping: OCR H556 Module 2.2 — Making measurements and analysing data, specifically the graphical analysis of experimental data with error bars and worst-fit lines. This lesson is central to the Practical Endorsement and to every Paper-3 graph-and-extract-a-physical-quantity question, including PAG 1 (measurement of physical quantities) and PAG 2 (mechanics) where gradient extraction is the final numerical step. (Refer to the official OCR H556 specification document for exact wording.)

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.

This lesson covers: drawing error bars; finding the line of best fit; determining the worst acceptable line; and extracting the absolute and percentage uncertainties on gradient and intercept.

Key definition. An error bar is a graphical representation of the absolute uncertainty on a plotted point, drawn as a line segment extending from $\text{value} - \Delta$ to $\text{value} + \Delta$ along the relevant axis.

Why Graphs Beat Formula Propagation

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

It relies on single measurements at each point, so systematic errors can hide undetected.
It produces a single number with no insight into how the data scatter against 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 compared to a single-point calculation.
Yields both a value and an uncertainty directly from the slope.

For these reasons, every major OCR practical ends with a graph whose gradient carries the final numerical answer. Paper 3 routinely tests precisely this skill.

Error Bars

An error bar extends $\pm \Delta$ above and below the plotted point (vertical bar) or $\pm \Delta$ left and right (horizontal bar). For most school practicals only the y-axis error bars are drawn, because the y-axis is usually the quantity with the larger percentage uncertainty.

Rules for drawing error bars

Draw error bars for the axis (or axes) with the larger percentage uncertainty. If the x-axis uncertainty is comparable, draw both.
Each bar extends $\pm$ the absolute uncertainty for that point. Bars are centred on the point.
Bars are drawn to scale, not at arbitrary length.
If a bar is too small to see at the chosen scale, write a note: "error bars on $L$ are smaller than the plotted symbol".
If the uncertainty varies from point to point — e.g. instrument resolution is fixed but the quantity changes — bars will have different lengths at different points.

Example — constructing error bars

A student measures the extension of a spring under various loads. Force is measured with $\pm 0.1\,\text{N}$ ; extension with $\pm 0.5\,\text{mm}$ .

$F\,/\,\text{N}$	$x\,/\,\text{mm}$
$1.0 \pm 0.1$	$5.2 \pm 0.5$
$2.0 \pm 0.1$	$9.8 \pm 0.5$
$3.0 \pm 0.1$	$15.3 \pm 0.5$
$4.0 \pm 0.1$	$20.5 \pm 0.5$
$5.0 \pm 0.1$	$25.0 \pm 0.5$

On a graph of $x$ (y-axis) against $F$ (x-axis), each point has a vertical error bar of total length $1.0\,\text{mm}$ . The horizontal $0.2\,\text{N}$ bars are small compared to the range of $F$ and might not be drawn, but their existence should be noted.

A Sketch — Best Fit, Min Slope, Max Slope

Inline SVG capturing the three lines you must internalise:

The steeper worst-fit line clips the top of the first error bar and the bottom of the last; the shallower worst-fit line clips the bottom of the first and the top of the last. The one further from the best-fit gradient is the worst-acceptable line.

The Line of Best Fit

A line of best fit is a single straight line drawn through the plotted data such that it passes through, or as close as possible to, as many points (and their error bars) as possible. Formally, the rigorous procedure is least-squares regression (minimising the sum of squared vertical deviations). For hand-drawn A-Level graphs, an "eyeball" judgement based on error bars is what OCR mark schemes expect — provided that judgement is consistent with the rules below.

Rules for drawing the line of best fit

Use a transparent ruler so you can see the points through it.
Position the line so that it passes through (or near) as many error bars as possible, with points roughly equally scattered above and below.
If theory predicts a particular intercept (e.g. Hooke's law predicts $c = 0$ ), check whether the data support that prediction rather than forcing it.
Draw lightly with a sharp pencil first, then firm up.
Extend the line across the full range of the data — do not stop at the last data point if you need to read off an intercept.
Do not join the dots. A physics graph is not a dot-to-dot; the scatter should not appear in the fitted line.

Exam Tip: OCR examiners check the line of best fit with a transparent ruler. A line that clearly misses several error bars, or has most points on one side, loses marks for "best fit". The fix is a sharp pencil and a slow hand.

The Worst Acceptable Line

The worst acceptable line (sometimes called the worst-fit line) is the line of steepest — or shallowest — gradient that still passes through the error bars of the first and last plotted points (and ideally most of the points in between). It exists in two candidate orientations:

Steeper than best fit: through the top corner of the first error bar and the bottom corner of the last (i.e. $(x_{\min}, y_{\min} + \Delta)$ to $(x_{\max}, y_{\max} - \Delta)$ if $y$ increases with $x$ , or vice versa).
Shallower than best fit: through the bottom corner of the first and the top corner of the last (reversing the corners above).

You compute both gradients and take the one further from the best fit as the worst-acceptable case. That candidate gives the uncertainty in the gradient (and in the intercept).

graph LR
    A[Plot data with error bars] --> B[Draw best-fit line by eye]
    B --> C[Identify outermost error-bar corners]
    C --> D[Draw steeper worst-fit candidate]
    C --> E[Draw shallower worst-fit candidate]
    D --> F{Which gradient is further<br/>from best fit?}
    E --> F
    F --> G[Use that gradient as worst<br/>Δm = |m_best − m_worst|]

Extracting the Gradient and Its Uncertainty

Having drawn the best-fit line, read off $m_{\text{best}}$ — the gradient — using a large triangle:

The triangle should span at least half the plotted range. Small triangles amplify reading errors.
Read endpoints on the line, not on individual data points (avoid the temptation to pick two data points; they may not lie on your best-fit line).
Quote units: gradient units are (unit of $y$ ) divided by (unit of $x$ ).

Read $m_{\text{worst}}$ from the worst-fit candidate identified above. Then

$\Delta m = |m_{\text{best}} - m_{\text{worst}}|, \qquad \%m = \frac{\Delta m}{m_{\text{best}}}\times 100\%.$

Worked Example 1 — Hooke's law

A student plots extension $x$ (mm) against force $F$ (N) for a spring:

Best-fit line passes through $(0,0)$ and $(5.0, 25.0)$ : $m_{\text{best}} = 25.0/5.0 = 5.0\,\text{mm N}^{-1}$ .
Worst-acceptable line passes through $(0,0)$ and $(5.0, 26.5)$ : $m_{\text{worst}} = 5.30\,\text{mm N}^{-1}$ .

Calculate the gradient with absolute and percentage uncertainty, and hence the spring constant.

Solution.

$\Delta m = |5.30 - 5.00| = 0.30\,\text{mm N}^{-1}, \qquad \%m = \frac{0.30}{5.0}\times 100 = 6.0\%.$

The gradient is $x/F = 1/k$ , so

$k = \frac{1}{m} = \frac{1}{5.0 \times 10^{-3}} = 200\,\text{N m}^{-1}.$

The reciprocal has the same percentage uncertainty, so $\%k = 6.0\%$ . $\Delta k = 0.06 \times 200 = 12\,\text{N m}^{-1}$ . Quote $k = (200 \pm 10)\,\text{N m}^{-1}$ (uncertainty to 1 s.f.).

Extracting the Intercept and Its Uncertainty

For the y-intercept:

Read $c_{\text{best}}$ from the best-fit line at $x = 0$ .
Read $c_{\text{worst}}$ from the worst-fit line at $x = 0$ .

Then

$\Delta c = |c_{\text{best}} - c_{\text{worst}}|.$

The intercept's physical meaning depends on the experiment: zero-point offset in a Hooke's law plot, room-temperature resistance in an $R$ - $\theta$ plot, threshold frequency in a photoelectric plot, and so on.

Worked Example 2 — intercept uncertainty

A graph of resistance $R$ against temperature $\theta$ for a metal wire shows:

Best-fit line: gradient $0.040\,\Omega\,^{\circ}\text{C}^{-1}$ , y-intercept $10.0\,\Omega$ .
Worst-fit line: gradient $0.045\,\Omega\,^{\circ}\text{C}^{-1}$ , y-intercept $9.4\,\Omega$ .

Calculate the uncertainty in the y-intercept.

Solution. $\Delta c = |10.0 - 9.4| = 0.6\,\Omega$ . Quote $c = (10.0 \pm 0.6)\,\Omega$ .

The y-intercept represents the wire's resistance at $0\,^{\circ}\text{C}$ ; its uncertainty is the uncertainty in the extrapolation back to that temperature.

When the line does not pass through the origin

Theory may predict $c = 0$ (Hooke's law, for instance), but the measured intercept comes out non-zero. This can indicate

a zero error (e.g. the extension was not zero when $F = 0$ because the pointer was misaligned);
a systematic error in one of the measurements (calibration offset);
onset of non-linear behaviour outside the data range.

If the uncertainty range of the intercept does not include zero, you have quantitative evidence of a systematic problem. If it does include zero, you cannot distinguish a true non-zero intercept from experimental noise.

Converting Graph Uncertainties to Final Answers

Most OCR practicals use a graph to extract a gradient, which is then substituted into a physics formula to give the final answer. The percentage uncertainty on the gradient then propagates through that formula using the rules from the previous lesson.

Worked Example 3 — Young modulus from a graph

A student plots stress $\sigma$ against strain $\varepsilon$ for a wire and finds

$m_{\text{best}} = 2.0 \times 10^{11}\,\text{Pa}$ .
$m_{\text{worst}} = 2.2 \times 10^{11}\,\text{Pa}$ .
Additional percentage uncertainty on cross-sectional area from micrometer readings: $4\%$ .

Graphical Treatment of Errors

Graphical Treatment of Errors

Why Graphs Beat Formula Propagation

Error Bars

Rules for drawing error bars

Example — constructing error bars

A Sketch — Best Fit, Min Slope, Max Slope

The Line of Best Fit

Rules for drawing the line of best fit

The Worst Acceptable Line

Extracting the Gradient and Its Uncertainty

Worked Example 1 — Hooke's law

Extracting the Intercept and Its Uncertainty

Worked Example 2 — intercept uncertainty

When the line does not pass through the origin

Converting Graph Uncertainties to Final Answers

Worked Example 3 — Young modulus from a graph

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