Graphical Analysis and Evaluation of Experimental Data

Plotting data on a graph and extracting physical quantities from the resulting line is the single most-tested skill in AQA Paper 3 and in the practical-endorsement evidence record. A well-drawn graph distils a long table of measurements into a clear visual statement about the underlying physics; it provides a gradient and intercept that can be related to physical constants; and the spread of the data about the line gives a direct, intuitive measure of the experimental uncertainty. Conversely, a sloppy graph — wrong axes, missing units, poorly-judged best-fit line — can throw away most of the value of an otherwise careful experiment.

This lesson covers the standard graphical workflow at A-Level: choosing the right axes (often after linearising a non-linear relationship), drawing best-fit and worst-fit lines, extracting the gradient and intercept with their uncertainties, using error bars correctly, and reading log-linear or log-log plots to detect power-law relationships. We will also look at the evaluation skills — judging whether a hypothesis is supported, identifying anomalies, and writing the conclusion that AQA Paper 3 questions demand.

Spec mapping: This lesson addresses content from AQA 7408 §3.1 on graphical methods for analysing experimental data: drawing lines of best fit, extracting gradient and intercept with their uncertainties, plotting and interpreting error bars, and using log-linear or log-log plots to investigate power-law relationships (refer to the official AQA specification document for exact wording).

Synoptic links:

• Required practicals 2, 4, 6, 8, 10: every linear-fit practical (resistivity, Young modulus, simple harmonic motion, etc.) demands gradient extraction with uncertainty.
• Capacitor discharge (§3.7): a log-linear plot of ln V vs t gives a straight line with gradient −1/RC; this is a canonical log-plot question.
• Stefan's law (option topic): a log-log plot of intensity vs temperature gives a gradient of 4, identifying the fourth-power relationship.

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The Standard Graph Workflow

For an A-Level practical the expected workflow is:

1. Choose axes so the relationship becomes a straight line (linearise if necessary).
2. Plot data with appropriate scales, axis labels (with units) and clear data points.
3. Add error bars showing the absolute uncertainty in each measurement.
4. Draw a line of best fit by eye through the data, balancing points above and below.
5. Draw a worst-fit line for uncertainty estimation.
6. Extract gradient and intercept with their uncertainties from the best/worst-fit lines.
7. Relate to physics — connect the gradient/intercept to a physical constant via the rearranged equation.
8. Quote the result with uncertainty and discuss whether the data supports the hypothesis.

flowchart LR
    DATA[Raw data table] --> LIN[Linearise<br/>choose y vs x axes]
    LIN --> PLOT[Plot points<br/>with axis labels and units]
    PLOT --> EB[Add error bars<br/>showing absolute Δy and Δx]
    EB --> BF[Draw best-fit line<br/>balance scatter]
    BF --> WF[Draw worst-fit line<br/>steepest or shallowest within error bars]
    WF --> GRAD[Extract gradient<br/>m_best, m_worst]
    GRAD --> DELTA[Δm = m_best − m_worst]
    DELTA --> CONN[Connect gradient to physics<br/>e.g. m = 1/RC]
    CONN --> REP[Report constant with uncertainty<br/>discuss consistency with theory]
    style PLOT fill:#27ae60,color:#fff
    style CONN fill:#1f4e79,color:#fff

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Choosing Axes and Linearising

Many physical relationships are not naturally linear, but can be transformed into a straight line by choosing appropriate quantities to plot. The standard transformations:

Original relationship	Plot...	Gradient gives	Intercept gives
y = mx + c	y vs x	m	c
y = kx²	y vs x²	k	0
y = k/x	y vs 1/x	k	0
y = ke⁻ᵏˣ	ln y vs x	−k	ln k₀
y = kxⁿ	ln y vs ln x	n	ln k

The exponential and power-law transformations are particularly important at A-Level because they identify which law underlies the data — does a quantity decay exponentially or follow a power law?

Worked example — capacitor discharge

For a capacitor discharging through a resistor, V = V₀ e^(−t/RC). Taking the natural log: ln V = ln V₀ − t/RC. Plotting ln V (y-axis) against t (x-axis) gives a straight line with gradient −1/RC and intercept ln V₀.

If the experiment gives a straight line on this plot, the exponential law is confirmed. The time constant RC is extracted from −1/gradient, and if R is known, C is calculated.

Worked example — power law

If a student suspects the period of a pendulum is T = kL^n, taking the log gives ln T = ln k + n ln L. Plotting ln T against ln L should give a straight line of gradient n. The theoretical value is n = 0.5 (since T ∝ √L), so the experimental gradient is compared to 0.5 to test the hypothesis.

Deep Worked Example — Power-Law Linearisation with Uncertainty Propagation

Suppose a student investigates whether the resistance R of a thermistor varies with temperature T (in kelvin) as a power law R = a T^n, where a and n are unknown constants. The student records seven (T, R) data pairs:

T / K	R / Ω
290	1180
300	980
310	820
320	690
330	580
340	495
350	420

The aim is to (i) confirm that a power-law model fits the data, (ii) extract n and a with their uncertainties, and (iii) compare with the theoretical thermistor model.

Step 1 — linearise. Take natural logs of both sides:

ln R = ln a + n × ln T

Defining y = ln R and x = ln T, the relationship is the straight line y = (ln a) + n × x, with gradient n and intercept ln a.

Step 2 — transform the data. Compute ln T and ln R for each pair:

ln T	ln R
5.670	7.073
5.704	6.888
5.737	6.709
5.768	6.537
5.799	6.363
5.829	6.205
5.858	6.040

The data should be plotted on standard linear graph paper (no log axes needed once the logs have been pre-computed) — y axis from about 6.0 to 7.1, x axis from about 5.66 to 5.86. The scale choice gives ≥60% paper coverage.

Step 3 — draw best-fit and worst-fit lines. By inspection (a least-squares fit gives the same result, but A-Level expects by-eye), the best-fit line passes through approximately (5.670, 7.073) and (5.858, 6.040). The best-fit gradient is:

n_best = (6.040 − 7.073) / (5.858 − 5.670) = −1.033 / 0.188 = −5.50

The worst-fit line, pulled to the steeper extreme through the error bars (assumed ±0.02 on each ln value, consistent with a few-percent uncertainty in the raw R measurement), would pass through points 0.02 below the top-left and 0.02 above the bottom-right. The worst-fit gradient is then approximately:

n_worst = (6.060 − 7.053) / 0.188 = −0.993 / 0.188 = −5.28

The absolute uncertainty in the gradient is therefore Δn = |n_best − n_worst| = |−5.50 − (−5.28)| = 0.22.

Step 4 — extract a from the intercept. The intercept of the best-fit line (at x = 0, i.e. ln T = 0, i.e. T = 1 K) is c_best = y − n × x = 7.073 − (−5.50)(5.670) = 7.073 + 31.19 = 38.26. So ln a = 38.26 and a = exp(38.26) ≈ 4.1 × 10¹⁶ Ω K^(−n). The very large number reflects the extrapolation from typical lab temperatures (290–350 K) back to 1 K, which is far outside the data range and not directly physically meaningful — but the value of a is needed to reproduce the curve at any other temperature.

Step 5 — propagate uncertainty. The uncertainty in n is Δn = 0.22 from step 3. For the intercept, repeating the best/worst-fit procedure gives Δc ≈ 1.3 (the small numerical change in y at x = 0 amplifies on extrapolation). Hence Δa / a = Δ(ln a) = Δc = 1.3, so a is known only to within a factor of e^1.3 ≈ 3.7 — three significant figures of a are not justified, and any quoting beyond one significant figure is misleading.

Step 6 — physical interpretation. The best-fit exponent n = −5.5 ± 0.2 says that R falls extremely steeply with T — much steeper than a simple inverse relationship n = −1. This is consistent with the standard NTC-thermistor model R = R₀ exp(B/T), which is not a pure power law in T but happens to mimic one over a narrow temperature range. The conclusion an A* student would reach: the power-law fit is empirically good on the (290 K, 350 K) interval, but the underlying physics is exponential, not power-law. To confirm this, the student should plot ln R against 1/T (the exponential-in-1/T linearisation) and see whether the fit is better than the power-law fit, with smaller residuals. The diagnostic use of two competing linearisations is the AO3 move that lifts this analysis above a Grade C.

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Plotting Data Correctly

A correctly-plotted A-Level graph has all of:

• Axes labelled with quantity and unit in the form "Length / m" or "Resistance / Ω". The "/ unit" notation strips the unit so the numbers on the axis are dimensionless.
• Scales chosen to fill at least 60% of the available paper space (markschemes commonly penalise scales using less than half the graph paper).
• Scales linear and sensibly spaced — usually 1, 2, 5 or 10 per major division. Avoid 3 or 7 per division.
• Data points marked clearly with a small cross or dot — neither a large blob nor an indistinct smudge.
• Each point labelled with its error bars in both directions where the uncertainty is significant.
• A title stating what the graph shows.

graph TD
    A[Choose linearisation] --> B[Define x and y]
    B --> C[Tabulate transformed data]
    C --> D[Choose scale: 1, 2, 5 or 10]
    D --> E[Label axes with quantity / unit]
    E --> F[Plot points with crosses]
    F --> G[Add error bars]
    G --> H[Draw best-fit line]
    style E fill:#27ae60,color:#fff

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Error Bars

Error bars represent the uncertainty in each measurement and are plotted as short line segments extending above and below each data point (for y-uncertainty) and to either side (for x-uncertainty).

• For a data point at (x, y) with uncertainties Δy and Δx, the vertical error bar runs from (x, y − Δy) to (x, y + Δy), and the horizontal error bar runs from (x − Δx, y) to (x + Δx, y).
• If the error bars are smaller than the size of the data-point marker, they are not normally drawn — the marker itself represents the uncertainty.
• The error bar shows the range within which the true value is expected to lie at the stated confidence level (usually 1 standard deviation, ~68% confidence; but at A-Level the half-range convention is used without explicit confidence statement).

Described diagram — graph with error bars. Imagine a scatter plot with x running from 0 to 10 along the bottom and y from 0 to 20 up the side. Five data points are plotted, each as a small cross. At each cross, a short vertical line extends about 1 unit above and 1 unit below — these are the y-error bars. Two of the points also have short horizontal segments extending ±0.5 units — these are the x-error bars on those two readings, where the x-uncertainty was larger. A best-fit line runs through the data, passing through every error bar.

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Best-Fit and Worst-Fit Lines

The line of best fit is drawn by eye to pass as close as possible to all data points, with roughly equal numbers of points above and below, and (ideally) passing through every error bar.

The worst-fit line (sometimes "worst acceptable line") is the steepest or shallowest line that still passes through every error bar — it represents the most extreme gradient consistent with the data. There are two possible worst-fit lines (one steeper, one shallower than best-fit); the convention is to choose whichever gives a larger |gradient − best-fit gradient|.

The uncertainty in the gradient is then:

Δm = |m_best − m_worst|

And similarly for the intercept:

Δc = |c_best − c_worst|

Worked example — gradient extraction with uncertainty

A student plots V (y-axis) against I (x-axis) for a resistor and obtains a straight line through the origin. From the best-fit line: m_best = (12.4 − 0) / (3.10 − 0) = 4.00 Ω. From the worst-fit line (slightly steeper): m_worst = (12.4 − 0) / (3.05 − 0) = 4.07 Ω.

Δm = |4.00 − 4.07| = 0.07 Ω.

So R = (4.0 ± 0.1) Ω.

Tip. Choose the two end points of the gradient calculation to be as far apart as possible — this minimises the rounding-error in the gradient and is markscheme-rewarded.

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Identifying Anomalies

An anomalous reading is a data point that lies clearly outside the trend established by the other points. Anomalies typically arise from transient experimental errors (a misread scale, a power-supply fluctuation, a draught, a momentary loss of concentration).

The accepted treatment at A-Level:

1. Identify the anomaly by marking it explicitly on the graph (often with a circle around it).
2. Investigate whether the cause can be traced (e.g. check the lab notebook for that reading; check if it occurred at a particular time).
3. Re-measure if possible — return to the experiment and repeat.
4. If re-measurement is not possible, exclude the anomaly from the best-fit-line analysis, and note the exclusion explicitly in the conclusion.

The exam-paper rule: never delete an anomaly silently. The credit-earning move is to identify, explain (where possible) and either re-take or exclude with justification.

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Log-Linear and Log-Log Plots

When a relationship is exponential or a power law, plotting on logarithmic axes is the diagnostic.

Log-linear (semilog) plot — for exponential relationships