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Uncertainty in a gradient and an intercept

Edexcel IAL Physics · WPH16 Practical Skills in Physics II

When a constant comes from a gradient, the scatter of the points about the line tells you how uncertain it is. A worst acceptable line turns that into a number.

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Best line and worst line

The IAL specification does not name this method, and recent WPH16 mark schemes have asked for a gradient from a large triangle and for compounded percentage uncertainties rather than for worst lines. It is still a sound way to discuss uncertainty quantitatively; if a question describes a method, use that one.

Passing it on

If a quantity is calculated from the gradient, its percentage uncertainty comes from the gradient's percentage uncertainty using the usual rules. If g = 4π2 / gradient, g has the same percentage uncertainty as the gradient. If E = gradient x l / A, add the percentage uncertainties of the gradient, l and A.

What scatter tells you

Pearson's uncertainties appendix says there is no expectation that error bars are added to graphs. Large scatter about the line suggests large random errors; a line that misses the origin when theory says it should not suggests a systematic error. A gradient taken from a line through many points is more reliable than any single calculation.

Worked example

The best-fit gradient is 4.05 s2/m and the steepest acceptable line has gradient 4.21 s2/m. Find the percentage uncertainty in the gradient and in g = 4π2 / gradient.

  1. Percentage uncertainty = |4.21 - 4.05| / 4.05 x 100 = 4.0%
  2. g = 4π2 / 4.05 = 9.75 m/s2, also ± 4.0%
  3. Absolute uncertainty in g = 0.39 m/s2, which is 0.4 m/s2 to 1 s.f., so give g to 1 d.p.

Answer: g = 9.7 ± 0.4 m/s2

Key idea: Worst acceptable line through the spread of points. Gradient uncertainty = |worst - best| / best x 100%; intercept uncertainty = |worst - best|. Carry the percentage into anything calculated from it.

Check yourself

  1. A best-fit gradient is 0.512 V/A and the worst acceptable gradient is 0.548 V/A. Find the percentage uncertainty in the gradient, to 2 s.f.
    Show answer

    7.0 %. |0.548 - 0.512| / 0.512 x 100 = 7.0%.

  2. The best-fit line has intercept 1.52 V and the worst line has intercept 1.47 V. What is the absolute uncertainty in the intercept, in V?
    Show answer

    0.05 V. |1.47 - 1.52| = 0.05 V.

  3. A graph of T2 against m has best gradient 1.58 s2/kg with a 4.0% uncertainty. Find k = 4π2 / gradient in N/m, to 3 s.f.
    Show answer

    25.0 N/m. k = 4π2 / 1.58 = 25.0 N/m.

  4. For k = 25.0 N/m with the gradient's 4.0% uncertainty, find the absolute uncertainty in k, in N/m.
    Show answer

    1.0 N/m. k has the same 4.0% uncertainty: 0.040 x 25.0 = 1.0 N/m.

  5. E = gradient x l / A. The gradient has 3.5% uncertainty, l 0.1% and A 2.6%. Find the percentage uncertainty in E.
    Show answer

    6.2 %. 3.5 + 0.1 + 2.6 = 6.2%.

  6. Which describes a worst acceptable line?
    • A line through the first and last points
    • A line through the origin
    • The steepest or shallowest straight line that still fits the spread of the points
    • A curve through every point
    Show answer

    The steepest or shallowest straight line that still fits the spread of the points. It shows how much the gradient could reasonably vary.

  7. Points on a graph show large scatter about the best-fit line. What does this suggest?
    • Large random errors in the measurements
    • A large systematic error
    • The axes are the wrong way round
    • The gradient is zero
    Show answer

    Large random errors in the measurements. Random errors scatter points either side of the line.

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Last updated 8 November 2026. Spotted a mistake? Tell us.