How to find the uncertainty in a gradient
Plot every point with its error bar. Draw the line of best fit, then the worst acceptable line: the steepest or the shallowest straight line that still passes through every error bar. Find both gradients with triangles that cover more than half the line. The uncertainty in the gradient is the difference between the two gradients (the IB takes half the difference between the steepest and shallowest lines). Divide by the best gradient for the percentage uncertainty, and carry it to the constant: g = 2 ÷ gradient has the same percentage uncertainty as the gradient.
How answers are marked: a best-fit gradient counts when its line passes through every error bar, give or take half a small square. A worst line counts when it passes through every bar and is at least half way from the best line to the extreme line. Answers worked from your own earlier answers count. On paper the grid is blank so you choose the scales; on screen, Plot the points for me draws them.
Example: g by free fall
In set GRAPH2 (Cambridge 9702), the ball takes 0.289 ± 0.008 s to fall 0.400 m, so t² = 0.289² = 0.0835 s² and Δ(t²) = 2 × 0.289 × 0.008 = 0.0046 s². The best-fit line through (0.200, 0.04139) and (1.200, 0.2452) has a gradient of 0.2038 s² m⁻¹. The shallowest line through every error bar has 0.1901, so the uncertainty is 0.014 s² m⁻¹, or 0.014 ÷ 0.2038 = 6.87%. Then g = 2 ÷ 0.2038 = 9.81 m s⁻² with the same percentage uncertainty: 9.81 × 6.87% = 0.67, so g = (9.81 ± 0.67) m s⁻². Open set GRAPH2
Board conventions
| Board | Uncertainty in the gradient | Calculated values in the table |
|---|---|---|
| Cambridge 9702 | gradient of best fit − gradient of the worst acceptable line | n or n + 1 significant figures; logs to that many decimal places |
| AQA | percentage: |best − worst| ÷ best × 100% | n significant figures; uncertainties in logs and sines are not tested, so those experiments are left out |
| OCR | |worst − best|, from the steepest or shallowest line through all the bars | n significant figures (one more is not penalised); logs to n decimal places |
| IB | (steepest − shallowest) ÷ 2 | n or n + 1 significant figures (this site’s rule) |
n is the number of significant figures in the least precise reading. The IB physics guide is for IB schools only and could not be read for this site, so the IB preset follows the half-range method that IB revision guides teach. Check it with your teacher.
Questions
How do you calculate the uncertainty in a gradient?
Draw the line of best fit and a worst acceptable line through all the error bars. Find both gradients with large triangles. The absolute uncertainty is the difference between the two gradients. Divide it by the best gradient and multiply by 100 for the percentage uncertainty.
What is a worst acceptable line?
The steepest or the shallowest straight line that still passes through every error bar. Cambridge asks you to draw it dashed or label it. It often joins the top of one end bar to the bottom of the other, but check that it passes through every bar in between.
How do you draw error bars?
Work out the uncertainty for each point, often half the range of repeated readings. Draw a line from the value minus the uncertainty to the value plus the uncertainty, with short caps at the ends. Add a horizontal bar too if the x value has a real uncertainty.
What is the uncertainty in ln x or lg x?
For ln x it is Δx ÷ x. For lg x it is 0.434 × Δx ÷ x. Working out ln (x + Δx) − ln x gives almost the same answer. AQA says uncertainties in logs and trig functions are not tested.
How many decimal places should a log have?
For Cambridge, as many decimal places as the reading has significant figures, or one more. So lg 76.5 is 1.884 or 1.8837. Put the unit inside the log in the heading: lg (L / cm).
How do you find the uncertainty in the y-intercept?
Find the intercept of the best-fit line and of the worst line. If the x-axis does not start at zero, calculate each one with c = y − mx from a point on that line. The uncertainty is the difference between the two intercepts.
How do I find the percentage uncertainty in g from the gradient?
If g comes from the gradient alone, as g = 2 ÷ gradient or 4π² ÷ gradient, it has the same percentage uncertainty as the gradient. If another measurement is involved, add its percentage uncertainty, doubled when it is squared, like a wire diameter in A = πd²/4.
Can I give my class the same data?
Yes. Share with class copies a link with the set code, and everyone who opens it gets the same readings. To stop copying, print a class set: each copy has the same experiments in the same order with different numbers, and its own key.
Sources, read October 27, 2026: Cambridge International AS & A Level Physics 9702 syllabus for 2025 to 2027 (Paper 3 and Paper 5 practical skills: error bars, worst acceptable line, gradient and intercept uncertainty, logs); AQA A-level Physics practical handbook (uncertainties from gradients, error bars, significant figures); OCR Physics A and B practical skills handbook and OCR H556/03 mark schemes for June 2019 and June 2022 (worst line through all error bars, |worst − best|). g = 9.81 m s⁻²; constantan 4.9 × 10⁻⁷ Ω m.