A gradient pulled off a graph is only as trustworthy as the data behind it. Every plotted point carries an uncertainty, shown as an error bar — and those bars let you work out how uncertain your gradient and intercept really are. This is the skill that ties the whole Tool 3 toolkit together, and it’s a Paper 1B favourite.
📚 What you need to know
An error bar shows the absolute uncertainty of a plotted point
Shorter bars = more precise; longer bars = less precise
A relationship is linear if a straight line fits through all the error bars
To find the uncertainty in a gradient, draw a best and a worst line of fit
The worst line is the steepest or shallowest that still fits the error bars
An error bar is a small line drawn through a data point, extending by the point’s uncertainty above and below (or side to side). It’s a visual statement of “the true value lies somewhere in this range.”
A short error bar means a precise measurement.
A long error bar means a less precise measurement.
Bars don’t all have to be the same size — each point can have its own uncertainty.
A common misconception is that every error bar on a graph must be identical. Not so! Each measurement can be more or less precise than the others, so its bar can be longer or shorter. If a question gives you different uncertainties for different points, draw them at different lengths — that’s the honest thing to do.
Is the relationship linear?
A favourite Paper 1B question asks whether data supports a proposed relationship. The rule of thumb:
The relationship is linear if you can draw a single straight line that passes through every error bar.
It’s non-linear if a smooth curve (but not a straight line) is needed to pass through all the bars.
Finding the uncertainty in a gradient
To turn error bars into an uncertainty on your gradient, you draw two lines:
The best line of fit — passing as close to the points as possible.
The worst line of fit — the steepest or shallowest line that still passes through all the error bars.
Both the best (red) and worst (dashed) lines pass through every error bar; the difference in their gradients gives the uncertainty.
Once you have both gradients, the percentage uncertainty in the gradient is the difference between them, relative to the best gradient:
Percentage uncertainty in a gradient% unc = (|best − worst gradient| ÷ best gradient) × 100
WE 1
From an extension–force graph, the best line of fit runs from (0, 6) to (80, 26) mm, and the worst line from (0, 4.7) to (80, 27) mm. Find the percentage uncertainty in the gradient.
Step 1 — best gradient
(26 − 6) / (80 − 0) = 0.25 mm N−1Step 2 — worst gradient
(27 − 4.7) / (80 − 0) = 0.28 mm N−1Step 3 — percentage uncertainty
(|0.25 − 0.28| / 0.25) × 100 = 12%gradient uncertainty ≈ 12%The worst line here is the steepest one that still fits the bars. Its gradient differs most from the best line, so it sets the uncertainty.
Uncertainty in the intercept
The same idea works for the y-intercept. Read where the best and worst lines cross the y-axis, and the spread between them gives the uncertainty in the intercept — either as a percentage, or as half the difference between the maximum and minimum intercepts for an absolute value.
💡 Top tips
Error bars can be different lengths — draw each to its own uncertainty.
Both best and worst lines must pass through all the error bars.
The worst line is the steepest OR shallowest that still fits.
Linear if a straight line fits every bar; non-linear if a curve is needed.
Apply the same best/worst method to the intercept.
⚠ Common mistakes
Drawing all error bars the same length when uncertainties differ
A worst line that misses some error bars
Forgetting to divide by the best gradient in the % formula
Confusing the steepest and shallowest as always the worst
Reading gradients off data points instead of the drawn lines
Quick recap:Error bars show each point’s uncertainty (and can differ in length). A relationship is linear if a straight line fits every bar. For the gradient’s uncertainty, draw a best and a worst line (steepest or shallowest that still fits) and use % unc = |best − worst| ÷ best × 100. The same method gives the intercept’s uncertainty.
That completes the Tool 3 maths toolkit — from rearranging equations and handling vectors right through to squeezing an honest uncertainty out of a graph. These skills aren’t a topic you revise once; they’re the machinery behind every calculation, every practical, and your internal assessment. Keep them sharp and the physics itself becomes the easy part.
Error bars and best-fit lines a mystery?
Book a free meeting and we’ll practise drawing worst-fit lines and turning error bars into gradient uncertainties — exactly the technique Paper 1B and the IA reward most.
