You’ve learned to read a gradient off a line — but how sure can you be of it? Every point carries a margin of doubt, and those margins feed straight through to the gradient and intercept you extract. Error bars make the doubt visible, and by drawing a “best” and a “worst” line through them you can put a proper ± on your final result. This is the skill that turns a graph from a rough sketch into evidence — and it’s exam and IA gold.
📘 What you need to know
An error bar shows the absolute uncertainty of a point, drawn above/below (and sometimes left/right) of it
Shorter bars mean more precise measurements; longer bars mean less precise
A valid line of best fit should pass through all the error bars
Draw a best-fit line (closest to the points) and a worst-fit line (steepest or shallowest that still fits all the bars)
The percentage uncertainty in the gradient comes from comparing the best and worst gradients
The same method gives the uncertainty in the y-intercept
A relationship is consistent with the data if a line can pass through every error bar
What error bars show
An error bar is a small line drawn through a data point that marks the range the true value could lie in — the point’s absolute uncertainty. It can run vertically (uncertainty in y), horizontally (uncertainty in x), or both, forming a little cross. The length of the bar is the whole story: a short bar means a precise measurement, a long bar a less precise one.
Error bars mark each point’s uncertainty. Shorter bars mean more precise measurements.
A common misconception is that all the error bars on a graph must be the same length. They don’t! Each point has its own uncertainty, so a graph can carry a mix of short and long bars — that’s not a mistake, it’s honest data. Some measurements simply come out more precise than others.
Best and worst-fit lines
To turn error bars into an uncertainty on your gradient, you draw two lines. The best-fit line passes as close as possible to all the points. The worst-fit line is the steepest — or shallowest — straight line that still manages to thread through every error bar. The gap between their gradients is your uncertainty.
The best-fit line runs closest to the points; the worst-fit line is the steepest that still passes through every error bar.
Uncertainty in the gradient
Once you have both gradients, the percentage uncertainty is the fractional difference between them. It compares how far the worst gradient strays from the best.
Percentage uncertainty in gradient
% uncertainty = |best gradient − worst gradient| ÷ best gradient × 100
🧭 Finding the gradient uncertainty
Plot the points and draw a vertical error bar on each
Draw the best-fit line as close as possible to all the points
Draw the worst-fit line — steepest or shallowest — that still passes through every bar
Calculate both gradients, then apply the percentage-uncertainty formula
Best gradient
&
Worst gradient
→ compare →
% uncertainty
Uncertainty in the intercept
The intercept works exactly the same way. Read the y-intercept of both the best and worst lines, and the fractional difference gives the percentage uncertainty in the intercept.
Percentage uncertainty in intercept
% uncertainty = |best intercept − worst intercept| ÷ best intercept × 100
Quick recap: error bars show each point’s uncertainty; a good line passes through all of them; draw a best and a worst (steepest/shallowest) line, then compare their gradients (and intercepts) to get the percentage uncertainty.
WE 1
On a force–extension graph, the best-fit line passes through (0, 6.0 mm) and (80 N, 26.0 mm). The steepest worst-fit line that still fits all the error bars passes through (0, 5.0 mm) and (80 N, 27.0 mm). Find the percentage uncertainty in the gradient.
Work out each gradient with Δy ÷ Δx, then compare them.
Best gradient
(26.0 − 6.0) ÷ (80 − 0)
= 20.0 ÷ 80 = 0.25 mm N⁻¹Worst gradient
(27.0 − 5.0) ÷ (80 − 0)
= 22.0 ÷ 80 = 0.275 mm N⁻¹Percentage uncertainty
|0.25 − 0.275| ÷ 0.25 × 100
= 10%So the gradient is quoted as 0.25 ± 10%, i.e. 0.25 ± 0.03 mm N⁻¹.
WE 2
For the same graph, the best-fit line has a y-intercept of 6.0 mm and the worst-fit line has a y-intercept of 5.0 mm. Find the percentage uncertainty in the intercept.
Apply the same fractional-difference idea to the two intercept values.
Compare intercepts
best = 6.0 mm, worst = 5.0 mm
Percentage uncertainty
|6.0 − 5.0| ÷ 6.0 × 100
= 1.0 ÷ 6.0 × 100≈ 17%The intercept is less precisely defined than the gradient here — its bigger percentage uncertainty reflects that.
💡 Top tips
The worst line must still fit every bar — it’s the most extreme line that remains consistent, not just any steep line
Read gradient points off the lines, using convenient values on the axes for easy arithmetic
A relationship is supported if a straight (or smooth) line can pass through all the error bars
Quote the final uncertainty to 1 significant figure, matching the value’s precision to it
⚠ Common mistakes
Expecting all error bars to be the same length — each point has its own uncertainty
Drawing a worst-fit line that misses some bars — it must pass through every one
Dividing by the worst gradient instead of the best in the percentage formula
Forgetting that a line passing through all the bars means the data supports the proposed relationship
That wraps up Tool 3: Mathematics! You’ve built the full toolkit — algebra, vectors, units, uncertainties and graphing — that underpins every calculation and every experiment across the whole IB Physics course. Keep these skills sharp and the physics itself gets a great deal easier.
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