The big idea: Real measurements always scatter a little. To get a clear answer you plot the points and draw one straight line of best fit through them.
You then read the physics off the line, not off any single point:
- the gradient (slope) of the line → a physics quantity - the intercept (where it crosses the y-axis) → another physics quantity
[Diagram: phys-best-fit] - Available in full study mode
Words to know: Best-fit line: the single straight line that passes as close as possible to all the points, with roughly as many points above it as below.
Error bar: the short vertical line through a point showing its uncertainty — the point could really lie anywhere along it.
Gradient: how steep the line is = rise ÷ run (the change in y divided by the change in x).
To read the gradient, pick two points on the line (not data points) that are far apart, then divide the rise by the run.
- gradient (slope) of the best-fit line
- rise — change in the y-value between two points on the line
- run — change in the x-value over the same interval
- gradients of the steepest / shallowest lines through the error bars
To get the uncertainty in the gradient, draw the steepest and the shallowest straight lines that still pass through all the error bars. The spread of their gradients gives the uncertainty:
[Diagram: phys-best-fit] - Available in full study mode
If the gradient is your final quantity, propagate the uncertainty: Often the quantity you want is the gradient itself (or a simple multiple of it). If you then divide or take a power of it, the data booklet gives the rules for combining uncertainties — quote the fractional uncertainty and carry it through.
- the final calculated quantity
- the measured quantities it is built from
- absolute uncertainty in y (same unit as y)
- fractional uncertainty in y (no unit)
- the power a quantity is raised to
Worked example — gradient and its uncertainty from a graph
On the extension-vs-load graph above, the best-fit line passes through (0, 0) and (5.0 N, 10.0 cm). The steepest line through the error bars has gradient 2.1 cm N⁻¹ and the shallowest has 1.9 cm N⁻¹. Find the gradient and its absolute uncertainty.
Solution
- Read the gradient as rise ÷ run, using the two points on the best-fit line. Formula first:
- Put in the numbers read off the line (rise = 10.0 cm, run = 5.0 N):
- Now the uncertainty — half the spread of the steepest and shallowest gradients. Formula first:
- Substitute the two gradients (2.1 and 1.9):
Final answer
gradient = (2.0 ± 0.1) cm N⁻¹ — the line gives 2.0 cm of extension per newton, known to about ±0.1.
Practice with real exam questions
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How this is tested: Paper 1B almost always builds a graph step by step: plot a point, draw the best-fit line, read the gradient, then turn it into a physics quantity with its uncertainty.
- Plot / draw: add a processed point or draw the best-fit (and the max-gradient) line. - Determine: read the gradient as rise ÷ run, then identify it as a quantity (e.g. a spring constant, a refractive index, a speed). - Uncertainty: use the steepest/shallowest lines to quote the gradient — and hence the quantity — as value ± uncertainty.
Classic trap: reading the gradient off data points instead of off the line, or forgetting that the gradient (not a single point) is what carries the answer.
IB-style question — a spring constant from the gradient
A student stretches a spring and plots extension e (in metres) against load F (in newtons). The best-fit line is straight through the origin with gradient (0.020 ± 0.001) m N⁻¹. The spring constant k is defined by F = ke, so k = 1 ÷ gradient. Determine k and its absolute uncertainty.
Solution
- The gradient is e ÷ F, so k is its reciprocal. Write the relationship first:
- Put in the gradient (m = 0.020 m N⁻¹):
- A reciprocal is a power (n = −1), so the fractional uncertainty is unchanged in size. Formula first:
- Work out the fractional uncertainty (Δm = 0.001, m = 0.020):
- Turn it into an absolute uncertainty: Δk = 0.05 × k:
Final answer
k = (50 ± 3) N m⁻¹ — read off the gradient, with the uncertainty rounded to one significant figure.