Why do we try to get straight-line graphs?

A straight line is easy to fit and check, and its gradient and intercept can be read accurately and linked directly to physical quantities.

When should I use a log–log graph?

When testing a suspected power law y = kx^n. The gradient gives the power n.

How do I find the uncertainty in a gradient?

Draw the steepest or shallowest line that still passes through all error bars; the difference from the best-fit gradient is the uncertainty.

What does a non-zero intercept tell me?

If theory predicts a line through the origin, a non-zero intercept suggests a systematic error in one of the variables.

Back to course

1.1Working as a PhysicistCore

Graphs and Analysis

A table of readings shows what you measured. A graph shows what it means. Nearly every practical in the course ends with a straight-line graph, because a straight line turns a messy set of data into two numbers — a $gradient$ and an $intercept$ — that each have a physical meaning.

35 min Topic 1 · Working as a Physicist

What you'll be able to do

Plot graphs with appropriate scales, labels and lines of best fit
Rearrange equations into the form $y = m x + c$ to identify gradient and intercept
Use logarithmic graphs to test for power law and exponential relationships
Use error bars and worst-fit lines to find the uncertainty in a gradient

Drawing a good graph

Label each axis with the quantity and unit, in the form "length / m". The independent variable, the one you control, goes on the $x$ -axis.

Choose scales so that the points fill at least half the grid in each direction, using sensible increments such as 1, 2 or 5 per square. Do not force the origin onto the graph unless it is needed.

Plot points precisely as small crosses, then draw a single smooth $line of best fit$ with roughly equal numbers of points on either side. Do not join the dots. Identify and ignore anomalous points, but say that you have done so.

Tip — When measuring a gradient, use a triangle covering more than half the line, and read coordinates from the line itself, not from data points.

Linearising equations

Compare the equation with $y = m x + c$ . Whatever you plot on the $y$ -axis must match $y$ ; whatever multiplies the $x$ variable is the gradient; any remaining constant term is the intercept.

For a pendulum, $T = 2 π l / g$ is not linear in $l$ . Squaring gives $T^{2} = \frac{4 π ^{2}}{g} l$ , so plotting $T^{2}$ against $l$ gives a straight line through the origin with gradient $4 π^{2} / g$ .

For e.m.f. and internal resistance, $V = ε - r I$ is already linear: plotting $V$ against $I$ gives gradient $- r$ and $y$ -intercept $ε$ .

1Gradient

= 4 π^{2} / g

g = 4 π^{2} /4.05 = 39.48/4.05

Answer

9.75 m s^{- 2}

Logarithmic graphs

For a $power law$ $y = k x^{n}$ , taking logs gives $lo g y = n lo g x + lo g k$ . A graph of $lo g y$ against $lo g x$ is a straight line with gradient $n$ and intercept $lo g k$ .

For an $exponential$ $y = y_{0} e^{- λ t}$ , taking natural logs gives $ln y = - λ t + ln y_{0}$ . A graph of $ln y$ against $t$ is a straight line with gradient $- λ$ .

Choosing which to use is itself a test: if $ln y$ against $t$ is straight, the relationship is exponential; if $lo g y$ against $lo g x$ is straight, it is a power law.

lo g y = n lo g x + lo g k

Power law

y = k x^{n}

ln y = ln y_{0} - λ t

Exponential decay.

1Gradient

n = (2.30 - 1.10) / (0.80 - 0.20) = 2.0

2Intercept:

lo g k = 1.10 - 2.0 \times 0.20 = 0.70

k = 1 0^{0.70}

Answer

n = 2.0

k = 5.0

Tip — Logarithms have no units, but label the axis with the quantity inside, for example "ln (A / Bq)".

Error bars and gradient uncertainty

$Error bars$ extend above and below (or either side of) each point by its absolute uncertainty. A line of best fit should pass through all or most of them.

To find the uncertainty in the gradient, draw the $worst acceptable line$ — the steepest or shallowest straight line that still passes through all the error bars.

The uncertainty in the gradient is the difference between the best-fit and worst-fit gradients. Its percentage uncertainty then feeds into the final result.

If a line expected to pass through the origin does not, even allowing for error bars, that indicates a systematic error — such as measuring pendulum length to the top of the bob instead of its centre.

1Gradient uncertainty

= 0.20

, i.e.

0.20/4.05 = 4.9%

g = 9.75 m s^{- 2}

carries the same percentage uncertainty.

0.049 \times 9.75 = 0.48

Answer

g = 9.7 \pm 0.5 m s^{- 2}

Equation recap

y = m x + c

Straight line.

lo g y = n lo g x + lo g k

Power law.

ln y = ln y_{0} - λ t

Exponential.

Δ m = ∣ m_{best} - m_{worst} ∣

Gradient uncertainty.

Common mistakes to avoid

Joining data points dot to dot.

Draw one smooth line of best fit.

Plotting

T

against

l

and expecting a straight line.

Plot

T^{2}

against

l

to linearise.

Using a small triangle for the gradient.

Use a triangle spanning at least half the line.

Assuming the line must pass through the origin.

Only force it through if the physics requires it and the data support it.

Key takeaways

Label axes with quantity and unit; fill the grid; draw a line of best fit.
Rearrange equations to $y = m x + c$ and match gradient and intercept.
Log–log graphs test power laws; $ln y$ – $t$ graphs test exponentials.
Error bars show each point’s uncertainty.
Gradient uncertainty is the difference between best and worst fit gradients.

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Real past-paper questions on Graphs and Analysis, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Graphs and Analysis

Drawing a good graph

Linearising equations

Logarithmic graphs

Error bars and gradient uncertainty

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Graphs and Analysis

Drawing a good graph

Linearising equations

Logarithmic graphs

Error bars and gradient uncertainty

Practise this topic

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