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Error bars and the uncertainty in a gradient

Why the graph has to show the uncertainty

At Level 3 the graph must carry the uncertainty, not just the values. That is what the criteria mean by "graphical analysis including a consideration of uncertainties".

An error bar is a line drawn through a plotted point extending one uncertainty above and one below it, with a small cap at each end.
Draw error bars on the axis whose uncertainty is significant — usually the vertical axis. If the horizontal uncertainty is negligible in comparison (say, less than a fifth of the vertical one), say so and leave it off.
The uncertainties plotted must be the propagated ones for the linearised quantity, not the raw ones.
- If you plot $T^2$ and $T$ has a $2\%$ uncertainty, the error bar on $T^2$ is $4\%$ — the power rule.

Getting an uncertainty on the gradient

Draw the line of best fit through the points.
Draw the steepest line that still passes through every error bar.
Draw the shallowest line that still passes through every error bar.
Find the gradient of each extreme line, and take half the difference:

$\delta m = \frac{m_{\text{steepest}} - m_{\text{shallowest}}}{2}$

Quote the result as $m \pm \delta m$ , with the uncertainty to one significant figure.

Steepest gradient 1.94

Shallowest gradient 1.74

Both lines pass through every error bar, so the gradient is 1.84 ± 0.10. Widen them and the uncertainty grows for no reason — quote the tightest pair that still works.

Drag each line and watch which points turn red. A line only counts as an extreme if it passes through every error bar — one miss and it is not usable.
Push the lines wider than they need to be and the uncertainty grows for no reason. The correct answer is the tightest pair that still work, because a needlessly large uncertainty makes the comparison with theory meaningless.

Rules that markers look for

Both extreme lines must pass through all the error bars, not most of them.
Both extreme lines should be plausible fits — usually pivoted about the middle of the data rather than hinged at one end.
Gradients must be read from points on the lines, taken far apart.
The gradient uncertainty must be consistent with the scatter in the data: if the points are tightly grouped, a huge quoted uncertainty is as wrong as a tiny one.

Turning a gradient uncertainty into a quantity uncertainty

If a physics quantity is calculated from the gradient, the percentage uncertainty carries through by the ordinary rules.
- Gradient $m = 4.03 \pm 0.15$ s² m⁻¹ has $\%\delta m = \dfrac{0.15}{4.03}\times 100 = 3.7\%$ .
- For a pendulum, $g = \dfrac{4\pi^2}{m}$ , so $g$ inherits the same $3.7\%$ (dividing by a quantity passes its percentage uncertainty straight through; $4\pi^2$ is exact).
- $g = \dfrac{4\pi^2}{4.03} = 9.79$ m s⁻², and $3.7\%$ of $9.79$ is $0.36$ , so $g = 9.8 \pm 0.4$ m s⁻².

Worked ExampleFrom error bars to a value of g with its uncertainty

A pendulum investigation gives a graph of $T^2$ against $L$ with error bars. The line of best fit has gradient $4.03$ s² m⁻², the steepest acceptable line has gradient $4.21$ and the shallowest $3.87$ . Find the gradient with its uncertainty, and use $T = 2\pi\sqrt{L/g}$ to find $g$ with its uncertainty.

Step 1 — Find the uncertainty in the gradient

$\delta m = \frac{m_{\text{steepest}} - m_{\text{shallowest}}}{2} = \frac{4.21 - 3.87}{2} = 0.17$

So the gradient is $m = 4.03 \pm 0.17$ s² m⁻¹.

Step 2 — Express it as a percentage uncertainty

$\%\,\delta m = \frac{0.17}{4.03} \times 100 = 4.2\%$

Step 3 — Relate the gradient to $g$

Squaring the theory, $T^2 = \dfrac{4\pi^2}{g}L$ , so the gradient of $T^2$ against $L$ is $m = \dfrac{4\pi^2}{g}$ , giving

$g = \frac{4\pi^2}{m} = \frac{39.48}{4.03} = 9.80 \text{ m s}^{-2}$

Step 4 — Carry the percentage uncertainty into $g$

$4\pi^2$ is an exact number, so it contributes no uncertainty. Dividing by $m$ passes $m$ 's percentage uncertainty straight through:

$\%\,\delta g = 4.2\% \quad \Rightarrow \quad \delta g = 0.042 \times 9.80 = 0.41$

$\textbf{g = 9.8 ± 0.4 m s}^{\textbf{−2}}$

Step 5 — Compare with the accepted value

The accepted value $9.81$ m s⁻² lies comfortably within $9.4$ to $10.2$ m s⁻², so the data supports the theory.

Error bars and the uncertainty in a gradient

Covers AS91521

Why the graph has to show the uncertainty

At Level 3 the graph must carry the uncertainty, not just the values. That is what the criteria mean by "graphical analysis including a consideration of uncertainties".

An error bar is a line drawn through a plotted point extending one uncertainty above and one below it, with a small cap at each end.
Draw error bars on the axis whose uncertainty is significant — usually the vertical axis. If the horizontal uncertainty is negligible in comparison (say, less than a fifth of the vertical one), say so and leave it off.
The uncertainties plotted must be the propagated ones for the linearised quantity, not the raw ones.
- If you plot $T^2$ and $T$ has a $2\%$ uncertainty, the error bar on $T^2$ is $4\%$ — the power rule.

Getting an uncertainty on the gradient

Draw the line of best fit through the points.
Draw the steepest line that still passes through every error bar.
Draw the shallowest line that still passes through every error bar.
Find the gradient of each extreme line, and take half the difference:

$\delta m = \frac{m_{\text{steepest}} - m_{\text{shallowest}}}{2}$

Quote the result as $m \pm \delta m$ , with the uncertainty to one significant figure.

Steepest gradient 1.94

Shallowest gradient 1.74

Both lines pass through every error bar, so the gradient is 1.84 ± 0.10. Widen them and the uncertainty grows for no reason — quote the tightest pair that still works.

Drag each line and watch which points turn red. A line only counts as an extreme if it passes through every error bar — one miss and it is not usable.
Push the lines wider than they need to be and the uncertainty grows for no reason. The correct answer is the tightest pair that still work, because a needlessly large uncertainty makes the comparison with theory meaningless.

Rules that markers look for

Both extreme lines must pass through all the error bars, not most of them.
Both extreme lines should be plausible fits — usually pivoted about the middle of the data rather than hinged at one end.
Gradients must be read from points on the lines, taken far apart.
The gradient uncertainty must be consistent with the scatter in the data: if the points are tightly grouped, a huge quoted uncertainty is as wrong as a tiny one.

Turning a gradient uncertainty into a quantity uncertainty

If a physics quantity is calculated from the gradient, the percentage uncertainty carries through by the ordinary rules.
- Gradient $m = 4.03 \pm 0.15$ s² m⁻¹ has $\%\delta m = \dfrac{0.15}{4.03}\times 100 = 3.7\%$ .
- For a pendulum, $g = \dfrac{4\pi^2}{m}$ , so $g$ inherits the same $3.7\%$ (dividing by a quantity passes its percentage uncertainty straight through; $4\pi^2$ is exact).
- $g = \dfrac{4\pi^2}{4.03} = 9.79$ m s⁻², and $3.7\%$ of $9.79$ is $0.36$ , so $g = 9.8 \pm 0.4$ m s⁻².

Worked ExampleFrom error bars to a value of g with its uncertainty

Step 1 — Find the uncertainty in the gradient

$\delta m = \frac{m_{\text{steepest}} - m_{\text{shallowest}}}{2} = \frac{4.21 - 3.87}{2} = 0.17$

So the gradient is $m = 4.03 \pm 0.17$ s² m⁻¹.

Step 2 — Express it as a percentage uncertainty

$\%\,\delta m = \frac{0.17}{4.03} \times 100 = 4.2\%$

Step 3 — Relate the gradient to $g$

Squaring the theory, $T^2 = \dfrac{4\pi^2}{g}L$ , so the gradient of $T^2$ against $L$ is $m = \dfrac{4\pi^2}{g}$ , giving

$g = \frac{4\pi^2}{m} = \frac{39.48}{4.03} = 9.80 \text{ m s}^{-2}$

Step 4 — Carry the percentage uncertainty into $g$

$4\pi^2$ is an exact number, so it contributes no uncertainty. Dividing by $m$ passes $m$ 's percentage uncertainty straight through:

$\%\,\delta g = 4.2\% \quad \Rightarrow \quad \delta g = 0.042 \times 9.80 = 0.41$

$\textbf{g = 9.8 ± 0.4 m s}^{\textbf{−2}}$

Step 5 — Compare with the accepted value

The accepted value $9.81$ m s⁻² lies comfortably within $9.4$ to $10.2$ m s⁻², so the data supports the theory.

Error bars and the uncertainty in a gradient

Why the graph has to show the uncertainty

Getting an uncertainty on the gradient

Rules that markers look for

Turning a gradient uncertainty into a quantity uncertainty

Step 1 — Find the uncertainty in the gradient

Step 2 — Express it as a percentage uncertainty

Step 3 — Relate the gradient to ggg

Step 4 — Carry the percentage uncertainty into ggg

Step 5 — Compare with the accepted value

Error bars and the uncertainty in a gradient

Why the graph has to show the uncertainty

Getting an uncertainty on the gradient

Rules that markers look for

Turning a gradient uncertainty into a quantity uncertainty

Step 1 — Find the uncertainty in the gradient

Step 2 — Express it as a percentage uncertainty

Step 3 — Relate the gradient to ggg

Step 4 — Carry the percentage uncertainty into ggg

Step 5 — Compare with the accepted value

Step 3 — Relate the gradient to $g$

Step 4 — Carry the percentage uncertainty into $g$

Step 3 — Relate the gradient to $g$

Step 4 — Carry the percentage uncertainty into $g$