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Comparing your result quantitatively with the theory

What makes a comparison quantitative

The conclusion is the criterion that separates Achieved from Merit in this standard, and the difference is entirely in whether the comparison with theory is numerical.

Achieved: state the equation or quantity from the graph, and compare it with the theory.
Merit: make the comparison quantitative, including consideration of uncertainties.

Four ways to compare with a theory

Choose whichever your investigation supports — you do not need all four.

Comparison	Use when
Type of relationship — the shape found versus the shape predicted	always; this is the minimum
Derived quantity versus accepted value — e.g. your $g$ against $9.81$ m s⁻²	the theory contains a known constant
Experimental gradient versus theoretical gradient	the theory predicts a specific gradient value
Experimental intercept versus expected intercept	the theory predicts a specific intercept (often zero)

The overlap test

This is the comparison Merit is asking for, and it takes one sentence.

Write your experimental range: value $\pm$ uncertainty, expanded into a minimum and a maximum.
Ask whether the theoretical value lies inside that range.
State the answer explicitly.

Inside the range → the data is consistent with the theory.
Outside the range → the data does not support the theory as measured, and you must then discuss why — either the theory does not apply here, or there is a systematic error your uncertainty does not cover.

Percentage difference

A useful supporting number, but never a substitute for the overlap test:

$\%\text{ difference} = \frac{|\text{experimental} - \text{theoretical}|}{\text{theoretical}} \times 100$

Compare this with your percentage uncertainty. If the difference is smaller than the uncertainty, the discrepancy is fully explained by experimental uncertainty.
If the difference is much larger than the uncertainty, that is a real disagreement — say so.

Wording that works

"The gradient obtained was $4.03 \pm 0.17$ s² m⁻¹, giving $g = 9.8 \pm 0.4$ m s⁻². The accepted value of $9.81$ m s⁻² lies within this range, and the percentage difference of $0.1\%$ is far smaller than my $4.2\%$ uncertainty, so my data supports the theory $T = 2\pi\sqrt{L/g}$ over the range of lengths tested ( $0.20$ m to $1.00$ m)."

That sentence contains the quantity, its uncertainty, the theoretical value, the overlap test, the percentage comparison, the verdict, and the range. That is a full Merit conclusion.

Worked ExampleDeciding whether the data supports the theory

A student tests $F = \dfrac{kq_1q_2}{r^2}$ by measuring force against separation. Linearising by plotting $F$ against $1/r^2$ gives a gradient of $2.9 \times 10^{-6} \pm 0.4 \times 10^{-6}$ N m². The theoretical gradient, calculated from the charges used, is $3.4 \times 10^{-6}$ N m². Does the data support the theory?

Step 1 — Write out the experimental range

$2.9 \times 10^{-6} \pm 0.4 \times 10^{-6} \Rightarrow \text{from } 2.5 \times 10^{-6} \text{ to } 3.3 \times 10^{-6}\text{ N m}^2$

Step 2 — Apply the overlap test

The theoretical value $3.4 \times 10^{-6}$ N m² lies just outside the top of that range. The data does not quite support the theoretical gradient.

Step 3 — Quantify how far outside

$\%\text{ difference} = \frac{|2.9 - 3.4|}{3.4}\times 100 = 15\%$

$\%\text{ uncertainty} = \frac{0.4}{2.9}\times 100 = 14\%$

The difference ( $15\%$ ) is slightly larger than the uncertainty ( $14\%$ ) — a marginal disagreement, not a dramatic one.

Step 4 — Interpret honestly

The measured gradient is consistently low, and the shape of the relationship — a straight line against $1/r^2$ — is confirmed. So the inverse-square form of the theory is supported, while the magnitude is not quite. A systematic effect that reduces the measured force would explain this: charge leaking away from the spheres during the experiment (likely in humid air) would make every force reading too small, and a separation measured between the sphere surfaces rather than their centres would make every $r$ too small and every $1/r^2$ too large, both of which lower the gradient.

Step 5 — State the conclusion