What the investigation has to deliver

Everything the report has to contain

The requirements build in three layers, and each layer assumes the one below it. Reading all three before you start is the single most efficient thing you can do.

Layer	What it contains
The investigation itself	Collect data over a reasonable range and number of values; determine uncertainties in the raw data; do a graphical analysis including a consideration of uncertainties; conclude with the equation or quantity from the graph, compared with the theory
Rigour on top of it	Describe how you controlled other significant variables; use techniques that improve accuracy; determine the uncertainty in one of the graphed variables; express the uncertainty in the relationship on the graph; and make a quantitative comparison with the theory, including uncertainties
Critical thinking on top of that	Discuss issues critical to the investigation — about two substantial points

How much data "a reasonable number of values" means

• Use at least five different values of the independent variable. Five is the floor, not the target — below five, no line through the points is convincing enough to identify the relationship, let alone test it.
• Spread those values over as wide a range as the apparatus allows. A non-linear relationship looks straight over a short interval, so a narrow range hides the very thing you are testing.
• Repeat each measurement and keep the repeats visible in the raw data table. The scatter between them is where your uncertainty comes from, so hiding it behind a single averaged column throws away your own evidence.

Where Level 3 differs from Level 2

At Level 2 you find whatever relationship the data shows. At Level 3 you are given a theory and must test it. Three consequences follow.

• Uncertainty is compulsory, at every stage. Even the most basic version of this work needs uncertainties in the raw data and a graphical analysis that considers them. There is no route through it that avoids uncertainty.
• The conclusion must compare with theory. Not "the relationship is $T \propto \sqrt{L}$", but "the measured gradient was $4.03 \pm 0.15$ s² m⁻¹ against a predicted $4.03$ s² m⁻¹, so the data supports the theory."
• That comparison has to be quantitative. Saying your result is "close to" the theory is not enough — you must say whether the theoretical value lies inside your uncertainty range, and quote both numbers.

The shape of a full report

1. Aim and theory — state the theory you were given and the relationship it predicts, including the predicted power or gradient.
2. Method — how you changed the independent variable, what you measured, and how you measured it. Include the techniques you used to improve accuracy.
3. Controlled variables — which ones matter, how you held them constant, and how they would have affected the result if they had not been.
4. Raw data table — every reading, with repeats, with the uncertainty in each measured quantity stated.
5. Processed data table — the linearised quantities, with their propagated uncertainties.
6. Graph — linearised axes, error bars, line of best fit, and the steepest and shallowest acceptable lines.
7. Analysis — gradient with its uncertainty, and the physics quantity derived from it with its uncertainty.
8. Conclusion — the relationship or quantity found, compared quantitatively with the theoretical value.
9. Discussion — two substantial critical issues.

The theory is given to you

• The two variables have a non-linear relationship according to a physics theory that is given to you. You are not expected to derive it.
• Your job is to convert that theory into a testable prediction: a power to compare with a gradient, or a constant to compare with a value.