Plan an investigation that can answer its question

Choose an instrument for the size and speed of the signal

A practical plan starts with the quantity to determine and the physical relationship linking it to measurable variables. State the apparatus in enough detail to show that it can resolve the expected signal: a range, resolution, relevant length, sampling interval or input resistance may matter. A named instrument alone does not demonstrate a workable method.

A digital meter with finer resolution is not automatically more accurate. Check its zero, reference calibration and appropriate range; a temperature probe can also lag behind the bath it measures. A high-resolution data logger sampled too slowly may miss a short pulse. Choose the complete measurement chain for the expected spatial and time scales.

Describe where measurements begin and end. A pendulum length runs from pivot to the bob’s centre of mass, while a gas volume may include connecting tubing. Explain how alignment, thermal contact or a timing fiducial will be established. Such details make a plan reproducible and reduce identifiable biases; generic promises to be careful do not.

Resolution uncertainty for one reading, under the specification convention: δx = ½ instrument resolution

Symbols and units

• δx is absolute uncertainty in the same unit as the instrument reading
• resolution is the size of one resolvable scale step.

How the equation explains the idea

1. Under the stated half-resolution convention, a 0.02 mm step gives ±0.01 mm for one reading. This addresses reading resolution rather than calibration or response delay.
2. If a displacement is the difference of two independent readings, both contribute. State whether a worst-case sum or a statistical combination is appropriate instead of assigning a single-reading uncertainty without justification.

Worked example

Compare two timing choices

1. An oscillation takes about 0.8 s and manual start–stop uncertainty is estimated as 0.2 s.
2. Timing one period gives about 25% timing uncertainty.
3. Timing 20 periods takes 16 s and gives about 1.25% before counting or drift effects.

A longer timed interval is useful if the oscillation remains stable and cycles are counted correctly.

Keep this idea. Match the instrument’s range, resolution and response to the signal you expect.

Decide what varies, what stays fixed and what is repeated

Identify the independent variable you deliberately vary and the dependent variable you measure. Other variables must be controlled because the proposed model assumes them fixed, not because every practical answer needs a memorised list. For pressure versus volume, the amount of gas and absolute temperature must stay fixed; for capacitor decay, resistance and capacitance should remain stable.

Choose a range wide enough to distinguish competing relationships while remaining safe and within the model’s limits. Several well-spaced values reveal curvature more effectively than many almost-identical readings. Repeat readings where random variation matters and repeat a reference condition after the series to detect drift. Repetition cannot remove a shared calibration offset or a temperature change correlated with the independent variable.

A plan should state how the results will answer the question: the axes, expected relation, gradient or intercept and how a constant follows. Identify a concrete hazard and a practical control, such as limiting pressure within rated apparatus or shielding a swept mechanical path. Online datasets support reasoning; actual practical skills require suitable supervised equipment and local risk assessment.

Experimental claim = stated model + controlled conditions + measured relation + uncertainty comparison

Worked example

A pressure–volume series

1. Use six safely spaced volume settings, retaining the same sealed gas sample.
2. After each adjustment wait for stable temperature and pressure, then record both.
3. Repeat selected volumes and return to the first volume to check for leaks or thermal drift.

A drift check answers a different question from repeated readings at one setting.

Keep this idea. Use a range to test the relationship, repeats for scatter and reference returns for drift.

Improve the cause of a discrepancy

A set of closely clustered results is precise but can still be inaccurate if a shared offset shifts it from the accepted value. Sensitivity is different again: a sensor with a large voltage change per kelvin is sensitive, but noise or poor calibration can still limit the temperature result. Use each term to describe a distinct property rather than as interchangeable praise.

Audit raw tables before fitting a line. Column headings should name the quantity and unit, raw readings should reflect the instrument’s resolution, and derived values should not imply unsupported precision. Check prefix conversions, especially squared dimensions. The required prefix set is G 10⁹, M 10⁶, k 10³, c 10⁻², m 10⁻³, μ 10⁻⁶ and n 10⁻⁹; case changes meaning.

An anomalous reading is a reason to investigate, not a licence to delete inconvenient evidence. Check transcription, repeat the condition and inspect a plausible physical cause. Explain an improvement through the error it reduces: a marker reduces inconsistent timing positions, thermal contact reduces probe lag, and a zero check addresses offset. Report the original and corrected record transparently.

Sensitivity = Δoutput/Δinput; percentage uncertainty = 100δx/|x|

Symbols and units

• Sensitivity is output change per input change with units output-unit/input-unit
• δx is positive absolute uncertainty in x units.

How the equation explains the idea

1. A larger response per input change makes an input difference easier to resolve for a given output-reading uncertainty. This can improve resolution without correcting an offset.
2. For a fixed δx, percentage uncertainty 100δx/|x| grows as the measurement approaches zero. Repeats reduce uncertainty in a mean for random noise but do not remove a persistent calibration bias.

Worked example

A biased temperature probe

1. A probe reads 25.4, 25.4 and 25.5 °C in a reference bath at 25.0 °C.
2. The small range suggests good repeatability, but the offset suggests a calibration problem.
3. Averaging gives 25.43 °C; it does not remove the approximately +0.43 °C bias.

Calibration targets the systematic effect; repeats target scatter.

Keep this idea. Tie each proposed improvement to an identified source of uncertainty or bias.