The Scientific Method in Practice

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Scientific method is a structured sequence: observation → hypothesis → experiment → analysis → conclusion. A hypothesis is a testable, falsifiable statement; an experiment must manipulate one independent variable, measure one dependent variable, and hold all controlled variables constant. All measurements use the SI system, built on seven base units: metre (m) for length, kilogram (kg) for mass, second (s) for time, kelvin (K) for temperature, ampere (A) for current, mole (mol) for amount, and candela (cd) for luminous intensity. Prefixes scale these units: kilo (k) = 10³, centi (c) = 10⁻², milli (m) = 10⁻³, micro (µ) = 10⁻⁶. Three core formulas:

density = mass ÷ volume (ρ = m/V, units kg/m³)
speed = distance ÷ time (v = d/t, units m/s)
percentage error = |experimental − accepted| ÷ accepted × 100%

Accuracy is closeness to the true value; precision is reproducibility of repeated measurements. Always quote a result to the same number of significant figures as the least precise measurement.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

The Scientific Method in Practice

The scientific method is not a single rigid procedure but a cycle: a researcher makes an observation, formulates a hypothesis (a specific, testable prediction), designs and runs a controlled experiment, analyses the data (often with mean, range, and a graphical plot), and draws a conclusion that either supports or refutes the hypothesis. A single failed prediction is enough to falsify a hypothesis — it does not need to be “proved,” only to survive testing.

Identifying Variables

Every experiment has three categories of variables:

Independent variable: the factor deliberately changed by the experimenter (plotted on the x-axis).
Dependent variable: the factor measured as the response (plotted on the y-axis).
Controlled variables: every other factor held constant so the change in the dependent variable can be attributed to the independent one.

A valid conclusion must state how the dependent variable changed with the independent variable, not merely that “something happened.”

SI Units and Prefixes

The Système International rests on seven base units, each with a defined standard (e.g. the kilogram is defined by a Planck constant, the metre by the speed of light). Derived units — such as the newton (kg·m/s²) for force and the joule (kg·m²/s²) for energy — are combinations of base units. SI prefixes let you scale a unit by powers of ten without redefining it: 1.50 km = 1500 m, 250 µm = 2.50 × 10⁻⁴ m. Always convert to matching units before substituting into a formula.

Accuracy, Precision, and Error

Accuracy quantifies how close a measurement is to the accepted (true) value and is reported via percentage error. Precision describes the spread of repeated measurements — a set of readings clustering tightly together is precise, regardless of where the cluster sits relative to the true value. A common error pattern: precise measurements (small spread) can still be highly inaccurate if the instrument is poorly calibrated (a systematic error, not a random one).

Significant Figures

Rules to count them:

All non-zero digits are significant.
Zeros between non-zero digits are significant.
Trailing zeros after a decimal point are significant.
Leading zeros (e.g. 0.0042) are never significant.
Trailing zeros with no decimal point are ambiguous (write 4.20 × 10³, not 4200, to show two sig figs).

When multiplying or dividing, the answer keeps the fewest sig figs of any value used; when adding or subtracting, the answer keeps the decimal places of the least precise value.

Typical CAT Question Patterns

Expect items asking you to convert units between SI prefixes (e.g. mm → m, mg → kg), calculate percentage error from a given experimental and accepted value, identify which variable is independent/dependent/controlled in a described experiment, or state the correct number of sig figs for a calculator reading. Short numerical answers dominate; experimental-design multiple choice usually offers three or four plausible distractors.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Systematic vs Random Error

A deeper layer of measurement science: random errors cause readings to scatter around the true value and can be reduced by repeating the measurement and averaging. Systematic errors shift every reading in the same direction (a miscalibrated balance always reads 0.02 g high) and are reduced by recalibrating or using a different instrument. Increasing the sample size tackles random error only; it cannot remove a systematic bias. Percentage error captures the combined effect of both, which is why careful experimental design — not just more trials — is essential.

Worked Example: Density of an Irregular Solid

A student measures a small stone: mass m = 12.40 g (4 sig figs), volume by water displacement = (24.7 − 20.0) mL = 4.7 mL (2 sig figs). Density = m/V = 12.40 ÷ 4.7 = 2.6 g/cm³ (rounded to 2 sig figs, matching the volume). Compare to the accepted 2.65 g/cm³ for a sample of similar composition: percentage error = |2.6 − 2.65| / 2.65 × 100% = 1.9%. Notice how the imprecise volume dominates the final uncertainty.

Common Mistakes

Reporting all ten digits of a calculator display — the answer is only as reliable as the least precise input.
Treating “controlled variable” as something that can vary; it must be held constant across all trials.
Converting one quantity in a formula (say mass to kg) but forgetting to convert another (volume still in cm³ instead of m³). Dimensional analysis before substituting prevents this.
Quoting a mean value without its range or uncertainty, then comparing it to an accepted value without computing percentage error.

Connections to Other Topics

This topic underpins every quantitative science topic that follows — mechanics relies on consistent length and time units, thermal physics on the kelvin, chemistry on the mole. Understanding percentage error also recurs in Topic 6 (Experimental Techniques) and in any uncertainty-propagation questions in Topic 4. Treat the precision/accuracy distinction as the foundation for interpreting any graph or data table throughout the syllabus.

Practice Prompts

A length is reported as 0.003450 km. How many significant figures does this measurement carry, and what is its value in millimetres?
An experiment yields a mean value of 9.2 s for a quantity whose accepted value is 10.0 s. State the percentage error and comment on whether the experiment is accurate, precise, or neither, given that individual readings were 9.1, 9.3, and 9.2 s.

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The Scientific Method in Practice