• Calculations: leave space in tables for means and derived quantities (like extension or area) that can't be measured directly.
• Graphs: pick sensible scales that fill at least half the grid, label axes with quantity and unit using a forward slash, plot points to within half a small square, and draw a smooth line/curve of best fit (not forced through the origin).
• Units: everything reduces to 7 SI base units (kg, m, s, A, K, mol, cd); other units like N, J, Pa are "derived" from these using their defining equations.
• Relationships: directly proportional (y ∝ x, straight line through origin) vs inversely proportional (y ∝ 1/x, decreasing curve) vs inverse square law (y ∝ 1/x²).
• Gradients: calculated from a large triangle drawn on the line of best fit — never from two raw data points, and never a tiny triangle.
• Reducing errors: watch for parallax error, missing fiducial markers, zero errors, poor-resolution equipment, and unwanted heating effects.
• Improvements: data loggers, cameras, and computer modelling reduce human error and improve reproducibility.
• Uncertainty: can be absolute, fractional, or percentage — and there are specific rules for combining uncertainties when adding, multiplying, or raising to a power.

Why raw data isn't enough

Think of an experiment in two stages: collecting the raw numbers, and then processing them into something useful. A table full of raw readings tells you almost nothing on its own — you need to turn it into means, calculated quantities, and eventually graphs. That's what this whole chapter is about.

The golden rule when designing a data table: always leave space for calculations. If you're taking three repeat current readings, leave a column for the average. If you need "extension," which can't be measured directly, leave columns for initial length, final length, and the calculated extension.

The Mean

• Any anomalous (clearly wrong/outlier) readings should be ignored before calculating the mean.
• The mean must be given to the same number of significant figures as the individual readings used to calculate it — don't suddenly gain precision you don't actually have!

Calculating quantities you can't measure directly

Sometimes the thing you actually need isn't something a ruler or ammeter can read straight off. Two classic examples:

• Area of a wire: you can't "measure" area directly with a micrometer — but you can measure radius r, then calculate area using A = πr².
• Extension in Hooke's Law: you measure the wire's initial length and final length with a ruler, then calculate extension as x = final length − initial length.

Three things examiners always check

When you plot a graph in an exam or in the lab, examiners are silently checking three things every single time: your scale choice, your axis labelling, and how carefully you plotted the points. Get sloppy on any one of these and you lose marks even if your physics understanding is perfect.

Choosing a scale

• The scale must be big enough that your data uses at least half the grid in both the x and y directions — don't squeeze all your points into a tiny corner.
• Use scales that are easy to read and plot — multiples of 1, 2, 5, or 10. Avoid multiples of 3 (try plotting 3.7 on a scale that goes up in 3s — it's a nightmare).
• Scales should increase outward and upward from the origin.

Labelling the axes

Plotting the points & drawing the line

• Points must be accurate to within half a small square — this is a real, checked tolerance, not just "roughly there."
• Use a sharp pencil so points are thin and clear, and so they don't get lost under your line of best fit.
• You need at least six points plotted, with any major outliers clearly identified (but still shown).
• The line (or curve) of best fit should have roughly equal numbers of points above and below it — a clear plastic ruler helps you judge this by eye.
• Don't force the line through the origin unless the data genuinely supports it. Physics doesn't care about your assumptions — let the data speak.
• The line should be smooth — never a jagged dot-to-dot "frequency polygon" style connection.

Everything traces back to 7 base units

Here's a genuinely satisfying fact about physics: no matter how complicated a unit looks (Pascals, Joules, Newtons, Ohms...), it can always be broken down into combinations of just seven SI base units. Every other unit in physics is "derived" — built mathematically from these seven.

Deriving units from definitions

To find the base units of a derived quantity, start from the equation that defines it:

Useful unit conversions

The three shapes you must recognise instantly

Graphs are how physicists see the relationship between two variables. There are three shapes you need to recognise on sight, because exam questions love asking "what does this graph tell you about the relationship?"

Turning "∝" into a real equation

A proportionality symbol (∝) can be turned into an actual equation by replacing it with "=" and adding a constant k:

• Direct: y ∝ x becomes y = kx
• Inverse: y ∝ 1/x becomes y = k/x

This constant k is really useful — if you calculate k for every row of your data table and it comes out roughly the same each time, that's solid proof the relationship really does hold.

Getting the gradient right (and getting marks for it)

Straight-line graphs are so useful in physics because their gradient is a real, physically meaningful constant — often exactly the quantity you're trying to find in the experiment.

The rules examiners actually enforce

• Draw a large triangle directly on the line of best fit — it should take up more than half the graph. Tiny triangles lose marks even if the final number is correct.
• Use points that lie on the line of best fit — never raw data points that happen to sit off the line.
• Show your full substitution working, not just the final answer.
• The gradient's units are the y-axis unit divided by the x-axis unit — e.g. for extension (m) against force (N), the gradient's units are N m⁻¹.
• Always check the axis scale and powers of ten. If an axis is labelled "Force F × 10³ / N," then a plotted value of "5" actually means 5 × 10³ N = 5000 N, not 5 N!

The y-intercept

The y-intercept is the value where the line crosses the y-axis (i.e., where x = 0). Read it directly off the graph, being careful to interpret the scale correctly — don't just guess "0" out of habit; sometimes the line genuinely doesn't pass through the origin, and that intercept often has physical meaning (e.g., in y = mx + c style equations from rearranged physics formulas).

Errors you can spot — and reduce

Random errors can never be fully eliminated — that's just the nature of measurement — but their effect can be reduced by taking repeat readings and averaging. Beyond that, there's a checklist of specific, commonly-tested error sources you should be able to name and fix.

Why fiducial markers matter

Imagine timing a pendulum's swing by eye. If you start/stop your stopwatch at the extremes of the swing (where it's momentarily still), your reaction time error is largest relative to how slow the pendulum is moving. But if you time it passing a fixed marker at the bottom of its swing — where it's moving fastest — your timing error has the smallest possible impact on accuracy, because the pendulum "blurs past" the marker quickly and consistently.

Making experiments more reliable & reproducible

Improvements aren't just "be more careful" — examiners want specific, technical suggestions. The most common category is switching from manual/human data collection to digital tools, because this removes human error like reaction time and subjectivity in judging readings.

Tools worth naming in an exam

• Data loggers: electronic devices with sensors that automatically monitor and record data (temperature, pressure, voltage, current) over time. They're more accurate, faster, and more reliable than manual logging, and can feed straight into a computer for tables, averages, and graphs.
• Cameras: useful for experiments that happen too fast to read a scale in real time. Take a photo burst, then read the scale afterward from stills. If frame rate is known, you can even calculate velocity from consecutive frames.
• Spreadsheet software (e.g. Excel): processes large volumes of data efficiently — invaluable for graph plotting and repeated calculations.
• Computer modelling: lets you speed up time to predict future outcomes of an experiment, and export polished graphs/charts straight into a report.

Benefits of data loggers / ICT specifically

• Higher accuracy of readings
• Reduction of human error (reaction time, subjective judgement)
• Can log over very long periods (e.g. hourly readings for days)
• Can log over very short periods too fast for a human to catch
• Reduces safety risk in extreme conditions (e.g. measuring boiling water temperature remotely)

Reproducibility

An improved method should ideally be reproducible — meaning another scientist, following the same method, should get the same result. A good way to test this in coursework-style discussion is to imagine applying your method to a different material or source, and checking whether it still gives sensible, consistent results.

No measurement is ever perfectly exact

Every measurement has some built-in doubt attached to it. Uncertainty isn't about "being wrong" — it's an honest, quantified statement of "here's how much this value could plausibly vary." There are three ways to express it:

Rules for finding uncertainty in different situations

• Single reading: ± half the smallest division on the scale.
• A measurement (involving two readings, e.g. length = final − initial): at least ±1 smallest division.
• Repeated data (e.g. the mean of several readings): half the range, i.e. ± ½(largest − smallest value).
• Digital readings: ± the last significant digit shown, unless the equipment states otherwise.

Combining uncertainties

Uncertainty in repeated readings — worked example

A student gets angular frequency readings: 0.154, 0.153, 0.159, 0.147, 0.152 rad s⁻¹.