Processing: Representing Data

Syllabus mapping

Working Scientifically outcome — Processing (part 2 of 2). Covers organising processed data into tables and graphs, choosing an appropriate graphical representation, and linearising a relationship so it can be read from a straight-line graph. See Precision and Uncertainty for the first half of Processing — error, uncertainty, and propagating uncertainty through a calculation.

Exact NESA outcome code TODO — confirm against the official syllabus PDF (Resources) before treating any wording on this page as verbatim NESA text.

You need to know: how to build a summary table from raw data, how to construct a graph that's actually readable, and how to turn a curved relationship into a straight line — because a straight line is far easier to check against a prediction than a curve is.

Assumed knowledge check

Before this page makes sense, you should be comfortable with:

• Graphs — reading and constructing line graphs, finding gradient and area under a graph

Core

From raw data to a summary table

Conducting covered building a raw data table as you collect it — every individual repeat, untouched. A summary/processed table is built afterwards, and looks different: one row per independent-variable value, with a mean column (and, where relevant, an uncertainty column) rather than every individual repeat spelled out.

Worked example — raw to summary

Raw data (three repeats of pendulum period at each length):

Length (m)	\(T_1\) (s)	\(T_2\) (s)	\(T_3\) (s)
0.30	1.11	1.09	1.10
0.50	1.42	1.45	1.41

Summary table:

Length (m)	Mean \(T\) (s)	\(\Delta T\) (s)
0.30	1.10	0.01
0.50	1.43	0.02

The summary table is what you actually plot — a graph built directly from a raw table with three overlapping columns of near-identical numbers is unreadable.

Building a graph that's actually readable

A handful of conventions turn a scribble into something a marker (or you, a week later) can read at a glance:

• Title — what the graph shows, not just axis names restated ("Period of a pendulum vs length", not "Graph 1").
• Labelled axes with units — every axis names its quantity and its unit, e.g. "Length (m)", not just "L".
• Independent variable on the x-axis, dependent on the y-axis — this is the convention every marker expects.
• A sensible scale — spread the data across as much of the grid as reasonably possible; don't compress five data points into one corner of the page. The scale does not have to start at zero unless you specifically need to read a y-intercept.
• A line or curve of best fit, not dot-to-dot — a best-fit line/curve represents the underlying relationship, smoothing over the scatter caused by random error in individual points. Connecting dots in sequence treats every wiggle as real, which it almost never is.
• Error bars, where uncertainty has been calculated for each point — a vertical line through each point extending \(\pm\) the uncertainty, showing the range within which the true value plausibly sits. Often skipped for time in a Year 11 prac write-up, but it's a genuine Working Scientifically skill and worth including once uncertainty has actually been calculated (see Precision and Uncertainty).

Linearising a relationship

Many of the relationships in this course are not straight lines when plotted directly — but a straight line is far easier to work with than a curve: it's easy to see whether data actually fits it, and its gradient and intercept can be read off directly and compared to a predicted value. Linearising means choosing what to plot on each axis so that a relationship which is naturally curved becomes a straight line.

The general approach: if a relationship is \(y = kx^n\), plotting \(y\) against \(x^n\) (instead of against \(x\)) gives a straight line through the origin with gradient \(k\).

Worked example — linearising the pendulum relationship

The pendulum relationship is \(T = 2\pi\sqrt{L/g}\), which is a curve if you plot \(T\) against \(L\) directly (a square-root shape). Squaring both sides:

\[ T^2 = \frac{4\pi^2}{g}L \]

This is now in the form \(y = kx\), with \(y = T^2\), \(x = L\), and gradient \(k = \dfrac{4\pi^2}{g}\). Plotting \(T^2\) (s²) on the y-axis against \(L\) (m) on the x-axis gives a straight line through the origin, and rearranging the gradient:

\[ g = \frac{4\pi^2}{\text{gradient}} \]

If the best-fit line's gradient measures \(4.02\ \text{s}^2\text{m}^{-1}\):

\[ g = \frac{4\pi^2}{4.02} \approx 9.82\ \text{m s}^{-2} \]

This is the second method referenced in Motion Relationships's suggested prac (two independent methods of determining g) — comparing this graphically-determined value against the free-fall method's result (see Precision and Uncertainty) is exactly the kind of cross-check Working Scientifically is trying to build.

Other relationships already flagged elsewhere on this site that need the same treatment before they're straight lines:

• Inverse-square light intensity (Light and Sound): \(I \propto \dfrac{1}{r^2}\) is a curve if \(I\) is plotted against \(r\), but plotting \(I\) against \(\dfrac{1}{r^2}\) gives a straight line through the origin.
• Refractive index (Wave Behaviours): Snell's Law, \(n_1\sin\theta_1 = n_2\sin\theta_2\), is already linear once you plot \(\sin\theta_1\) against \(\sin\theta_2\) directly — no transformation needed, which is exactly why that's the recommended graph for a more reliable value of \(n\) than any single-trial calculation.
• Constant acceleration (Motion Relationships): \(v^2 = u^2 + 2as\) is linear if \(v^2\) is plotted against \(s\), with gradient \(2a\) — useful whenever \(v\) is easier to measure than \(t\).

Advanced

Choosing the transformation when it isn't obvious. Not every relationship arrives pre-labelled as \(y=kx^n\). Given an unfamiliar formula, isolate the two quantities you can actually measure, then algebraically rearrange until one is a constant multiple of some power (or combination) of the other — exactly as done above for the pendulum. If a relationship involves two variables you're changing (rather than one IV and one DV), you generally need to hold one constant while linearising against the other, one variable at a time.

Extension

Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.

📎 Depth study idea

Not every relationship linearises with a simple power transformation — exponential relationships (radioactive decay, capacitor discharge, damped oscillations) need a logarithmic axis instead. For \(y = Ae^{-kx}\), taking the natural log of both sides gives \(\ln y = \ln A - kx\) — a straight line if you plot \(\ln y\) against \(x\), with gradient \(-k\) and y-intercept \(\ln A\). This is standard practice in senior physics and any lab-based degree, and log-linearising a genuinely exponential data set (rather than a syllabus-standard power relationship) is a strong, technically substantial depth study angle if your investigation happens to produce decay-shaped data.

Video/visual resources

• 🎥 Khan Academy — TODO: source a graphing/line-of-best-fit explainer. Must cover linearising a relationship specifically (plotting \(y\) against \(x^n\) rather than \(x\)), not just basic best-fit-line mechanics — that's the actual hard skill on this page. Essential.
• 🎥 Physics High — TODO: check — plausible, since linearising graphs (e.g. \(T^2\) vs \(L\) for the pendulum prac) is a genuinely exam-relevant physics prac-write-up skill.

Check yourself

1. Explain why a graph of \(T\) against \(L\) for a pendulum is curved, but a graph of \(T^2\) against \(L\) is a straight line.

   Answer

   \(T = 2\pi\sqrt{L/g}\) means \(T\) is proportional to \(\sqrt{L}\), a square-root curve, not a straight line. Squaring both sides gives \(T^2 = \dfrac{4\pi^2}{g}L\), which is exactly the linear form \(y = kx\) with \(y = T^2\) and \(x = L\) — so plotting the squared quantity removes the curve.

2. A student investigating light intensity vs distance (see Light and Sound) plots \(I\) against \(\dfrac{1}{r^2}\) and gets a straight line through the origin with gradient \(48\). What does this gradient represent, physically?

   Answer

   Since \(I = k \cdot \dfrac{1}{r^2}\) is the linearised form of \(I \propto \dfrac{1}{r^2}\), the gradient is the constant of proportionality \(k\) — physically related to the power output of the light source. A straight line through the origin (rather than a curve, or a line that doesn't pass through the origin) is itself evidence that the inverse-square relationship holds for this data.

3. A student has raw data with five repeated period measurements at each of six different pendulum lengths. Describe what their summary table should contain, and what they should plot to get a straight-line graph.

   Answer

   The summary table should have one row per length (six rows), with columns for length, mean period (averaged over the five repeats), and the uncertainty in that mean (e.g. half-range — see Precision and Uncertainty) — not all thirty individual raw readings. To get a straight line, they should plot \(T^2\) (not \(T\)) on the y-axis against length on the x-axis.