Analysing: Trends and Relationships¶
Syllabus mapping
Working Scientifically outcome — Analysing (part 1 of 2). Covers identifying trends and relationships in processed data, comparing a result against a known/accepted value, and recognising the difference between interpolation and extrapolation. See Evaluating Data and Conclusions for the second half of Analysing — reliability, validity, limitations, and drawing a conclusion.
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 describe what a graph is actually telling you in precise language, and how to compare an experimental result against an accepted value properly — not just "close" or "not close."
Core¶
Describing a trend precisely¶
"The graph goes up" isn't an analysis — it doesn't say how it goes up, which is usually the entire point of the investigation. Once a relationship is linearised (see Representing Data), describe it in terms of proportionality:
- Directly proportional — a straight line through the origin; doubling \(x\) doubles \(y\).
- Linear, but not directly proportional — a straight line with a non-zero y-intercept; a change in \(x\) still produces a predictable change in \(y\), but \(y\) isn't simply a multiple of \(x\).
- Inversely proportional — \(y\) decreases as \(x\) increases, in a \(y \propto 1/x\) pattern (a curve when plotted directly; a straight line through the origin when plotted against \(1/x\)).
A trend statement should name the pattern and point at the evidence for it: "\(T^2\) is directly proportional to \(L\), shown by the straight line of best fit passing through the origin" is analysis; "the graph is a straight line" alone is just description.
Interpolation vs extrapolation¶
- Interpolating — reading a value from within the range of data actually collected. Generally safe, since you have evidence the relationship holds there.
- Extrapolating — reading a value from beyond the tested range. Riskier: you're assuming the same relationship continues past the last data point, which isn't guaranteed. Many syllabus relationships genuinely do break down outside a certain range (e.g. Hooke's Law fails past a spring's elastic limit; SUVAT assumes constant acceleration, which stops applying the moment that stops being true).
Worked example
A spring extension–force graph is linear from 0 N to 4 N. Reading off the extension at 2 N is interpolation — safe. Using the same gradient to predict extension at 15 N is extrapolation, and specifically risky here, since most springs stop obeying Hooke's Law well before that.
Comparing a result to an accepted value¶
Where a syllabus quantity has a known accepted value (e.g. \(g = 9.8\ \text{m s}^{-2}\), the speed of light, a material's refractive index), a result is compared using percentage error, not percentage uncertainty (see Precision and Uncertainty — these are two different calculations that get mixed up constantly):
Worked example
An experimental value of \(g = 10.0 \pm 0.9\ \text{m s}^{-2}\) (from Precision and Uncertainty's free-fall worked example), against the accepted value \(9.8\ \text{m s}^{-2}\):
This alone doesn't say whether the result is "good" — that depends on comparing it against the uncertainty, not just eyeballing the percentage error. Since the accepted value \(9.8\) falls inside the experimental range \(10.0 \pm 0.9\) (i.e. between \(9.1\) and \(10.9\)), the result agrees with the accepted value within experimental uncertainty — a stronger, more specific statement than "the percentage error was small."
Percentage error vs percentage difference¶
Percentage error assumes there's a known accepted value to compare against. When comparing two experimental results against each other instead — such as the two independent methods for determining \(g\) in Motion Relationships — neither one is "correct," so the comparison is a percentage difference, using the mean of the two as the reference point:
Advanced
When "agreement within uncertainty" isn't enough. Two results can agree within their stated uncertainties and still both be systematically wrong in the same direction — uncertainty ranges only capture random error, not systematic error (see Precision and Uncertainty). A result that agrees with an accepted value within uncertainty is good evidence the random error was reasonably estimated; it says nothing about whether a systematic error was present, which is why a full evaluation (see Evaluating Data and Conclusions) has to consider both separately.
The four boxes below are optional, tertiary-level statistical tools — genuinely beyond this syllabus, but exactly the kind of thing the strongest students (often also doing Extension 1/2 Maths) reach for in a depth study. Pick whichever is relevant; each stands alone and builds on the mean/standard deviation content from Precision and Uncertainty.
Extension — Standard error of the mean: how much to trust your average
Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.
📎 Depth study idea
Standard deviation, \(\sigma\), describes how spread out individual readings are. It does not, by itself, say how confident you should be in the mean of those readings — and that's a different, genuinely useful question. The standard error of the mean (SEM) answers it:
SEM shrinks as \(n\) grows, even though \(\sigma\) doesn't — averaging more repeats narrows your confidence in the mean, without making any individual reading more precise. This is why "just do more trials" is genuinely good advice, and SEM is the number that shows why.
Worked example
Five pendulum period readings: 1.98, 2.01, 1.95, 2.03, 1.99 s (the same data set used in Precision and Uncertainty).
Mean \(\bar{T} = 1.992\) s. Standard deviation (calculator, \(\sigma_{n-1}\) key): \(\sigma \approx 0.030\) s.
Report as \(T = 1.992 \pm 0.014\) s. Compare this to the simpler half-range estimate used in Precision and Uncertainty, \(\pm 0.04\) s — SEM is smaller and more rigorous, because it's specifically quantifying confidence in the mean, not just describing the full spread of the raw readings.
Extension — The one-sample t-test: is your result really different from the accepted value?
Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.
📎 Depth study idea
Comparing a mean to an accepted value by checking whether the accepted value "falls inside" \(\bar{x} \pm SE\) (as done earlier on this page) is a reasonable rough check, but it's not a formal test. The one-sample t-test makes it rigorous, by calculating how many standard errors the accepted value is away from your mean:
where \(\mu\) is the accepted value. You then compare the size of \(t\) to a critical value from a t-distribution table, based on the degrees of freedom (\(df = n - 1\)) and how confident you want to be (commonly 95%). If \(|t|\) is larger than the critical value, the difference is statistically significant — unlikely to be down to random chance alone. If \(|t|\) is smaller, your result is consistent with the accepted value.
Worked example
Five determinations of \(g\) by free fall: 9.9, 10.1, 9.7, 10.0, 9.8 m s⁻². Mean \(\bar{x} = 9.90\), standard deviation \(\sigma \approx 0.16\), so \(SE = 0.16/\sqrt{5} \approx 0.071\).
Comparing against the accepted value \(\mu = 9.8\):
With \(df = 4\), the critical t-value for 95% confidence (from a standard t-table) is \(2.776\). Since \(|1.41| < 2.776\), the difference is not statistically significant — this result is consistent with the accepted value, and the ~1% gap is plausibly just random scatter.
Extension — The two-sample t-test: comparing two experimental results
Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.
📎 Depth study idea
The percentage-difference calculation earlier on this page is a quick comparison between two experimental values, but like the accepted-value comparison above, it doesn't say whether the difference is significant or just random scatter. A simplified two-sample t-test extends the same idea to compare two means, each with their own spread and number of repeats:
(This is a simplified approach using \(df \approx n_1 + n_2 - 2\). A full statistics course uses Welch's t-test, which adjusts the degrees of freedom more precisely when the two samples' spreads differ — worth knowing that refinement exists if you take this further in a depth study.)
Worked example
Two independent methods for determining \(g\) (see Motion Relationships): free fall gives \(\bar{x}_1 = 9.7\), \(s_1 = 0.15\), \(n_1 = 5\); pendulum gives \(\bar{x}_2 = 10.1\), \(s_2 = 0.18\), \(n_2 = 5\).
With \(df \approx 8\), the critical t-value for 95% confidence is \(2.306\). Since \(|3.81| > 2.306\), the two methods give significantly different results — worth going back to Evaluating Data and Conclusions to hunt for a systematic error in one (or both) methods, rather than shrugging it off as "close enough."
Extension — Chi-squared goodness-of-fit
Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.
📎 Depth study idea
Comparing an experimental slope against a theoretical one gets more rigorous treatment at university via chi-squared goodness-of-fit testing — a single statistic that weighs how far every data point sits from a predicted line, accounting for each point's individual uncertainty, rather than comparing one final number to one accepted value. Worth researching if a depth study collects enough data points (10+) that a single percentage-error comparison feels like it's throwing away information the rest of the data set actually has.
Video/visual resources¶
- 🎥 Khan Academy — TODO: source a percentage error / comparing results explainer. Must distinguish percentage error (vs an accepted value) from percentage difference (between two experimental results) — the two are easy to conflate and use different reference denominators. Essential.
- 🎥 Physics High — TODO: check — plausible, since comparing an experimental result to an accepted value is a standard part of most HSC prac write-ups.
Check yourself¶
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An experiment gives a refractive index of glass as \(n = 1.44 \pm 0.05\) (see Wave Behaviours). The accepted value for that type of glass is \(1.52\). Calculate the percentage error, and state whether the result agrees with the accepted value within uncertainty.
Answer
\(\%\text{ error} = \dfrac{|1.44-1.52|}{1.52}\times100\% \approx 5.3\%\). The experimental range is \(1.39\) to \(1.49\), which does not include \(1.52\) — so the result does not agree with the accepted value within experimental uncertainty, despite the percentage error being fairly small.
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Explain why reading the extension of a spring at a force of 1 N, from a best-fit line built using data between 0 N and 5 N, is interpolation rather than extrapolation.
Answer
Interpolation means reading a value that falls within the range of data actually collected. Since 1 N sits between the tested range of 0 N and 5 N, this is interpolation — the relationship has direct evidence covering that point, unlike a prediction for, say, 8 N, which would be extrapolation.
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Two independent methods give \(g = 9.7\ \text{m s}^{-2}\) and \(g = 10.1\ \text{m s}^{-2}\). Calculate the percentage difference between them.
Answer
Mean \(= \dfrac{9.7+10.1}{2} = 9.9\). \(\%\text{ difference} = \dfrac{|9.7-10.1|}{9.9}\times100\% \approx 4.0\%\).
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(Stretch — uses the Extension boxes above.) Five repeated readings give a mean of \(24.6\) with standard deviation \(\sigma = 0.5\) and \(n = 5\). Calculate the standard error of the mean, and explain in one sentence why it's smaller than \(\sigma\).
Answer
\(SE = \dfrac{\sigma}{\sqrt{n}} = \dfrac{0.5}{\sqrt{5}} \approx 0.22\). It's smaller than \(\sigma\) because SEM quantifies confidence in the mean of several readings, not the spread of any individual reading — averaging multiple repeats narrows the uncertainty in the mean even though it doesn't change how spread out the individual readings themselves are.