DPST1021 Chap.15 Measurement, Uncertainty and Straight-Line Graphs
Measurement, Uncertainty and Straight-Line Graphs
Physics 1A treats laboratory work as a core strand, not an add-on. Eight two-hour labs run through the term, from equilibrium of rigid bodies to standing waves in a tube, and the first course learning outcome asks you to analyse results with reliable estimates of uncertainty.
Before any pre-lab exercise opens, an online laboratory skills quiz must be passed with at least 9 out of 12, and the problem solving workshops in the first two weeks focus on uncertainties, significant figures and introductory experimentation.
The chapter follows the two quick-start guides the course provides.
One covers uncertainty: reporting a value with an absolute or relative uncertainty, estimating it from a single instrument reading or from repeated trials, and carrying it through sums, differences, products and quotients, with different rules for dependent and independent quantities.
The other covers linear regression: rewriting a physical law in the straight-line form Y = AX + B, choosing what to plot on each axis, and extracting a constant such as g or a spring stiffness from the gradient.
Graphs join the two ideas.
Error bars on each point allow a steepest and a shallowest line of worst fit, and half the spread of their gradients becomes the uncertainty in the gradient of the best line, with the intercept handled the same way. These habits are what demonstrators look for when they grade each lab as Pass, Borderline or Fail, and a total of 6.5 out of 8 is needed to pass the lab component.
What this chapter covers
- 01
Absolute and relative uncertainty: report a measurement as a value plus a range in its own units, or as a fraction or percentage of the value.
- 02
Single readings: take the stated instrument accuracy where one is given, otherwise use half of the smallest graduation on the scale.
- 03
Repeated trials: use the standard deviation for a large set and half the range for fewer than about ten trials.
- 04
Propagation rules: combine absolute uncertainties for sums and relative ones for products, switching to quadrature when the quantities are independent.
- 05
Linearising a law: rearrange an equation into Y = AX + B so the plotted graph is straight and the wanted constant sits inside the gradient.
- 06
Worst-fit lines: draw the steepest and shallowest lines through the error bars and quote half their difference as the gradient uncertainty.
Worked example · free
Specific heat of water from a heater run
- 1Temperature change: 25.8 − 21.0 = 4.8 °C. The two readings are independent, so Δ(ΔT) = √(0.1² + 0.1²) = 0.14 °C.
- 1Best value: c = Q/(mΔT) = 5000/(0.250 × 4.8) = 4167 J/(kg·K).
- 1Relative uncertainties: ΔT gives 0.14/4.8 = 2.9%, mass gives 0.001/0.250 = 0.4%, heat gives 2.0%.
- 1Independent factors combine in quadrature: √(2.9² + 0.4² + 2.0²) = 3.6%, so Δc = 0.036 × 4167 = 149 J/(kg·K).
- 1Round the uncertainty to one significant figure, 100 J/(kg·K), and match the value: c = (4.2 ± 0.1) × 10³ J/(kg·K).
Key terms
- Absolute uncertainty
- The likely range of a measured value expressed in the same units as the value itself.
- Relative uncertainty
- The absolute uncertainty divided by the measured value, often written as a percentage.
- Independent uncertainties
- Errors arising from separate readings or instruments, so they do not rise and fall together.
- Line of worst fit
- A line drawn as steep or as shallow as the error bars allow, used to bound the best-fit gradient.
- Linearisation
- A rearrangement of a physical law into the form Y = AX + B so that measured data plot as a straight line.
Measurement, Uncertainty and Straight-Line Graphs FAQ
Should I use the standard deviation or half the range for my trials?
For a small set, roughly fewer than ten trials, half the range works better, because a standard deviation from three or four values is itself unreliable. With a larger set the standard deviation is the better estimate. Either way, state which method you used so a demonstrator can follow the reasoning on the page.
Why is quadrature smaller than simply adding the uncertainties?
Independent errors are unlikely to all push the same way at once, so some partly cancel. Adding the squares and taking the root reflects that, giving a combined value between the largest single term and the plain sum. Direct addition is reserved for dependent errors, which really do move together.
What if my worst-fit lines cannot pass through every error bar?
That is acceptable. The aim is lines through most of the bars, even if some only graze an edge. A point far outside any reasonable line deserves a second look: recheck the reading, or repeat the measurement, rather than forcing the line to include it.
Does a straight graph prove my equation is correct?
It shows the data are consistent with the linear form within the uncertainties, which is strong support but not proof. Check the intercept too: if theory predicts zero, the uncertainty range of the intercept should include zero, and a clear offset often points to a systematic error such as a misread starting length.
Exam move
Treat the laboratory skills quiz as the first checkpoint: work through its videos with both quick-start guides open, and aim well above the 9 out of 12 threshold so the pre-lab exercises unlock without delay. Then make a one-page decision chart for yourself covering single readings, repeated trials, sums and products, and graphs, and test it on numbers from your own lab manual after each session.
Practise linearising unfamiliar laws, for example a cooling curve or a spring period, until you can name the axes and the gradient in under a minute. Before each lab, finish the pre-lab quiz early, because scoring below half turns that week into a Fail. Finally, rehearse quoting a result with the uncertainty rounded first and the value matched to it.
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