Difficulty: Introductory | Prerequisites: basic statistics (mean, standard deviation), algebra
The lab component of PHYS 123 is not just about getting the right answer; it tests whether you can report a result properly, design an experiment with clear variables, and present data in a way that lets you test a model. These skills show up as multiple-choice questions on every midterm. If you skipped the labs, this section will catch you up on the rules the course uses.
Report measurements as (best estimate) ± (uncertainty), matching the decimal places. Choose independent and dependent variables that mirror what you controlled and measured. To test a power-law or root-law model, transform your axes so the expected relationship becomes a straight line.
Best estimate
The mean (average) of your measurements. This is your single best guess at the true value.
Uncertainty
A measure of the spread in your data. In PHYS 123 you typically use the standard deviation of the mean or a similar measure. The uncertainty tells you how confident you are in your best estimate.
Significant figures in uncertainty
The uncertainty is reported to one significant figure. The best estimate is then rounded to match the decimal place of the uncertainty.
Independent variable
The quantity you deliberately change in an experiment. It goes on the horizontal axis of your plot.
Dependent variable
The quantity you measure as a result of changing the independent variable. It goes on the vertical axis.
Linearisation
The process of transforming your axes so that a non-linear theoretical relationship plots as a straight line. If the data fall on a straight line after the transformation, the model is supported.
The PHYS 123 rule:
Compute the mean of your measurements.
Compute the uncertainty (standard deviation of the data, or whatever measure the course specifies).
Round the uncertainty to one significant figure.
Round the mean to the same decimal place as the uncertainty.
Report as: mean ± uncertainty, with units.
Worked example: Four measurements of frequency: 37.54, 36.47, 38.13, 37.81 Hz.
Mean = (37.54 + 36.47 + 38.13 + 37.81) / 4 = 149.95 / 4 = 37.4875 Hz
The spread is roughly 0.7 Hz (computed from standard deviation of these values).
Uncertainty rounded to one significant figure: 0.7 Hz.
Mean rounded to match the tenths place: 37.5 Hz.
Answer: 37.5 ± 0.7 Hz.
Common mistakes: reporting too many decimal places in the mean (37.49 ± 0.7 is wrong because the mean has hundredths but the uncertainty only goes to tenths), or rounding the uncertainty to two significant figures.
The key principle: match what you control and what you measure.
In Lab A1, the team varied the mass of the hanging weight (which sets the tension) and measured the frequency. In Lab A2 (the simulation), the analogue of tension is the spring constant. So the correct pairing is: frequency as dependent variable, spring constant as independent variable.
The question is not just "what could you change?" but "what is the analogue of what you changed in the physical experiment?"
If your model is A = k/√B, you want to transform the axes so the relationship becomes y = mx (a straight line through the origin).
A = k × (1/√B), so if you plot A on the vertical axis and 1/√B on the horizontal axis, you get a straight line with slope k.
Plotting A vs B would give a curve, not a line.
Plotting √A vs B or A vs √B would also give curves.
The general strategy: isolate the dependent variable on one side, and whatever multiplies the constant on the other side becomes your horizontal axis variable.
Students report the mean to more decimal places than the uncertainty. If your uncertainty is ± 0.7, your mean should end at the tenths place, not the hundredths.
Students confuse "one significant figure in the uncertainty" with "one decimal place." The number 0.07 has one significant figure; the number 0.70 has two.
Students pick independent and dependent variables based on what is easiest to plot rather than what mirrors the experimental design.
Students linearise by transforming the wrong variable. If the model is A = k/√B, you transform B (plot 1/√B), not A.
⚠️ The uncertainty-reporting question appears on nearly every PHYS 123 midterm. Memorise the rule: one sig fig in the uncertainty, then match the mean.
⚠️ Variable-choice questions require you to think about the analogy between the physical experiment and the simulation. What physical quantity does each simulation parameter correspond to?
⚠️ Linearisation questions test whether you can algebraically rearrange a model into y = mx form and identify which transformed quantities go on which axis.
True or false: an uncertainty of 0.03 has one significant figure. (True.)
Fill in the blank: to linearise y = k√x, plot y on the vertical axis and ______ on the horizontal axis. (√x)
True or false: the dependent variable is the one you deliberately change. (False. The dependent variable is the one you measure.)
Q: Measurements of frequency: 37.54, 36.47, 38.13, 37.81 Hz. Which is the correct way to report the result: 37.49 ± 0.01, 37.5 ± 0.7, 37 ± 1, 37.49 ± 0.7, or 37.5 ± 1?
A: 37.5 ± 0.7 Hz. The uncertainty rounds to one significant figure (0.7), and the mean rounds to the tenths place to match.
Q: In Lab A1, frequency was measured as the mass of a hanging weight was varied. In the simulation (Lab A2), what should the dependent and independent variables be?
A: Frequency as dependent variable, spring constant as independent variable. The spring constant in the simulation is the analogue of the tension (set by the hanging mass) in the physical experiment.
Q: The model is A = k/√B. How do you linearise the plot?
A: Plot A on the vertical axis and 1/√B on the horizontal axis. The resulting graph should be a straight line through the origin with slope k.
These lab skills are not specific to waves. The uncertainty rules, variable selection, and linearisation techniques apply to every physics lab course you will take. Linearisation in particular reappears whenever you need to test a theoretical model against experimental data, from E&M labs to advanced optics.
Uncertainty, error bars, significant figures, best estimate, standard deviation, mean, dependent variable, independent variable, linearisation, linearized plot, power law, log-log plot, straight-line fit, PHYS 123 lab, UW physics lab, Lab A1, Lab A2, spring-mass simulation, standing wave experiment