Data Linearization, Proportionality, and Anomalies
Data linearization is the process of choosing transformed quantities (such as T2, 1/d2 or ln R) so that a non-linear relationship plots as a straight line, from which the gradient and intercept can be found; proportionality means that two quantities have a constant ratio, so their graph is a straight line through the origin; an anomaly is a data point that does not fit the pattern and should be checked and justified, not hidden. In brief: a straight line is the easiest shape to judge by eye and to fit, so power laws y = kxn are plotted as y against xn or as log y against log x (gradient n), and exponentials y = Ae−λx as ln y against x (gradient −λ).
Many laws in physics are not linear, but they can be tested by a suitable change of variables. This guide explains proportionality, how to linearize power laws and exponentials, how to deal with anomalous results, compares the common forms in a table, works through numerical examples (inverse-square law, radioactive decay and pendulum data) and corrects common mistakes.
What is proportionality?
Two quantities are directly proportional (y ∝ x) if y = kx: the ratio y/x is constant, doubling x doubles y, and the graph is a straight line through the origin with gradient k. Inversely proportional (y ∝ 1/x) means that xy is constant, and the graph of y against x is a hyperbola, while y against 1/x is a straight line through the origin. A straight-line graph that does not pass through the origin (y = mx + c, with c ≠ 0) shows a linear relationship but not a proportional one: doubling x does not double y. To test proportionality, plot the data and check that the best-fit line passes through the origin within the uncertainty, or calculate y/x for each point to see that it is constant within the uncertainty. A line through the origin is a necessary test: a curve that looks straight over a small range can still be wrong over a large range.
How do you linearize a relationship?
- Power law y = kxn with known n: plot y against xn. The gradient is k, the line passes through the origin. Examples: T2 against L (n = 1 for T2 ∝ L), I against 1/d2 for the inverse-square law, and Ek against v2.
- Power law with unknown n: take logarithms: log y = log k + n log x. A plot of log y against log x is a straight line with gradient n and intercept log k. This also tests whether the data follow a power law at all.
- Exponential y = Aebx: take natural logarithms: ln y = ln A + bx, a plot of ln y against x with gradient b (negative for decay) and intercept ln A. Examples: radioactive decay, capacitor discharge, absorption of radiation. The same works with log10, where the gradient is b log10e.
- Other forms: a quantity of the form y = a + b/x is plotted against 1/x; the equation of a lens 1/u + 1/v = 1/f against 1/u gives a line of gradient −1.
Once the equation is in the form y = mx + c, compare it with the theory term by term: the gradient and intercept contain the unknown physical constants. Uncertainties must be transformed: the percentage uncertainty of x2 is twice that of x, and the uncertainty in ln y is the fractional uncertainty of y (Δy/y).
How should anomalies be handled?
An anomalous result is a measurement that is well outside the expected random scatter, and it stands out on a graph as a point far from the best-fit line, or in repeated readings as one value far from the others. Steps: (1) check the calculations and the readings for a mistake; (2) repeat the measurement if possible; (3) consider the physical cause (a misread scale, a draft, a faulty connection); (4) if there is a justified reason, exclude it from the analysis, but record that it was excluded and why; (5) if it is not clear why it is anomalous, report it and perhaps analyze the data with and without it. Never remove a point only because it does not fit the hypothesis: an unexpected result may show new physics or an error in the model. An anomalous point is different from the natural scatter, and the error bars can show whether it is consistent with the line.
Linearization compared
| Relationship | Plot (y against x) | Gradient | Intercept | Example |
|---|---|---|---|---|
| y = kx | y against x | k | 0 | V = IR |
| y = kx2 | y against x2 | k | 0 | Ek = ½mv2 |
| y = k/x2 | y against 1/x2 | k | 0 | Inverse-square law |
| y = kxn (n unknown) | log y against log x | n | log k | T against L: n = 0.5 |
| y = Ae−λx | ln y against x | −λ | ln A | Radioactive decay |
Worked example: three linearizations
- Inverse-square law. A light meter gives an intensity I of 400, 100, 44 and 25 W/m2 at distances d = 0.50, 1.00, 1.50 and 2.00 m. The values of 1/d2 are 4.00, 1.00, 0.444 and 0.250 m−2. Each ratio I ÷ (1/d2) is 100, 100, 99 and 100 W: constant, so I is proportional to 1/d2. The graph of I against 1/d2 is a straight line through the origin with gradient 100 W (for a point source it equals P ÷ 4π, so P ≈ 100 × 4π ≈ 1,260 W).
- Exponential decay. The count rate R of a source is 800, 485, 294, 179 and 108 counts per second at t = 0, 10, 20, 30 and 40 s. The values of ln R are 6.685, 6.184, 5.684, 5.187 and 4.682. The gradient between the end points is (4.682 − 6.685) ÷ 40 = −0.0501 s−1, so the decay constant λ = 0.050 s−1 and the intercept ln R0 = 6.685 gives R0 = 800 counts/s. The half-life is ln 2 ÷ λ = 0.693 ÷ 0.0501 = 13.8 s.
- Unknown power. For the pendulum with L = 0.200 m and 1.000 m, T = 0.900 s and 2.010 s. The gradient of log T against log L between these points is (ln 2.010 − ln 0.900) ÷ (ln 1.000 − ln 0.200) = 0.803 ÷ 1.609 = 0.499, close to 0.5, so T ∝ L1/2, which is the same as T2 ∝ L.
- Anomaly. Four timings of 10 oscillations at L = 0.600 m are 15.6, 15.5, 15.7 and 17.2 s. The last is 1.6 s above the mean of the other three (15.6 s), while the others vary by only 0.1 s. If a stopwatch error is suspected, the timing is repeated. Including it, the mean would be 16.0 s (2.6 percent too high); excluding it with a reason, the mean is 15.6 s. In the report the rejected value and the reason are stated.
How did logarithmic methods develop?
John Napier published logarithms in 1614, and Henry Briggs soon produced tables of common logarithms. Slide rules based on logarithmic scales were used by scientists and engineers for about 350 years. Logarithmic graph paper (log-log and semi-log) allowed power laws and exponentials to be tested without calculating logarithms. Kepler’s third law (T2 ∝ r3) was found in 1619 by looking for a relation between planetary periods and distances, and a log-log plot of the planetary data shows the exponent 3/2. Today, spreadsheets perform the transformation and the fitting.
Which mistakes are common?
Mistake 1: Calling any straight line “proportional”
Proportionality needs a line through the origin. A line with an intercept is linear but not proportional.
Mistake 2: Using the wrong gradient meaning after linearization
On a plot of ln y against x the gradient is −λ, not λ (the sign matters), and on a plot of T2 against L it is 4π2/g, not g.
Mistake 3: Deleting anomalous points without a reason
Check them, repeat, and give the reason if they are excluded. Do not delete them just to improve the fit.
Mistake 4: Forgetting to transform the uncertainties
When squaring or taking logarithms, the error bars change: the percentage uncertainty doubles for a square, and Δ(ln y) = Δy/y.