Data Analysis & Model Comparison

A model makes a testable claim. Define the controlled variable, measured response, units, assumptions and tested range. In the spring experiment, m is the total attached mass, including the hanger. A massless Hookean spring predicts T² = (4π²/k)m. A spring with effective moving mass predicts an intercept. A calibration or mass offset can also cause an intercept, so the fitted line alone does not identify the mechanism.

Use one row for each measurement. State how standard uncertainties were obtained and whether measurements share a calibration error. In this lab, the input uncertainties are independent standard uncertainties in T² and mass error is negligible. If the measured quantity is T, propagate u(T²) ≈ 2T u(T). Constant uncertainty in T therefore does not give constant uncertainty in T². Outliers need investigation, not automatic deletion.

For small-amplitude motion in the linear spring regime, plotting T² against m gives slope a = 4π²/k and intercept b = a m_eff. The slope has units s²/kg, so k has units kg/s² = N/m. The effective mass is b/a, in kg. A uniform spring contributes approximately one third of its own mass in the usual approximation. The fit applies only over the mass and amplitude range tested.

When all standard uncertainties are equal, a = Σ(xᵢ − x̄)(yᵢ − ȳ)/Σ(xᵢ − x̄)² and b = ȳ − ax̄. With unequal uncertainties, use weighted means and weighted sums. Larger standard uncertainty means less weight. The through-origin constraint instead gives a₀ = Σwᵢxᵢyᵢ/Σwᵢxᵢ². It should be imposed because of defensible physics, not because the line looks neater. Significant x errors require a different fitting method.

Residuals retain the units of y. Curvature may suggest an omitted term; a tilt after forcing the origin may suggest an offset; changing spread may indicate nonconstant variance. None uniquely identifies the cause: calibration, correlations and chance also matter. Lack of an obvious pattern is useful evidence, not proof that the model is correct. With a free intercept, Σwᵢrᵢ = 0; only for equal weights does this imply Σrᵢ = 0.

This lab treats σᵢ as known standard uncertainties. Define S = Σwᵢ and Q = Σwᵢ(xᵢ − x̄_w)². Then uₐ² = 1/Q, u_b² = 1/S + x̄_w²/Q and cov(a, b) = −x̄_w/Q. Do not multiply this covariance by reduced χ². If a common unknown noise variance is instead estimated from residuals, use s² = Σrᵢ²/(N − 2) and uₐ = s/√Σ(xᵢ − x̄)²; that is a different assumption. For k, uₖ = (4π²/a²)uₐ. For b/a, include the covariance term; the slope and intercept are not independent.

The free-intercept line uses p = 2; the origin-constrained line uses p = 1. Under a correct model with independent Gaussian known errors, reduced χ² has expectation 1 and standard deviation √(2/ν). With few points it fluctuates substantially. Large values can reflect missing physics, underestimated uncertainties, outliers or correlation. Small values can reflect conservative uncertainties, correlations or chance. A value near 1 does not certify a model, and ν = 0 gives no reduced χ².

For independent estimates, divide their absolute difference by the standard uncertainty of that difference. If the estimates are correlated, include −2 cov(A, B) inside the square root. This standardized difference is not automatically a Student t-test. Two or three standard uncertainties are context-dependent heuristics, not universal rules that prove agreement or a real difference. The assumptions and uncertainty quality matter.

For these nested linear models with known independent Gaussian errors, Δχ² = (b/u_b)². Under the zero-intercept null, an extra fitted parameter reduces χ² by one on average; this is an expectation, not a guaranteed improvement or a decision rule. Compare the reduction with the added flexibility and use independent evidence. R² values must use the same definition to be comparable. In the worked example, both models have high centered R², but the constrained model has a much larger χ² and a residual tilt.

The worked example covers 0.100–0.500 kg. Small-amplitude, Hookean behaviour, negligible mass error and independent T² uncertainties are assumed. Large amplitudes, permanent deformation, damping and correlated calibration errors are not tested here. Report estimates with units, sensible rounding, uncertainties and residual evidence. A successful fit supports a model within those conditions; it does not justify extrapolation or establish causation.