Data Fitting and Regression

Least-squares data fitting, linear and nonlinear regression to extract physical parameters from measurements.

What it is

Curve fitting and statistical regression form the quantitative bridge connecting discrete experimental data points $(x_i, y_i \pm \sigma_{y_i})$ with theoretical physical laws $y = f(x; \boldsymbol{\theta})$, where $\boldsymbol{\theta} = (\theta_1, \dots, \theta_m)$ denotes the vector of unknown physical constants to determine (such as acceleration $g$, Planck's constant $h$, or internal resistance $r$).

The gold standard in experimental physics is the weighted least-squares method, derived from maximum likelihood estimation under independent Gaussian errors. It consists of minimizing the goodness-of-fit chi-squared objective function ($\chi^2$): $\chi^2(\boldsymbol{\theta}) = \sum_{i=1}^N \left( \frac{y_i - f(x_i; \boldsymbol{\theta})}{\sigma_{y_i}} \right)^2$ Each data point enters the objective weighted inversely by its experimental variance: $w_i = 1/\sigma_{y_i}^2$. High-precision data points exert strong mathematical pulling power on the fitted curve, while noisy points with broad error bars contribute marginally.

In the case of linear regression $f(x) = m x + c$, analytical minimization yields exact weighted normal equations for slope $m$ and intercept $c$, alongside their standard uncertainties and cross-covariance $\operatorname{cov}(m, c)$.

Why it appears in physics

In physical science, no fundamental parameter is reliably measured from an isolated reading. Physical laws are confirmed by verifying whether functional relationships hold across decades of dynamic range:

1. In Hooke's law ($F = k \Delta x$), spring stiffness $k$ is extracted as the slope of a linear fit between applied load and extension.
2. In the photoelectric effect ($V_0 e = h \nu - \Phi$), the fundamental ratio $h/e$ is obtained from the slope of stopping potential versus incident photon frequency.
3. In thermodynamics and nuclear physics, exponential and power laws are frequently linearized through logarithmic transformations (e.g. radioactive decay $N(t) = N_0 e^{-\lambda t} \implies \ln N = \ln N_0 - \lambda t$ or Arrhenius activation energy $\ln k = \ln A - E_a / RT$).
4. Goodness-of-fit evaluation via reduced chi-squared $\chi_\nu^2 = \chi^2 / (N - m)$ provides an objective arbiter: it certifies whether the physical model adequately describes the observations or if the hypothesis must be rejected.

Visual or geometric idea

Geometrically, the physical model $f(x; \boldsymbol{\theta})$ traces a family of curves across the Cartesian plane $(x, y)$. Each data point carries a vertical error bar defining its uncertainty range $\pm \sigma_{y_i}$.

The normalized residual $r_i = (y_i - f(x_i)) / \sigma_{y_i}$ measures the vertical distance between observation and curve in units of standard deviation. Weighted least squares deforms and shifts the curve to minimize the summed squared normalized distances. In a high-quality fit, the fitted line threads through approximately $68\%$ of the $1\sigma$ error bars, with residuals scattered randomly above and below without coherent curvature or trend.

Minimal symbolic language

1. Chi-squared loss function ($\chi^2$): $\chi^2 = \sum_{i=1}^N w_i [y_i - f(x_i)]^2, \qquad w_i = \frac{1}{\sigma_{y_i}^2}$
2. Unweighted linear normal equations ($w_i = 1$): $m = \frac{N \sum x_i y_i - \sum x_i \sum y_i}{N \sum x_i^2 - (\sum x_i)^2}, \qquad c = \bar{y} - m \bar{x}$
3. Reduced chi-squared (degrees of freedom $\nu = N - m$): $\chi_\nu^2 = \frac{\chi^2}{N - m}$
4. Parameter covariance matrix (linear model $\mathbf{y} = \mathbf{X} \boldsymbol{\theta}$): $\mathbf{V}_{\boldsymbol{\theta}} = (\mathbf{X}^T \mathbf{W} \mathbf{X})^{-1}, \qquad \sigma_m = \sqrt{V_{11}}, \quad \sigma_c = \sqrt{V_{22}}$
5. Coefficient of determination ($R^2$): $R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}$

What the result means

The optimal parameters $\hat{\boldsymbol{\theta}}$ yield the best unbiased estimate of the underlying physical constants. Interpreting the reduced chi-squared $\chi_\nu^2$ is crucial:

• $\chi_\nu^2 \approx 1$: The model fits the data well, and estimated error bars accurately reflect true random experimental scatter.
• $\chi_\nu^2 \gg 1$ (e.g. $\chi_\nu^2 > 3$): Poor fit. Either the physical model is invalid (e.g. linear model applied to quadratic physics), unmodeled systematic errors exist, or uncertainties were severely underestimated.
• $\chi_\nu^2 \ll 1$ (e.g. $\chi_\nu^2 < 0.2$): Suspicious. Experimental uncertainties were grossly overestimated, or data points were artificially smoothed.

When not to use it

1. Ordinary least squares when errors exist on both variables ($x$ and $y$): If horizontal uncertainties $\sigma_{x_i}$ are significant relative to $\sigma_{y_i}$, standard OLS underestimates slope $m$ (attenuation bias). Orthogonal distance regression (ODR) or the York estimator must be used.
2. Unweighted linear regression on transformed non-linear equations: When linearizing $y = A e^{B x}$ via logarithms, transformed variances scale as $\sigma_{\ln y} \approx \sigma_y / y$. Unweighted regression on $\ln y$ assigns excessive weight to tiny values of $y$ carrying massive relative uncertainties, distorting parameter extraction. Weighted regression with $w_i = y_i^2 / \sigma_{y_i}^2$ or direct nonlinear optimization (Levenberg-Marquardt) is mandatory.
3. Overfitting: Fitting an $(N-1)$-degree polynomial to $N$ data points forces $\chi^2 = 0$ but destroys predictive insight, producing wild oscillations between points (Runge's phenomenon). Model complexity must be rooted in physical mechanisms.

Operational summary

• Plot experimental data on a scatter plot displaying explicit error bars $\sigma_{y_i}$ (and $\sigma_{x_i}$ where applicable).
• Formulate the model function and determine linearity in parameters.
• For linear models, solve the weighted least-squares normal equations with weights $w_i = 1/\sigma_i^2$.
• Extract standard parameter errors $\sigma_m$ and $\sigma_c$ from the diagonal elements of the covariance matrix.
• Evaluate reduced chi-squared $\chi_\nu^2$ and inspect normalized residuals for systematic trends.

Weighted Linear Regression and Curve Fitting Procedure

Before calculating

Tabulate experimental observations in three columns: independent coordinate $x_i$, dependent coordinate $y_i$, and standard experimental uncertainty $\sigma_{y_i}$. Ensure all $\sigma_{y_i} > 0$. If measurement uncertainties are identical across all points (homoscedasticity), ordinary unweighted least squares may be used; if uncertainties vary between data points (heteroscedasticity), statistical weights $w_i = 1/\sigma_{y_i}^2$ must be calculated.

Step 1: identify the mathematical object

Formulate the linear mathematical model linking physical quantities: $y = m x + c$ Map the fit parameters $(m, c)$ to underlying physical constants:

• Ohm's law ($V = R I$): dependent variable $y = V$, independent variable $x = I$, slope $m = R$, theoretical intercept $c = 0$.
• Linear thermal expansion ($L = L_0 + L_0 \alpha \Delta T$): slope $m = L_0 \alpha$, intercept $c = L_0$.

Step 2: choose the operation

Compute the fundamental weighted moments of the dataset: $S = \sum_{i=1}^N w_i, \quad S_x = \sum_{i=1}^N w_i x_i, \quad S_y = \sum_{i=1}^N w_i y_i$ $S_{xx} = \sum_{i=1}^N w_i x_i^2, \quad S_{xy} = \sum_{i=1}^N w_i x_i y_i, \quad S_{yy} = \sum_{i=1}^N w_i y_i^2$ Compute the linear system determinant: $\Delta = S \cdot S_{xx} - (S_x)^2$

Step 3: preserve units and signs

• Statistical weights $w_i = 1/\sigma_{y_i}^2$ carry physical dimensions $[y]^{-2}$.
• The fitted slope $m$ carries units of $[y] / [x]$.
• The intercept $c$ carries units of $[y]$.
• The determinant $\Delta$ has units $[y]^{-4} [x]^2$. Preserve algebraic signs strictly in all cross-products $S_{xy}$.

Step 4: calculate

1. Optimal fit parameters: $m = \frac{S \cdot S_{xy} - S_x \cdot S_y}{\Delta}$ $c = \frac{S_{xx} \cdot S_y - S_x \cdot S_{xy}}{\Delta}$
2. Standard parameter uncertainties: $\sigma_m = \sqrt{\frac{S}{\Delta}}$ $\sigma_c = \sqrt{\frac{S_{xx}}{\Delta}}$ $\operatorname{cov}(m, c) = -\frac{S_x}{\Delta}$
3. Goodness of fit (Chi-squared and residuals): $\chi^2 = \sum_{i=1}^N w_i (y_i - m x_i - c)^2$ Degrees of freedom: $\nu = N - 2$. $\chi_\nu^2 = \frac{\chi^2}{N - 2}$

Step 5: interpret the result

• Physical inference: Derive the target constant from $m$ or $c$, propagating uncertainties via $\sigma_m$ or $\sigma_c$.
• Validation criterion $\chi_\nu^2$:
  • If $0.8 \leq \chi_\nu^2 \leq 1.5$: the model is valid and experimental uncertainties are well-estimated.
  • If $\chi_\nu^2 > 2$: inspect normalized residuals $r_i = y_i - (m x_i + c)$. Parabolic residual patterns indicate missing nonlinear physics (e.g. $x^2$ term).
• Final reporting: Report parameters with clear physical units and consistent decimal rounding matching uncertainty: $m = (\bar{m} \pm \sigma_m) \text{ [units]}$, $c = (\bar{c} \pm \sigma_c) \text{ [units]}$.

Self-check checklist

• Is determinant $\Delta > 0$? (If $\Delta = 0$, all $x_i$ values are identical, making a slope mathematically undefined).
• Do units of slope match $[y]/[x]$ and intercept match $[y]$?
• Did I verify that residuals do not show systematic curvature?
• Is reduced chi-squared $\chi_\nu^2$ close to unity?
• If theoretical physics requires $c = 0$ (e.g. Ohm's law), is the fitted intercept consistent with zero within $[-\sigma_c, +\sigma_c]$?

Guided Example: Determining Planck's Constant via the Photoelectric Effect

Physical Situation

In Millikan's classic photoelectric experiment, a cesium photocathode is illuminated with monochromatic light across 4 distinct optical frequencies $\nu$. For each frequency, the stopping potential $V_0$ required to quench the electronic photocurrent is recorded, alongside an experimental measurement uncertainty $\sigma_V = 0.02 \text{ V}$ across all data points:

• Point 1: $\nu_1 = 5.50 \times 10^{14} \text{ Hz}$, $V_{0,1} = 0.36 \pm 0.02 \text{ V}$
• Point 2: $\nu_2 = 6.20 \times 10^{14} \text{ Hz}$, $V_{0,2} = 0.65 \pm 0.02 \text{ V}$
• Point 3: $\nu_3 = 6.90 \times 10^{14} \text{ Hz}$, $V_{0,3} = 0.94 \pm 0.02 \text{ V}$
• Point 4: $\nu_4 = 7.60 \times 10^{14} \text{ Hz}$, $V_{0,4} = 1.23 \pm 0.02 \text{ V}$

The fundamental elementary charge is $e = 1.602 \times 10^{-19} \text{ C}$. The goal is to estimate Planck's constant $h$ and the cathode work function $\Phi$.

Mathematical Translation

Einstein's photoelectric equation is: $e V_0 = h \nu - \Phi \implies V_0 = \left( \frac{h}{e} \right) \nu - \frac{\Phi}{e}$ This maps to a straight line $y = m x + c$ where $y = V_0$, $x = \nu$, slope $m = h/e$, and intercept $c = -\Phi/e$. Since all measurement uncertainties are identical ($\sigma_i = 0.02 \text{ V}$), statistical weights are uniform: $w_i = 1/(0.02)^2 = 2500 \text{ V}^{-2}$. Using scaled coordinates $x$ in units of $10^{14} \text{ Hz}$: $x = [5.50, 6.20, 6.90, 7.60]$, $y = [0.36, 0.65, 0.94, 1.23]$.

Step-by-Step Calculation

1. Sample Means: $N = 4$ $\bar{x} = \frac{5.50 + 6.20 + 6.90 + 7.60}{4} = 6.55 \times 10^{14} \text{ Hz}$ $\bar{y} = \frac{0.36 + 0.65 + 0.94 + 1.23}{4} = 0.795 \text{ V}$
2. Centered Summations: $\sum (x_i - \bar{x})^2 = (-1.05)^2 + (-0.35)^2 + (0.35)^2 + (1.05)^2 = 2.45$ $\sum (x_i - \bar{x})(y_i - \bar{y}) = (-1.05)(-0.435) + (-0.35)(-0.145) + (0.35)(0.145) + (1.05)(0.435) = 1.015$
3. Slope and Intercept: $m = \frac{1.015}{2.45} \approx 0.4142857 \text{ V}/(10^{14} \text{ Hz}) = 4.143 \times 10^{-15} \text{ V}\cdot\text{s}$ $c = \bar{y} - m \bar{x} = 0.795 - (0.4142857 \times 6.55) = -1.9186 \text{ V}$
4. Statistical Uncertainties: $\sigma_m = \frac{\sigma}{\sqrt{\sum (x_i - \bar{x})^2}} = \frac{0.02}{\sqrt{2.45} \times 10^{14}} \approx 0.128 \times 10^{-15} \text{ V}\cdot\text{s}$ $\sigma_c = \sigma \sqrt{\frac{1}{N} + \frac{\bar{x}^2}{\sum (x_i - \bar{x})^2}} = 0.02 \sqrt{0.25 + 17.511} \approx 0.084 \text{ V}$
5. Recovering Physical Parameters: $h = m \cdot e = (4.143 \times 10^{-15} \text{ V}\cdot\text{s}) \times (1.602 \times 10^{-19} \text{ C}) = 6.637 \times 10^{-34} \text{ J}\cdot\text{s}$ $\sigma_h = \sigma_m \cdot e = 0.205 \times 10^{-34} \text{ J}\cdot\text{s}$ Work function: $\Phi = -c \cdot e = 1.919 \text{ eV} \pm 0.084 \text{ eV}$.

Reading the Result

The computed value for Planck's constant is $h = (6.64 \pm 0.21) \times 10^{-34} \text{ J}\cdot\text{s}$. The CODATA accepted reference value $h_{\text{CODATA}} = 6.62607015 \times 10^{-34} \text{ J}\cdot\text{s}$ falls cleanly within the $68\%$ confidence interval.

Common Error Avoided

Forcing the regression line through the origin ($c=0$). Imposing $c=0$ would erroneously assume electrons require zero binding energy to escape the photocathode, corrupting the slope and yielding an inaccurate value for $h$.

Inverse Example: Diagnosing Newton's Law of Cooling from Structured Residuals

Physical Situation

A student investigates the thermal cooling of a heated metallic sample in ambient air at $T_{\text{amb}} = 20.0^\circ\text{C}$. The student fits temperature $T(t)$ directly to a linear model $T(t) = m t + c$, obtaining $m = -0.50^\circ\text{C/min}$ and $c = 80.0^\circ\text{C}$ with a high correlation coefficient $R^2 = 0.94$. However, the residual plot shows negative residuals for $t < 20 \text{ min}$, strongly positive residuals for $20 < t < 60 \text{ min}$, and negative residuals again for $t > 60 \text{ min}$.

Given Result

Linear fit with an inverted parabolic residual signature (systematic deterministic curvature).

Analytical Reconstruction

Systematic curvature in residuals proves the linear model violates the underlying continuum physics. Newton's law of cooling governs the process: $\frac{dT}{dt} = -k (T - T_{\text{amb}}) \implies T(t) - T_{\text{amb}} = (T_0 - T_{\text{amb}}) e^{-k t}$ Linearizing by taking the natural logarithm of excess temperature $\Delta T = T(t) - T_{\text{amb}}$: $y^* = \ln(T(t) - 20.0) = \ln(T_0 - 20.0) - k t$ Plotting $\ln(T - 20.0)$ against time $t$ eliminates all residual curvature, scattering points randomly around zero and dropping reduced $\chi_\nu^2$ from $18.4$ down to $1.05$.

What Physical Question It Can Answer

Why is a high $R^2 = 0.94$ insufficient proof of linearity? The coefficient of determination $R^2$ measures explained variance but cannot detect systematic model bias. Residual analysis and reduced $\chi_\nu^2$ are mandatory to reveal whether the true physics is exponential rather than linear.

Boundary Verification

As $t \to \infty$, the false linear model predicts $T \to -\infty$ (violating absolute zero and thermal equilibrium), whereas the exponential model properly asymptotes to $T(t) \to T_{\text{amb}} = 20^\circ\text{C}$.

Common Mistakes in Data Fitting and Regression

1. Relying solely on $R^2$ to assess linearity

It seems reasonable because an $R^2 > 0.98$ delivers a deceptive sense of mathematical perfection. It fails because $R^2$ only measures variance reduction relative to a horizontal mean line; it is completely blind to systematic residual curvature (Anscombe's quartet demonstrates four datasets sharing identical $R^2 = 0.67$ where one is linear, one is a clean parabola, and another has an extreme outlier). It is detected by plotting residuals or computing $\chi_\nu^2$. It is corrected by always graphing residuals $r_i = y_i - f(x_i)$ and demanding $\chi_\nu^2 \approx 1$. Mini-example: fitting $y = x^2$ for $x \in [10, 20]$ to a straight line yields $R^2 > 0.99$, yet residuals form an obvious parabolic arch disproving physical linearity.

2. Using ordinary least squares when uncertainties exist on both coordinates

It seems reasonable because standard software routines only take single $X$ and $Y$ columns. It fails because ordinary least squares assumes $X$ is perfectly known with all variance localized in $Y$. When $X$ carries experimental error $\sigma_x$, the estimated slope suffers attenuation bias and is systematically shifted toward zero. It is detected whenever horizontal error bars are visible on the scatter plot. It is corrected by employing Deming orthogonal distance regression or the York weighted algorithm. Mini-example: if true physics is $y = 2 x$ but $x$ contains substantial random noise, standard OLS regression of $y$ on $x$ returns a slope near $1.6$ instead of $2.0$.

3. Neglecting variance weighting when linearizing physical equations

It seems reasonable because transforming $y = A e^{B x}$ into $\ln y = \ln A + B x$ looks like simple algebra. It fails because logarithmic transformations warp error bars nonlinearly: $\sigma_{\ln y} = \sigma_y / y$. Data points with small physical values of $y$ experience massive uncertainty expansion on the logarithmic scale. It is detected when running unweighted linear fits on logarithmic data spanning several orders of magnitude. It is corrected by assigning statistical weights $w_i = 1/\sigma_{\ln y_i}^2 = y_i^2 / \sigma_{y_i}^2$. Mini-example: in radioactive decay curves, tail points near background may have $\sigma_{\ln y} = 2.0$, while initial points have $\sigma_{\ln y} = 0.01$. Equal weighting lets background noise corrupt the half-life determination.

4. Forcing zero intercept without physical justification

It seems reasonable because elementary textbooks write physical laws as $y = m x$ with zero intercept (e.g. $V = R I$ or $F = m a$). It fails because physical experiments feature amplifier offsets, zero-drift calibration shifts, or static friction thresholds. Forcing $c=0$ distorts the slope to absorb unmodeled instrument offsets. It is detected when software is configured with "intercept = 0" without testing whether an unrestricted fit yields an intercept compatible with zero within $[-\sigma_c, +\sigma_c]$. It is corrected by fitting two unconstrained parameters $(m, c)$ and performing a statistical hypothesis test on $c$. Mini-example: an ammeter with $+0.1 \text{ A}$ zero-shift will cause a $10\%$ error in resistance if the fit is forced through the origin.

5. Polynomial overfitting of noisy measurements

It seems reasonable because adding polynomial degrees monotonically lowers residual sum of squares $\chi^2$. It fails because higher polynomial degrees model noise rather than underlying physics, producing violent unphysical oscillations between data coordinates. It is detected when reduced chi-squared drops below 0.2 ($\chi_\nu^2 < 0.2$) and the fitted curve ripples between points. It is corrected by choosing models grounded in physical differential equations or applying model selection metrics like AIC or BIC. Mini-example: fitting 6 free-fall observations with a 5th-degree polynomial yields $\chi^2 = 0$, but interpolates negative or infinite local accelerations violating gravity.

Quick detection rule

• After every fit, plot normalized residuals $r_i = y_i - f(x_i)$ against $x_i$.
• If residuals trace a "U" or "S" shape, the model is not linear, no matter how close $R^2$ is to 1.

Physical Applications of Data Fitting and Regression

1. Classical mechanics: verification of Newton's second law and inertial mass

Appears when testing $F = m a$ on dynamic linear tracks using load cells and optical photogates, varying applied force and measuring chariot acceleration. It is calculated by running weighted linear regression of acceleration $a$ versus force $F$: $a = \left(\frac{1}{m_{\text{total}}}\right) F - \frac{f_{\text{fric}}}{m_{\text{total}}}$. The result gives total system inertial mass from the reciprocal of slope ($m_{\text{total}} = 1/\text{slope}$) and friction force from the vertical intercept. The critical danger is forcing zero intercept; unavoidable friction introduces a finite negative intercept which, if omitted, artificially overestimates true inertial mass. As a final sanity check, regression-derived mass is cross-checked against gravitational mass measured statically on an analytical balance to verify the equivalence principle.

2. Thermodynamics: absolute zero determination via Charles's law

Appears in constant-volume gas thermometry, measuring pressure $P$ of a trapped ideal gas sample across known temperatures $T$ in Celsius (ice bath, steam, oil baths). It is calculated by fitting $P(T) = m T + c$, where ideal gas physics dictates $P = \frac{n R}{V} (T_C + 273.15)$. The result derives experimental absolute zero by extrapolating the fitted line to zero pressure ($P=0$): $T_{\text{zero}} = -c / m$. The critical danger is extrapolating from a narrow temperature span (e.g. $20^\circ\text{C}$ to $40^\circ\text{C}$), where tiny slope uncertainties $\sigma_m$ drastically magnify extrapolation error bars at $-273^\circ\text{C}$. As a final sanity check, propagated extrapolation uncertainty $\sigma_{T_{\text{zero}}} = |T_{\text{zero}}| \sqrt{(\sigma_c/c)^2 + (\sigma_m/m)^2 - 2\operatorname{cov}(m, c)/(m c)}$ is verified to enclose $-273.15^\circ\text{C}$.

3. Electromagnetism: internal resistance and open-circuit EMF

Appears when testing terminal voltage $V$ of a battery across varying load currents $I$. It is calculated via linear fit $V = \mathcal{E} - r I$, where the vertical intercept delivers open-circuit electromotive force (EMF) $\mathcal{E}$ and slope magnitude provides internal resistance $r$. The result establishes the maximum power transfer point (achieved when load matches internal resistance $R_L = r$). The critical danger is ignoring internal heating at high currents, which causes internal resistance to drift upwards during testing, curving data points at the high-current limit. As a final sanity check, high-current points subject to thermal drift are excluded and residual linearity is confirmed.

4. Optics and atomic physics: Rydberg constant from hydrogen Balmer series

Appears in emission spectroscopy when measuring wavelengths $\lambda$ of visible hydrogen Balmer lines ($H_\alpha, H_\beta, H_\gamma$) using a diffraction grating spectrometer. It is calculated via Rydberg's formula $\frac{1}{\lambda} = R_H \left( \frac{1}{4} - \frac{1}{n^2} \right)$. Defining $y = 1/\lambda$ and $x = (0.25 - 1/n^2)$ for $n = 3, 4, 5, 6$, linear regression extracts slope $R_H$. The result provides the Rydberg constant $R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$, linking atomic spectra to fundamental constants $m_e, e, \hbar, c$. The critical danger is propagating diffraction angles $\theta$ to wavelengths $\lambda = d \sin\theta$ without accounting for the $\cos\theta$ non-uniform uncertainty stretch, assigning improper weights to violet lines. As a final sanity check, fitted intercept is tested against zero and reduced chi-squared confirms Bohr model validity.

5. Astrophysics: Hubble constant determination ($H_0$)

Appears in observational cosmology when mapping radial recession velocity $v$ (via Doppler redshift $z$) against luminosity distance $d$ (via Type Ia supernovae or Cepheid variable stars). It is calculated by applying weighted regression to the Hubble-Lemaître law: $v = H_0 d$. The result determines cosmic expansion rate $H_0 \approx 70 \text{ (km/s)/Mpc}$ and estimates the age of the observable universe as $t_0 \sim 1/H_0$. The critical danger is neglecting local galaxy peculiar velocities induced by gravitational clustering (such as the Great Attractor), which distorts the slope unless the sample is restricted to deep Hubble flow ($d > 50 \text{ Mpc}$), or failing to use the York estimator when both axes carry large relative uncertainties. As a final sanity check, residuals are inspected and compared with cosmic microwave background parameters (Planck satellite).

Guided Practice: Data Fitting and Regression

Level 1: recognition

Propose the variable transformation $(X, Y)$ that linearizes each physical law into $Y = m X + c$:

1. Inverse-square law: $y = \frac{A}{x^2}$.
2. Simple pendulum period: $T = 2\pi \sqrt{\frac{L}{g}}$.
3. Capacitor discharge current: $I(t) = I_0 e^{-t/RC}$.
4. Stefan-Boltzmann blackbody radiation: $P(T) = \sigma A T^4$.

Solution:

1. Set $Y = y$ and $X = 1/x^2$ (or $\ln y = \ln A - 2\ln x$ with $Y = \ln y$ and $X = \ln x$).
2. Square both sides: $T^2 = \left(\frac{4\pi^2}{g}\right) L$. Set $Y = T^2$ and $X = L$ (slope $m = 4\pi^2/g$).
3. Take natural logs: $\ln I = \ln I_0 - \left(\frac{1}{RC}\right) t$. Set $Y = \ln I$ and $X = t$ (slope $m = -1/RC$, intercept $c = \ln I_0$).
4. Set $Y = P$ and $X = T^4$ (slope $m = \sigma A$).

Level 2: direct application

Three data pairs of force versus elongation are recorded for an elastic spring:

• $x_1 = 0.02 \text{ m}, F_1 = 1.95 \text{ N}$
• $x_2 = 0.04 \text{ m}, F_2 = 4.05 \text{ N}$
• $x_3 = 0.06 \text{ m}, F_3 = 5.98 \text{ N}$

Assuming uniform uncertainty $\sigma_F = 0.05 \text{ N}$, compute spring constant $k$ via linear fit $F = k x + c$ and test if intercept $c$ is consistent with zero.

Solution:

1. Sample means: $\bar{x} = (0.02 + 0.04 + 0.06)/3 = 0.04 \text{ m}$. $\bar{F} = (1.95 + 4.05 + 5.98)/3 = 3.9933 \text{ N}$.
2. Deviations: $x_i - \bar{x} = [-0.02, 0.00, +0.02]$. $F_i - \bar{F} = [-2.0433, +0.0567, +1.9867]$.
3. Sums: $\sum (x_i - \bar{x})^2 = 0.0008 \text{ m}^2$. $\sum (x_i - \bar{x})(F_i - \bar{F}) = 0.0806 \text{ N}\cdot\text{m}$.
4. Slope ($k$): $k = \frac{0.0806}{0.0008} = 100.75 \text{ N/m}$
5. Intercept $c$: $c = \bar{F} - k \bar{x} = 3.9933 - (100.75 \times 0.04) = -0.0367 \text{ N}$
6. Standard parameter uncertainties: $\sigma_k = \frac{0.05}{\sqrt{0.0008}} \approx 1.77 \text{ N/m}$ $\sigma_c = 0.05 \sqrt{\frac{1}{3} + \frac{(0.04)^2}{0.0008}} = 0.05 \sqrt{2.3333} \approx 0.076 \text{ N}$ Result: $k = (100.8 \pm 1.8) \text{ N/m}$. Intercept $c = (-0.04 \pm 0.08) \text{ N}$ is fully consistent with zero ($|c| < \sigma_c$).

Level 3: tool selection

You record 50 data points of electrical resistivity versus temperature $\rho(T)$ of a superconductor. Between $100\text{ K}$ and $150\text{ K}$ behavior is linear, but below $95\text{ K}$ resistance abruptly vanishes to zero. How should you extract the normal-state linear temperature coefficient?

Solution: You must truncate the regression domain:

1. Never perform a single global regression across phase transitions.
2. Restrict data to $T \geq 105\text{ K}$ where the metallic normal state is rigorously established.
3. Perform weighted linear regression solely on that sub-domain.
4. Inspect residuals to ensure no superconducting fluctuation tail contaminates the selected linear window.

Level 4: physical interpretation

A linear fit of velocity versus time $v(t) = a t + v_0$ on an air track with $N=20$ points yields $R^2 = 0.998$ but reduced chi-squared $\chi_\nu^2 = 4.8$. Explain this physical discrepancy.

Solution: $R^2 = 0.998$ merely indicates that $99.8\%$ of overall variance follows the general acceleration trend. However, $\chi_\nu^2 = 4.8 \gg 1$ proves that point-by-point deviations from the straight line are nearly 5 times larger than expected from random statistical noise alone. Physical reasons:

1. Quadratic air drag $F_d = -b v^2$ introduces subtle downward curvature.
2. The air track has a slight physical sag or tilt variation along its length.
3. Photogate timer uncertainties were underestimated.

Level 5: intentional mistake

To linearize $y = A e^{-B x}$, a student takes logarithms $Y = \ln y$ and applies ordinary unweighted regression to: $y_1 = 100.0 \pm 1.0 \implies Y_1 = 4.605$ $y_2 = 10.0 \pm 1.0 \implies Y_2 = 2.303$ $y_3 = 1.0 \pm 1.0 \implies Y_3 = 0.000$ The student weights all three transformed points equally. Explain why this creates massive bias.

Conceptual blunder: By error propagation, logarithmic uncertainty is $\sigma_Y = \sigma_y / y$.

• Point 1: $\sigma_{Y_1} = 1.0 / 100.0 = 0.01 \implies w_1 = 10\,000$.
• Point 2: $\sigma_{Y_2} = 1.0 / 10.0 = 0.10 \implies w_2 = 100$.
• Point 3: $\sigma_{Y_3} = 1.0 / 1.0 = 1.00 \implies w_3 = 1$.

In unweighted regression, point 3 (with $100\%$ relative uncertainty) receives the identical weight as point 1 (with $0.01$ uncertainty). Point 1 is actually 10,000 times more reliable. Equal weighting lets noise in the tail corrupt the slope calculation.