01 / CHAPTER 11
Graphs, data, and uncertainty
Use measurements to infer motion while reporting resolution and uncertainty honestly.
ANCHOR EQUATION
Sample standard deviation
In symbols: σ = √[Σ(xᵢ − x̄)²/(N − 1)]
A compact way to report the spread of repeated measurements.
A graph is a measurement model
A position trace is not the motion itself; it is sampled evidence with finite pixel size, frame rate, calibration, and tracking error. Differentiation amplifies high-frequency noise, so inference requires both a physical model and an uncertainty estimate.
Fit before differentiating
For repeated measurements xᵢ at times tᵢ, a straight-line fit x = x₀ + vt estimates constant velocity from the slope. Residuals rᵢ = xᵢ − x_fit(tᵢ) show whether the model misses curvature. A quadratic fit x = x₀ + v₀t + ½at² estimates acceleration; the coefficient of t² is ½a, not a.
Suppose a 10-frame fit gives x(t) = 0.42 + 1.80t + 0.35t² m. The inferred initial speed is 1.80 m/s and acceleration is 0.70 m/s². If residuals are ±0.02 m, reporting a = 0.70 ± 0.05 m/s² is more honest than many decimal places.
Precision is not accuracy
A high-frame-rate video can still be inaccurate if the scale marker is tilted or the camera moves. If residuals show a pattern, the error is likely model error rather than random measurement noise.
Work it: Create a three-point dataset from x = 0.5 + 2t + 0.4t² at t = 0, 1, 2 s. Fit the quadratic by hand and check the predicted x at 1.5 s; explain one source of experimental uncertainty.