Two-point sensor calibration and residual error

Fit two reference points, then use span validation to identify nonlinear calibration error.

Physics Phoebe: fit the references, then test an unused reading., your practice guide

Physics Phoebe: fit the references, then test an unused reading.
Predict the reading, then compare it with the measurement.

Python 3 in your browser (JupyterLite)

Python · no install

Fit two reference points, then use span validation to identify nonlinear calibration error.

Tier 2 · Web · paste-in setup · No account

Version tested: Python 3.12.7 / Pyodide 0.27.6 in JupyterLite 0.6.4; Chromium 148.0.7778.96; captureSource playwright:jupyterlite. Date: 2026-10-08.

Open the notebook in your browser and run each Python cell; no install or account is needed.

Three ways to run: use JupyterLite here with no install; run main.py locally from the downloadable lab folder; or open the same notebook in Google Colab.

Open in your browser (new tab)

Steps

Screens captured against Python 3 in your browser (JupyterLite) Python 3.12.7 / Pyodide 0.27.6 in JupyterLite 0.6.4; Chromium 148.0.7778.96; captureSource playwright:jupyterlite on 2026-10-08; the tool may have moved on — the text steps are the contract.

  1. Step 1

    Do
    In the JupyterLite notebook, run step 1 to inspect the synthetic equation and the two reference readings.
    You will see
    SYNTHETIC temperature fixture; seed=607; no random draws; raw(T)=1.4+0.88*T+0.0015*(T-50)^2; reference_C raw_units
    Why it matters
    The fixture and seed make the input reproducible.

  2. Step 2

    Do
    In the JupyterLite notebook, run step 2 to fit gain and offset from the two endpoints.
    You will see
    Two-point fit: corrected_C=gain*raw+offset; raw span=88.000; reference span=100.000 C; gain=1.136364 C/raw_unit; offset=-5.852273 C
    Why it matters
    A linear fit must interpolate the two references exactly.

  3. Step 3

    Do
    In the JupyterLite notebook, run step 3 to test the corrected values across the span.
    You will see
    Independent validation grid: endpoints were fit, interior was not; true_C raw corrected_C residual_C; 0 5.15 0.00 0.00
    Why it matters
    Unused interior points test whether the line generalizes.

  4. Step 4

    Do
    In the JupyterLite notebook, run step 4 to plot residual error versus true temperature.
    You will see
    Residual error plot data (corrected minus true); true_C residual_C plotted scale: each # ~0.5 C; 0 0.00
    Why it matters
    A residual plot reveals curvature hidden by endpoint checks.

  5. Step 5

    Do
    In the JupyterLite notebook, run step 5 to compare interior and endpoint error.
    You will see
    RESULT CARD: two-point calibration on synthetic readings; endpoints residuals=0.000, 0.000 C; interior mean |error|=3.125 C
    Why it matters
    Span validation is required before accepting a correction.

Chapter checks

These questions refer to the chapter’s examples. Use the return links to review their answers.

  1. A sensor reads 20% when the true value is 10%, and reads 80% when the true value is 90%. What is the gain (slope) coefficient?

    Return to the chapter’s knowledge check

  2. You need to calibrate a soil moisture sensor that will measure between 20% and 80% moisture in a greenhouse. Which reference points should you use?

    Return to the chapter’s knowledge check

Return to Sensor Calibration: Math and Validation · Browse Labs