Calculator
Example Data Table
This sample uses a near-linear sensor response with light noise.
| Reference x | Measured y |
|---|---|
| 0 | 0.010 |
| 1 | 0.520 |
| 2 | 1.020 |
| 3 | 1.500 |
| 4 | 2.030 |
| 5 | 2.510 |
Formula Used
Calibration curve fitting is done using least‑squares regression. Choose a model that matches the measurement physics and operating range.
Linear
y = a + b·x
b = (nΣxy − ΣxΣy) / (nΣx² − (Σx)²)
a = ȳ − b·x̄
Through Origin
y = b·x
b = Σxy / Σx²
Use only when theory requires zero offset.
Quadratic
y = a + b·x + c·x²
Coefficients are solved from normal equations using 3×3 elimination.
Fit quality metrics
- SSE = Σ(y − ŷ)²
- R² = 1 − SSE / SST, where SST = Σ(y − ȳ)²
- RMSE = √(SSE / df)
How to Use This Calculator
- Enter reference values as x and measured readings as y.
- Select a curve model that matches your instrument behavior.
- Optionally add prediction inputs to convert future readings.
- Press Generate Curve to see results above the form.
- Download CSV for spreadsheets or PDF for quick sharing.
FAQs
1) When should I use a quadratic curve?
Use it when the response bends with the range, such as saturation effects. Validate with residuals: if curvature removes patterns and improves R² without overfitting, it is a better choice.
2) What does RMSE tell me in calibration?
RMSE estimates the typical prediction error in the same units as y. Smaller RMSE means the fitted curve explains your readings more tightly, assuming the input data is stable and representative.
3) Why is “through origin” sometimes risky?
Forcing a zero intercept can hide real offsets from wiring, drift, or bias. Use it only if physics demands it and your low-end points support it, otherwise accuracy can degrade at small signals.
4) How many points do I need?
Use at least 5–7 points across the working range when possible. Two points define a line, but more points reduce sensitivity to noise and help detect nonlinearity or outliers.
5) What if my x values repeat?
Repeated x values are allowed, but if all x values are the same, a model cannot be fit. If repeats exist, consider averaging y at each x or measuring more distinct levels.
6) How do I interpret residuals?
Residuals are y − ŷ. Randomly scattered residuals suggest a good model. Residuals that trend or curve indicate model mismatch, drift, or a need for more careful sampling.
7) Can I estimate x from a measured y?
Yes, for linear models the tool can invert y to estimate x. For curved models, inversion may yield multiple solutions. If you need quadratic inversion, use numerical solving with a defined range.
8) Does a high R² guarantee accuracy?
Not always. R² measures fit within your dataset, not long-term stability. For accurate calibration, control temperature, warm-up, and measurement conditions, and re-check with independent verification points.