Example Data Table
This sample set mimics a nearly linear detector response. Replace with your own standards.
| Standard # | Concentration | Measured signal |
|---|---|---|
| 1 | 0.0 | 0.02 |
| 2 | 1.0 | 0.48 |
| 3 | 2.0 | 0.98 |
| 4 | 5.0 | 2.45 |
| 5 | 10.0 | 4.91 |
Formula Used
A calibration curve links an input quantity x (such as concentration, field strength, or photon flux) to a measured instrument response y (such as voltage, absorbance, or count rate). This calculator fits common models by least squares.
- Linear: y = m x + b
- Linear through origin: y = m x
- Quadratic: y = a x² + b x + c
- Logarithmic: y = a ln(x) + b (requires x > 0)
- Power: y = a x^b (requires x > 0 and y > 0)
For linear fits, the slope and intercept are computed from standard sums (Σx, Σy, Σxy, Σx²). For quadratic fits, the coefficients are found by solving the 3×3 normal equations. The fit quality is reported as R² and a standard error in y.
How to Use This Calculator
- Choose a model that matches your detector behavior.
- Enter calibration standards as x–y pairs in the points table.
- Optionally enter an unknown signal to solve for x.
- Optionally enter an unknown x to predict the expected signal.
- Press Solve Calibration Curve to view results above the form.
- Use Download CSV or Download PDF for reporting.
Good practice: include blanks, replicate points, and avoid extrapolation beyond the calibration range.
Detector response and the purpose of calibration
Calibration converts instrument signal into a physical quantity. Standards x paired with response y define sensitivity and background so unknowns can be interpreted. Many labs use 5–7 standards plus a blank, spanning the expected range and bracketing real samples.
Choosing standards and concentration spacing
Cover the full range you expect from samples, not only the middle. Space points tighter near the detection limit and around any curvature. A practical set is 0, low, mid, high, and an upper check. Repeat the blank and a mid‑point to estimate repeatability and catch drift.
Linear model when physics predicts proportionality
Linear behavior is common when output is proportional to input, such as low‑light photodiodes or linear amplifiers. In y = m x + b, m is sensitivity and b is background. Linear fits invert cleanly for unknowns and stay robust under moderate noise.
When forcing the curve through zero is appropriate
Use a through‑origin fit only when zero input must produce zero output after proper baseline correction. Forcing b = 0 with a nonzero blank biases m and shifts solved x. Compare both linear options and keep the one with smaller, pattern‑free residuals.
Nonlinear curvature and quadratic compensation
Saturation and gain compression create curvature at high input. A quadratic y = a x² + b x + c can approximate gentle bending over a limited region. Use it when residuals show systematic curvature under a linear fit. Add standards where the curve bends and avoid using the quadratic outside the fitted range.
Logarithmic and power behavior in sensors
Log and power models capture wide dynamic‑range behavior. Log uses y = a ln(x) + b and needs x > 0. Power uses y = a x^b and needs x > 0 and y > 0. Use them when multiplicative scaling dominates.
Residuals, R², and deciding if the fit is acceptable
This solver reports R² and a standard error in y. High R² is useful, but residual patterns matter more. Many workflows target R² ≥ 0.995 with residuals within a few times the noise. If residuals grow with x, improve coverage or restrict the valid range.
Using solved x and predicted y for real measurements
Enter an unknown y to solve x, or enter x to predict y for planning exposure, dose, or acquisition time. Always check the displayed range. If results fall outside, dilute, re‑measure, or extend the standards rather than extrapolating—especially for nonlinear models.
FAQs
How many calibration points should I use?
Use at least 5 points for linear work and 7–10 for nonlinear behavior. Add replicates at the blank and mid‑range to estimate repeatability and reduce sensitivity to a single outlier.
What does R² tell me in a calibration curve?
R² summarizes how much variance the model explains, but it can look high even with systematic bias. Always inspect residuals and ensure the curve is physically plausible across the full range.
When should I choose “linear through origin”?
Choose it only when a true zero input must produce zero output and your blank is effectively zero. If background exists, allowing an intercept usually gives less biased estimates for unknowns.
Why does the solver warn about values outside the range?
Solving or predicting outside the fitted x‑range is extrapolation. Small errors in coefficients can create large errors beyond the standards, especially for nonlinear models. Dilute samples or extend your standards instead.
Can I use the log or power model with zero values?
No. The log model requires positive x, and the power model requires positive x and y. If zeros occur, consider baseline correction, an offset model, or restrict the fit to strictly positive standards.
What should I do with an outlier point?
First verify the standard preparation and measurement. If an outlier is due to a known mistake, remove it and re‑measure. Otherwise, keep it and improve repeatability with replicates and better range coverage.
Why export both CSV and PDF?
CSV is best for spreadsheets and analysis pipelines, while PDF is convenient for reports and lab notebooks. Exporting preserves the chosen model, parameters, and the measured‑versus‑fit table for traceability.