Calibration Curve Calculator (unknown concentration, uncertainty, LOD & LOQ)

Understanding the calibration curve calculator — concept and principle

A calibration curve turns an instrument reading, such as absorbance, into a concentration. This calculator fits a least-squares straight line to your standards, reads the unknown sample backwards, and gives its standard uncertainty, a 95% confidence interval and the limits of detection and quantitation.

The fitted straight line turns a reading of 0.380 back into a concentration of 5.065, well inside the 0–10 range.

How it is calculated

Line y = a + b·x by least squares: b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)²; a = ȳ − b·x̄.
Unknown x₀ = (y₀ − a) ÷ b, multiplied by any dilution factor.
Residual standard deviation s = √(Σ residuals² ÷ (n − 2)).
u(x₀) = (s ÷ |b|) × √(1/k + 1/n + (y₀ − ȳ)² ÷ (b² Σ(x − x̄)²)), with k replicate readings.
95% interval = t(n − 2) × u(x₀); LOD = 3.3 s ÷ b; LOQ = 10 s ÷ b.

The principle

Many instruments respond in proportion to concentration over a working range (for example Beer–Lambert absorbance). Measuring standards of known concentration lets you find that proportion. The line is uncertain, and so is the unknown's reading; the uncertainty formula combines both, and it grows the farther the unknown is from the centre of the standards. LOD and LOQ express the smallest concentrations that can be detected or quantified reliably given the scatter of the line.

Worked example

Default inputs: standards 0, 2, 4, 6, 8, 10 with responses 0.002, 0.151, 0.297, 0.452, 0.598, 0.751; unknown 0.380; 1 reading; dilution 1.

Slope b ≈ 0.07487; intercept a ≈ 0.0008095; R² ≈ 0.9999.
x₀ = (0.380 − 0.00081) ÷ 0.07487 ≈ 5.065.
Standard error of x₀ ≈ 0.03358; 95% interval ≈ ± 0.09324 (t with 4 degrees of freedom).
LOD ≈ 0.1026; LOQ ≈ 0.3109.
The unknown is inside the standards, so no extrapolation.

Each point is the calculator re-run with only “Response of the unknown sample (mean)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Response of the unknown sample (mean)	Unknown concentration
0.095	1.258
0.19	2.527
0.285	3.796
0.38	5.065
0.475	6.333
0.57	7.602
0.76	10.14

How to read the result

Report the result as about 5.07 ± 0.09 in the standards' units at 95% confidence. A high R² alone does not prove linearity; also look at the residuals for curvature. Results below the LOQ should be reported as "detected, not quantified" or with caution.

Common mistakes

Extrapolating beyond the highest or lowest standard.
Relying on R² without checking residuals.
Forgetting the dilution factor.
Using too few standards; at least five or six spread across the range are common practice.

Keep in mind: Assumes a linear response with constant scatter. Regulated methods must follow their validation guidelines.

Frequently asked questions

How do you use a calibration curve to find an unknown concentration?

Fit the line to the standards and compute x₀ = (y₀ − intercept) ÷ slope.

What is a good R² for a calibration curve?

Values above 0.995 are common for well-behaved linear methods, but residual plots matter as well.

How are LOD and LOQ defined here?

LOD = 3.3 × residual standard deviation ÷ slope; LOQ = 10 × the same ratio, as in ICH Q2 guidance.

Sources: ICH Q2(R2), Validation of Analytical Procedures (2023) · Miller, J. N. & Miller, J. C., Statistics and Chemometrics for Analytical Chemistry, 6th ed. Pearson

Checked 9 Nov 2026. NIST/SEMATECH e-Handbook of Statistical Methods, §2.3.6.7.3 “Comparison of check standard analysis and propagation of error”: calibrated value X′ = (Y′ − â)/b̂ and its propagation-of-error uncertainty from s_Y′, s_â, s_b̂ and their covariance. LOD = 3.3σ/S and LOQ = 10σ/S with σ the residual standard deviation of the regression line: ICH Q2 validation guideline (standard values).

Your numbers