Calibration Curve Calculator: Unknown Concentration from a Standard Curve, with R², LOD and LOQ

How it works

Fit y = m·x + b by least squares: m = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)², b = ȳ − m·x̄ · unknown x = (ȳᵤ − b) ÷ m · sᵧ/ₓ = √(Σ(y − ŷ)² ÷ (n − 2)) · s(x) = (sᵧ/ₓ ÷ m) √(1/k + 1/n + (ȳᵤ − ȳ)² ÷ (m² Σ(x − x̄)²)) with k replicate readings · CI = ± t(n − 2) × s(x) · LOD = 3.3σ ÷ m, LOQ = 10σ ÷ m (σ = sᵧ/ₓ or the blank SD)

A calibration curve turns an instrument reading into a concentration. You measure standards of known concentration, fit a straight line through the points by ordinary least squares, and read the unknown off the line: x = (signal − intercept) ÷ slope. Paste the standards as two columns (concentration, signal) straight from a spreadsheet; commas, tabs or semicolons work, and a decimal comma is fine when columns are separated by a tab or semicolon. Enter one or several replicate readings of the unknown; they are averaged. Besides the concentration, the calculator reports how good the calibration is and how far to trust the answer: R² and r, the standard errors of the slope and intercept, the scatter about the line (sᵧ/ₓ), and a confidence interval for the unknown using the textbook formula and Student’s t with n − 2 degrees of freedom. The limits of detection and quantitation follow the ICH Q2 approach, 3.3σ ÷ slope and 10σ ÷ slope, with σ taken from the regression or from blank replicates if you enter them. The residual table back-calculates each standard so you can spot an outlier, and you are warned when the unknown is outside the calibrated range or below the LOQ. Under More options you can force the line through the origin, change the confidence level and apply a dilution factor. If you only know the equation, choose “Known equation”. Unlike the Beer–Lambert calculator, which uses one molar absorptivity, or the qPCR efficiency calculator, which only reads the slope of a dilution series, this tool fits your own standards. It fits a straight, unweighted line only; for curved or heteroscedastic data use a quadratic or weighted fit.

Worked example

Six standards of 0, 0.1, 0.2, 0.3, 0.4 and 0.5 give signals of 0.00, 12.36, 24.83, 35.91, 48.79 and 60.42. The means are x̄ = 0.25 and ȳ = 30.385, Sxx = 0.175 and Sxy = 21.124, so the slope is 21.124 ÷ 0.175 = 120.71 and the intercept is 30.385 − 120.71 × 0.25 = 0.2086: y = 120.71x + 0.2086, with R² = 0.99974. The scatter about the line is sᵧ/ₓ = 0.4033. The unknown was read three times (29.32, 29.16 and 29.51, mean 29.33), so x = (29.33 − 0.2086) ÷ 120.71 = 0.2413. Its standard deviation is (0.4033 ÷ 120.71) × √(1/3 + 1/6 + (29.33 − 30.385)² ÷ (120.71² × 0.175)) = 0.00236, and with t = 2.776 for 4 degrees of freedom the 95% interval is 0.2413 ± 0.0066. LOD = 3.3 × 0.4033 ÷ 120.71 = 0.0110 and LOQ = 0.0334.

Sources: Harvey, Analytical Chemistry 2.1, §5.4 Linear regression and calibration curves (LibreTexts) · Lab-Training: LOD/LOQ calculator (ICH Q2(R2) 3.3σ/S and 10σ/S) · Omni Calculator: Calibration curve calculator · Best-Calculators: Calibration curve (linear fit and unknowns)

Frequently asked questions