Understand the relationship
The reasoning behind the result
The standards define the ruler
b = Σ(xᵢ − x̄)(yᵢ − ȳ)/Σ(xᵢ − x̄)²; a = ȳ − bx̄
A calibration pairs known concentrations x with measured signals y. Ordinary least squares chooses slope b and intercept a to minimize the sum of squared vertical signal residuals. Concentration is treated as the known explanatory quantity, and all entered observations receive equal weight.
Keeping replicate rows preserves their contribution to the fit. It also lets you see scatter at the same concentration. This implementation does not fit a weighted, nonlinear or errors-in-both-axes model; choose a method appropriate to the experiment before interpreting its outputs.
Invert first, then correct the dilution
cmeasured = (ysample − a)/b; coriginal = cmeasured × DF
Subtract the intercept before dividing by the slope. This returns concentration in the diluted material that produced the signal. Multiplying by the dilution factor then returns the original-sample concentration.
The signal must follow the same blank correction and instrument settings used for the standards. Subtracting an additional blank only from the sample, or applying the dilution factor twice, changes the analytical question and produces a biased result.
A good R² is not a validation certificate
residualᵢ = yᵢ − (a + bxᵢ); sresidual = √[Σ(residualᵢ²)/(n − 2)]
Residuals show the direction and size of the difference between observed and fitted signal. A curved residual pattern can reveal a mismatch that a high coefficient of determination conceals. Two calibration points determine a line exactly but leave no residual degrees of freedom, so the residual SD is reported as unavailable.
The range status compares back-calculated diluted concentration with the entered calibration interval. Extrapolation is shown explicitly; it is not silently clipped to the nearest standard. Negative back-calculated concentrations are retained as signed estimates and are not interpreted as negative amounts of material.
When this replaces a separate assay page
Linear protein, Bradford and other concentration assays can use this workflow when their actual protocol supports a straight-line calibration over the entered range. The assay name alone does not justify a linear response.
A saturating immunoassay or a protocol specifying a logistic calibration needs its own model. The related ELISA tool handles an explicitly entered four-parameter logistic equation; it does not inherit this straight-line assumption.
Follow the numbers
From four standards to the original sample
- Standards (0, 0.05), (1, 0.25), (2, 0.45) and (3, 0.65) define y = 0.20x + 0.05, with x in mg/mL.
- A sample signal of 0.45 gives (0.45 − 0.05)/0.20 = 2 mg/mL in the measured diluted sample.
- A dilution factor of five gives 2 × 5 = 10 mg/mL in the original sample. The measured 2 mg/mL lies inside the 0–3 mg/mL calibration interval.
The range check applies before dilution correction, because the instrument measured the diluted sample.
Quick guide
How to use this calculator
- Choose measured standards or an existing linear equation, and name the concentration unit.
- Enter each standard measurement separately; do not silently average away replicate scatter.
- Enter sample signals processed in the same way as the standards and the dilution factor for each sample.
- Check calibration residuals and each sample's range status before using the original-sample concentration.
Calculation method
Calculation and interpretation
Connect actual calibration measurements to sample concentrations, with the fitted equation, residuals and range checks visible.
signal = slope × concentration + intercept; original concentration = (signal − intercept)/slope × dilution factor
Worked example
From four standards to the original sample
The range check applies before dilution correction, because the instrument measured the diluted sample.
signal = slope × concentration + intercept; original concentration = (signal − intercept)/slope × dilution factor
Supported inputs
Precision and limits
Measured response, not experimental validation
Calculations do not establish assay specificity, recovery, matrix compatibility, sample identity or biological meaning. Use standards, blanks and controls appropriate to the actual method.
Units and dilution
Use one concentration unit throughout a calibration. A dilution factor is original concentration divided by measured diluted concentration; multiply by it once when reporting the original sample.
Linear model only
Unweighted ordinary least squares with a fitted intercept. No forced-zero fit, 4PL/5PL fitting, confidence interval, outlier deletion or experimental acceptance threshold is inferred.
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