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Calibration Curve Calculator (Standard Curve)

Fit a calibration line by least squares with slope, intercept, R² and residuals, and read an unknown's concentration from its signal with a 95% interval.

Formula

y=slope⋅x+intercepty = \text{slope}\cdot x + \text{intercept}
slope=∑(x−xˉ)(y−yˉ)∑(x−xˉ)2\text{slope} = \dfrac{\sum (x-\bar{x})(y-\bar{y})}{\sum (x-\bar{x})^2}
x0=y0−interceptslopex_0 = \dfrac{y_0 - \text{intercept}}{\text{slope}}
sx0=s∣slope∣1m+1n+(y0−yˉ)2slope2∑(x−xˉ)2s_{x_0} = \dfrac{s}{|\text{slope}|}\sqrt{\dfrac{1}{m} + \dfrac{1}{n} + \dfrac{(y_0-\bar{y})^2}{\text{slope}^2\sum (x-\bar{x})^2}}
xx
concentration of a standard
yy
measured signal (absorbance, fluorescence, peak area)
xˉ, yˉ\bar{x},\ \bar{y}
means of the standards' concentrations and signals
R2R^2
fraction of the variation in signal explained by the line
ss
residual standard deviation of the points about the line, with n − 2 degrees of freedom
kk
number of replicate readings averaged for the unknown

How it works

A standard curve converts an instrument signal into a concentration. Standards of known concentration are measured, a line is fitted through them by ordinary least squares, and the line is inverted to find the concentration that would produce the unknown's signal.

R² close to 1 means the points lie near a straight line, but it does not prove the relationship is linear. Always look at the plot and the residuals: curvature at the high end is a sign that the assay is saturating, and residuals that run positive, then negative, then positive mean the wrong model was chosen.

The concentration read from the line is an estimate with an uncertainty, which comes from the scatter of the standards and the position of the unknown relative to their mean. It is smallest at the centre of the calibration range and widens towards the ends. The page gives the 95% interval for the unknown, and shows how averaging replicate readings of it narrows the interval.

Worked example

Six standards from 0 to 1000 give signals from 0.000 to 0.790. An unknown reads 0.350.

  1. Least-squares fit: y = 0.000787 x + 0.00634, with R² = 0.9998.
  2. x = (0.350 − 0.00634) / 0.000787 = 436.7.
  3. Residual SD s = 0.00506 signal units; the standard error of x is 6.95, so the 95% interval is 417.4 to 455.9.

The unknown is 436.7 (95% interval 417.4 to 455.9), in the same unit as the standards, and sits inside the calibrated range.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

  • The signal is proportional to concentration across the range of the standards.
  • Error lies in the signal, not in the concentrations of the standards.
  • Standards and unknowns were measured under the same conditions and in the same matrix.

Common mistakes

  • Extrapolating beyond the highest or lowest standard. Dilute the sample so it falls inside the curve.
  • Forgetting to multiply the result by the dilution factor of the sample.
  • Relying on R² alone. A curved relationship can still give an R² above 0.99.
  • Forcing the line through the origin when the standards do not support it, which biases the slope.

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Related equipment

Service documentation, failure modes and parts for the instruments this calculation is used with.