Calibration Line (Linear Regression)
x = concentration / standard value, y = instrument signal (e.g., absorbance). Separator: semicolon or whitespace.
Analytical context
The calibration line links concentration and measured signal. Linear regression computes the best-fit line \(y = m x + b\) from standards, enabling unknown sample back-calculation and model-quality checks.
\[ y = m x + b \]
Typical use cases:
- Photometry (absorbance vs concentration)
- Electrochemical sensors (signal vs concentration)
- Fast quality control with standard series
Crosslink: For method-validation limits, use the LOD/LOQ Calculator.
Formulas (MathJax)
\[ m = \frac{n\sum x_i y_i - (\sum x_i)(\sum y_i)}{n\sum x_i^2 - (\sum x_i)^2} \]
\[ b = \bar{y} - m\bar{x} \]
\[ \hat{y} = m x + b \]
\[ R^2 = \frac{\left(n\sum x_i y_i - (\sum x_i)(\sum y_i)\right)^2}{\left(n\sum x_i^2 - (\sum x_i)^2\right)\left(n\sum y_i^2 - (\sum y_i)^2\right)} \]
\[ x = \frac{y-b}{m} \]
Symbol legend
- \(x\): known standard value (e.g., concentration)
- \(y\): measured signal (e.g., absorbance, peak area)
- \(m\): slope of calibration line
- \(b\): intercept (signal offset)
- \(\hat{y}\): predicted signal
- \(R^2\): coefficient of determination (fit quality)
- \(n\): number of calibration points
- \(\bar{x},\bar{y}\): mean values of x and y
Detailed examples
Example 1 (build calibration):
From standards, regression gives \(m\approx0.108\) and \(b\approx0.011\).
Calibration function: \(y=0.108x+0.011\).
From standards, regression gives \(m\approx0.108\) and \(b\approx0.011\).
Calibration function: \(y=0.108x+0.011\).
Example 2 (unknown sample):
Measured signal \(y=0.300\).
Back-calculation: \(x=(y-b)/m\approx2.67\).
Measured signal \(y=0.300\).
Back-calculation: \(x=(y-b)/m\approx2.67\).
Example 3 (prediction):
For \(x=2.5\), predicted \(\hat{y}\approx0.281\).
Useful for expected signal checks of control standards.
For \(x=2.5\), predicted \(\hat{y}\approx0.281\).
Useful for expected signal checks of control standards.
Example 4 (fit quality):
When \(R^2\) is close to 1, linearity is strong.
Lower \(R^2\) suggests checking outliers, matrix effects, or non-linear behavior.
When \(R^2\) is close to 1, linearity is strong.
Lower \(R^2\) suggests checking outliers, matrix effects, or non-linear behavior.
Deeper insight
Intercept and blank correction
A non-zero intercept \(b\) may indicate blank signal, instrument offset, or matrix effect. This should be considered in method validation and reporting.
Number of calibration points
More standards generally improve robustness against measurement noise. Calibration points should cover the full working range of unknown samples.
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