Least-Squares Calibration and the Uncertainty of a Result Read From It
A calibration line is fitted, not drawn by eye, and the fit carries its own uncertainty into every result read off it. Knowing that uncertainty is what lets a result be quoted with a range rather than a number.
What a learner can do afterwards
- Fits a line by least squares and explains what quantity the fit minimises
- Uses the residuals to judge whether a straight line is the right model at all
- Calculates the uncertainty in a concentration read from the line and says where on the line it is smallest
- Explains why extrapolating beyond the standards is unsafe even when the line looks perfect
1 · Read
A calibration line is fitted, never drawn by eye. Least squares chooses the line that makes the sum of the squared vertical gaps as small as it can be, where each gap, called a residual, is the up or down distance from one point to the line. Positive and negative residuals always cancel, so their plain sum is zero for the fitted line.
Reading an unknown reverses the line. Suppose the fit is absorbance equals 2.5 times concentration plus 0.10, and a drink sample reads 1.35. Subtract the intercept and divide by the slope: 1.35 minus 0.10 is 1.25, divided by 2.5 is 0.50. Quote 0.50, and remember the result is surest near the middle of your standards, where the line is best pinned down.
Before trusting any reading, plot each residual against concentration and look at the shape. A formless cloud means the straight line fits; a clear U shape curved at both ends means it does not, and the linear fit should be rejected. The honest move is to fit a curve instead or to narrow the range until straight holds.
Never extrapolate beyond your standards, even when the line looks perfect. The fit only knows the stretch it was trained on, and chemistry loves to bend just past the last point.
Fit by least squares, read unknowns backwards through the line, check the residuals, and stay inside the standards.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.