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 minimises the sum of the squared vertical gaps, where each gap, called a residual, is the up or down distance from one point to the line. For an ordinary fit with a free intercept, positive and negative residuals cancel, so their plain sum is zero.
Reading an unknown reverses the line. Suppose the fit is absorbance equals 2.0 times concentration plus 0.10, and a drink sample reads 1.30. Subtract the intercept and divide by the slope: 1.30 minus 0.10 is 1.20, divided by 2.0 is 0.60. Quote 0.60 plus or minus its fitted uncertainty, smallest at the standards mean for a plain unweighted fit with exact 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.
The rewritten reading was written for this dataset. The reading 'R-squared intuition' comes from the supplied Khan Academy article record. License terms for that material have not been verified.
More to read
Checking unusual calibration measurements
A residual is the measured y value minus the value predicted by a fitted line. It records the vertical difference, positive above the line and negative below it. For a suitable linear model, residuals should scatter around zero without a clear pattern. A large residual flags a possible outlier.
Investigate unusual measurements before changing the dataset. A recording error or spilled sample may justify correcting or excluding a value. A genuine measurement may instead reveal a curved relationship or a region with different behavior. An awkward result alone is no reason to remove it. Check how it was measured and whether it belongs in the analysis.
Plot residuals against x to assess the fit. A widening fan suggests increasing variability. A curved pattern suggests that a straight line misses the relationship. An isolated large residual identifies a measurement needing closer investigation.
R-squared intuition
Imagine you had to guess the absorbance of a sample with no line at all. Your best guess would be the average of all the readings, and you would often be wrong by a lot. Now fit a least squares line and use it to make the guess instead. The line uses the concentration to sharpen the prediction, so your errors shrink. The number r squared tells you how much of that original error the line removes. If r squared is 0.81, the line has removed 81 percent of the guesswork error, and 19 percent is still left over. If r squared is close to 1, the points track the line tightly and the fit is strong. If it is close to 0, the line barely helps and the points are mostly scattered. Remember that r squared measures straight line fit only. A curved relationship can have a low r squared even when the data follows a beautiful smooth pattern. That is your cue to transform the data or pick a different model, not to force a straight line where one does not belong.
2 · Watch
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Where it sits
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Nothing; this is a starting point.
Optional connections
16 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.