Least-Squares Calibration and the Uncertainty of a Result Read From It · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Reading a result off a fitted line

Science · Chemistry · ages 19-20
Name ______________________   Date ____________

  1. For an ordinary least-squares line with a free intercept, the positive and negative residuals cancel out, so the plain sum of the residuals is zero.

    Circle one:   True   False

  2. You fit a calibration line through five standard readings by least squares. What exactly does the least-squares method make as small as it can?

    • The sum of the squared vertical gaps
    • The horizontal gaps between the points and the line
    • The largest single vertical gap only
    • The slope of the line

  3. What exactly does the least squares fit make as small as it can?

    • Summed squared vertical gaps
    • The horizontal gaps between the points
    • The slope of the line

  4. What is a residual?

    • The slope of the fitted line
    • The vertical distance from one point to the line
    • The largest standard value

  5. After fitting a line, you plot each residual against concentration. The points form a clear U shape, curved at both ends. What does this tell you?

    • The straight line is the wrong model, the relationship is curved
    • The fit is perfect and the residuals are random
    • One standard was measured wrong and should be deleted
    • The uncertainty of the line is zero

  6. Another fit gives reading equals 1.5 times concentration plus 0.20. A sample reads 1.70. Type the concentration.

    Answer: ______________

  7. A colorimeter calibration gives the least-squares line absorbance = 2.5 times concentration + 0.10. A drink sample reads an absorbance of 1.35. What is its concentration in the units of the standards? Round to two decimal places.

    Answer: ______________

  8. Your standards run from 1 to 10 ppm and the calibration line fits them almost perfectly. A sample reads far above the top standard, at an absorbance matching about 25 ppm. What should you do?

    • Extend the line and report 25 ppm without any comment
    • Trust the line because it fits the standards so well
    • Result unreliable; dilute into range
    • Average 25 ppm with the top standard

  9. Standards were run at 0, 2, 4, 6, and 8 ppm for a plain unweighted straight-line fit with even scatter and exact standards. On your calibration line, where is the uncertainty of a concentration read back from the absorbance the smallest?

    • At 4 ppm, the mean of the standard concentrations
    • At 0 ppm, the lowest standard
    • At 8 ppm, the highest standard
    • It is the same everywhere on the line

  10. Residuals curve and your unknown sits at the top end of the standards. What should you do?

    • Trust the line since it looks straight
    • Delete the curved residuals
    • Fit a curve or narrow; distrust end
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading a result off a fitted line W1-mt_L08U-Extq_-s1

  1. True · A property of this fit is that the residuals add to zero. That is why the method squares them instead of just adding them.
  2. The sum of the squared vertical gaps · Least squares chooses the line that minimises the sum of the squared vertical residuals, the up and down gaps between the data points and the line.
  3. Summed squared vertical gaps · Least squares minimises the squared up and down distances.
  4. The vertical distance from one point to the line · Each residual is one point's up or down miss.
  5. The straight line is the wrong model, the relationship is curved · Random scatter around zero means a straight line is fine. A pattern like a U shape means the true relationship bends, so a straight line is the wrong model.
  6. 1 · Subtract 0.20 and divide by 1.5 to get 1.0.
  7. 0.5 · Rearrange the fitted line to solve for concentration: (1.35 - 0.10) / 2.5 = 0.50.
  8. Result unreliable; dilute into range · A fit only tells you about the range you measured. Outside it you are guessing, because the relationship may bend or change.
  9. At 4 ppm, the mean of the standard concentrations · The readback uncertainty grows as you move away from the mean of the standards. It is smallest right at the centre, near 4 ppm here, for a plain unweighted fit with even scatter and exact standards.
  10. Fit a curve or narrow; distrust end · A curved pattern plus an edge reading is the riskiest combination.
Worksheet · LightMySky