Math Bridge: Calibration Span as a Lever

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Math BridgeSensorsStruggle-friendly runway

Why a narrow calibration span acts like a lever

One thread from two points to slope and offset, then to the chapter's 4.1% and 95.0% endpoint errors.

Phoebe, the physics guidePhoebe guides
The one targetDerive how a small reference error spreads across a range.
The chapter case45%/55% at raw 450/550, with 10 bad counts.
What it buys youChoose wide references and test holdout points.

See the relationship before changing it

Set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only high calibration raw count, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 550 counts.

  2. 2

    Name the relationship. span leverage = (high raw - 450) / 10 percent

  3. 3

    Substitute with units. (550 - 450) / 10 = 10.00 counts/%

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change high calibration raw count

Try Predict the direction of span leverage = (high raw - 450) / 10 percent. Test another high calibration raw count, then compare raw span per known percent.

550 counts
Chapter baseline
Raw span per known percent

Observe Moving the high point changes fitted span while the low fixture stays fixed. Reset high calibration raw count to 550 and compare raw span per known percent.

Explain Moving the high calibration point farther from the fixed low point spreads the known physical span across more raw counts, increasing counts per percentage point linearly.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only high calibration raw count moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only high calibration raw count. It is not a field prediction.

1. A calibration line has two jobs

The rule y=mx+b converts a raw ADC count x into a reported percentage y. The slope m says how much percentage one count represents. The offset b says what the line reports at raw zero.

Phoebe: Two reference points force one straight line. If either point moves, both the slope and the line's intercept can move.

2. Fit the correct points

1

Subtract known valuesΔy=55−45=10 percentage points.

2

Subtract raw countsΔx=550−450=100 counts.

3

Divide for slopem=Δy/Δx=10/100=0.1000% per count.

4

Solve the offsetb=45−0.1000(450)=0.

3. Move only the high raw point

A 10-count error changes the measured high point from 550 to 560. The known reference is still labelled 55%.

1

New raw span560−450=110 counts.

2

New slopem=10/110=0.0909% per count.

3

New offsetb=45−0.0909(450)=4.09%.

4. Try the reference error

m=(y_high−y_low)/(x_high+e−x_low); b=y_low−mx_low; y=mx+b

TryMove the high-reference error from 0 to 20 counts and watch both endpoints move.

Measured high raw
Fitted gain
Report at raw 0
Report at raw 1,000

ObserveAt 10 bad counts, the slope becomes 0.0909%/count; raw 0 reads 4.09% and raw 1,000 reads 95.00%.

ExplainThe widget refits m and b from the same two-point equations shown above, then evaluates that line at both endpoints.

Technical boundaries.

The span comparison keeps the low reference fixed and treats the calibration as a straight line with one erroneous point. It does not include reference uncertainty, repeated readings, independent validation points, nonlinearity, or proof of the allowed operating range.

reference uncertainty
Needs separate evidence
repeat readings
Needs separate evidence
independent holdout points
Needs separate evidence
nonlinearity checks
Needs separate evidence
an allowed operating range
Needs separate evidence

Use field evidence or a deeper model before release.

5. Evaluate the endpoints

1

At raw zeroy=0.0909(0)+4.09=4.09%, which the chapter rounds to 4.1%.

2

At raw 1,000y=0.0909(1,000)+4.09=95.00%.

The fit is exact at the two points it was told to trust, yet wrong far away. That is why a fit coefficient is not release evidence.

6. Widen the baseline

The same 10-count error is 10% of a 100-count span but only 1.25% of an 800-count span. References near 10% and 90% bracket the chapter's 20–80% operating range, so the system interpolates instead of extrapolating from the middle.

7. Check yourself

What is the correct slope through 450→45% and 550→55%?
Answer: (55−45)/(550−450)=10/100=0.1000% per count.
Why does a 10-count error make the new slope smaller?
Answer: The same 10 percentage points are divided by a wider measured span of 110 counts.
Why add holdout points?
Answer: The two fit points must lie on the fitted line; separate points reveal endpoint error and curvature.
Honesty boundary.

The span comparison keeps the low reference fixed and treats the calibration as a straight line with one erroneous point. It does not include reference uncertainty, repeated readings, independent validation points, nonlinearity, or proof of the allowed operating range.

45%
Low calibration reference percentage
55%
High calibration reference percentage
450
Raw count at low reference
550
Raw count at high reference
10-count error
Error in high calibration reading
0–1,000 raw scale
Full raw-count scale being extrapolated
4.1%
Erroneous calibrated output at raw zero
95.0%
Erroneous calibrated output at raw full scale
10% relative narrow-span error
Raw-count error relative to narrow calibration span
1.25% wide-span comparison come from the chapter
Raw-count error relative to wide calibration span

The page assumes one erroneous high raw point and an otherwise linear sensor; it does not claim two-point calibration removes curvature.

Return to the chapter with the maths unpacked →