The Narrow-Span Lever Arm
The Narrow-Span Lever Arm
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
The Narrow-Span Lever Arm
A soil moisture sensor is calibrated at just 45% and 55% moisture — only 10% apart on a 0-1000 ADC scale (raw readings 450 and 550) — and after deployment a 10-count slip at the high reference turns the true 0% point into 4.1% and the true 100% point into 95.0%. This audit asks the question that lever-arm invites: carried to full precision, how much of that swing is really a 9.09% gain shift, and how much shrinks away once the calibration span is widened?
Companion to the chapter Calibration Span Error and Validation — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is shifted high reference. The middle card applies this page's rule. The green card is fitted gain. Walk the arrows once: 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 shifted high reference, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 560 counts.
- 2
Name the relationship. gain = 10 percentage points / (high raw - 450)
- 3
Substitute with units. 10 / (560 - 450) = 0.090909%/count
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change shifted high reference
Try Predict the direction of gain = 10 percentage points / (high raw - 450). Test another shifted high reference, then compare fitted gain.
Observe The same reference slip changes the fitted slope more when the calibration span is narrow. Reset shifted high reference to 560 and compare fitted gain.
Explain The same reference slip changes the fitted slope more when the calibration span is narrow.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Fit references (450, 45%) and (550, 55%), then shift the high raw point to 560 before pressing Check audit.
Gain moves from 0.100000 to 0.0909091 and offset rises to 4.090909; widening the span to 800 counts shrinks the same slip near 1.23%.
A fixed 10-count reference error rotates the fitted line around the other point, so a shorter calibration baseline amplifies slope error.
Ada: The panel above says a 10-count slip at the high reference turns a true 0% into 4.1% and a true 100% into 95.0%. Let me carry every digit and check what a narrow span really costs.
- Correct fit through
(450, 45%)and(550, 55%): gain(55 - 45) / (550 - 450) = 10 / 100 = 0.100000, offset45 - 0.1 x 450 = 0. - Slip the high raw by +10 counts to 560: gain
10 / 110 = 0.0909091, offset45 - 0.0909091 x 450 = 4.090909. - True 0% (raw 0) now reads
0.0909091 x 0 + 4.090909 = 4.09%. - True 100% (raw 1000) now reads
0.0909091 x 1000 + 4.090909 = 95.00%. - The gain shifted by
(0.1 - 0.0909091) / 0.1 = 9.09%. The chapter’s quick “error / span” rule (10 / 100 = 10%) is the first-order estimate; carried exactly it is 9.09%. Widen the span to 800 counts and the same 10-count slip becomes10 / 810 = 1.23%(the chapter’s 1.25% shorthand).
The short reference span acts as a lever arm: the fit pivots on the low point, so a fixed reference error swings the far end in proportion to how short the baseline is. Bracket the operating range and the lever shortens; crowd the two references together and a single-count reference mistake is amplified into a several-percent field error at the extremes.
Every number above is taken from the chapter’s own material and re-derived step by step.
Technical boundaries: This span-sensitivity exercise assumes a linear sensor and exact low reference; it excludes noise on both points, nonlinear response, drift, saturation, hysteresis, and reference-percentage error.