Measurements · Module 1 of 10
Basic Concepts and Calibration
A measurement is an estimate of a true value, and it is only as trustworthy as its calibration. This module fixes the vocabulary of error and accuracy and shows how calibration ties an instrument's output to a known standard.
Readiness check
This module opens the course. Tick only what you can do closed-notes.
- Subtract a true value from a reading.
- Compute a relative error as a percentage.
- Find a slope from two points.
- Recall that a sensor turns a physical quantity into a signal.
- Distinguish repeatable from correct.
The core idea
A measurement estimates a true value, and its error is the reading minus that true value. Calibration applies known inputs from a standard and builds the calibration curve, whose slope is the static sensitivity. Accuracy, closeness to the true value, is separate from precision, the repeatability.
error e = reading − true valuerelative error = e / true valuestatic sensitivity K = Δoutput / ΔinputEvery instrument reports a number that stands in for a physical quantity it can never know exactly, so measurement is estimation. The gap between the reading and the true value is the error, and expressing it as a fraction of the true value gives the relative error, the form that lets you compare instruments. We learn an instrument's behaviour by calibration: apply a sequence of known inputs, traceable to accepted standards, and record the outputs. Plotting output against input gives the static calibration curve, and its slope is the static sensitivity K, how much output you get per unit input. The curve also reveals the range, the linearity, and an error band. A crucial distinction runs through all of this: accuracy is how close a reading is to the true value, while precision is how tightly repeated readings agree with each other. An instrument can be precise but inaccurate, clustering tightly around the wrong value, which calibration against a standard is exactly what corrects. Everything later in the course, dynamic response, statistics, and uncertainty, rests on this calibrated, error-aware view of a measurement.
The skills, taught in order
Five skills set the language the whole course uses.
1.1 The general measurement system
A measurement system has a sensor that responds to the quantity, a transducer that converts it to a signal, and output stages that condition and display it. Naming the stages clarifies where error and delay enter.
1.2 True value, error, accuracy, precision
Error is reading minus true value. Accuracy is closeness to the true value; precision is repeatability. Errors split into systematic, a consistent offset, and random, scatter. Calibration targets the systematic part.
| Term | Meaning | Fixed by |
|---|---|---|
| Accuracy | closeness to the true value | calibration |
| Precision | repeatability of readings | better instrument or averaging |
| Sensitivity K | output per unit input | set by design |
Accuracy and precision are independent; an instrument can have one without the other.
1.3 Calibration and sensitivity
A static calibration applies known inputs and records outputs, building the calibration curve. Its slope is the static sensitivity K = Δoutput / Δinput, which converts a future reading back into the measured quantity.
1.4 Range, span, and standards
The range is the interval of input the instrument accepts; the span is its width. Calibration inputs come from standards traceable to national references, which is what makes a reading defensible rather than arbitrary.
1.5 The experimental test plan
Good data starts before the instrument: decide what to measure, what to hold fixed, and what to randomize so uncontrolled effects do not bias the result. A test plan turns measurement into evidence.
Engineering connection: calibrating a load cell against known masses gives its sensitivity in mV per newton and its error band, the numbers a stress test later relies on.
Worked example 1: error and accuracy
Against a reference standard of 10.0 (normalised units), an instrument reads 9.6. Find the error and the relative error.
- ProblemFind the error and relative error for the reading in Figure 1.
- Given / findReading 9.6, true value 10.0. Find e and the relative error.
- AssumptionsThe standard's value is the accepted true value.
- Modele = reading − true; relative = e / true.
- Equationse = 9.6 − 10.0relative = e / 10.0
- Solvee = −0.4; relative = −0.4 / 10.0 = −4%.
- CheckThe reading is low by 0.4 out of 10, which is 4 percent, a systematic offset calibration could correct.
- ConclusionThe instrument reads 4 percent low here; a calibration correction would shift future readings up by that amount.
Worked example 2: static sensitivity
A temperature sensor is calibrated: an input span of 0 to 50 degrees Celsius produces an output of 0 to 100 mV. Find the static sensitivity and the predicted output at 30 degrees Celsius.
- ProblemFind the sensitivity and the 30 degree output for the calibration in Figure 2.
- Given / findInput span 0 to 50 degrees, output span 0 to 100 mV. Find K and output at 30 degrees.
- AssumptionsLinear calibration over the range.
- ModelK = Δoutput / Δinput; output = K × input.
- EquationsK = 100 / 50V = K × 30
- SolveK = 2 mV/°C; V = 2 × 30 = 60 mV.
- Check30 degrees is 60 percent of the 50 degree span, and 60 percent of 100 mV is 60 mV, matching.
- ConclusionThe sensor gives 2 mV per degree, so 30 degrees reads 60 mV, the calibration made quantitative.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Accuracy equals precision | Tight but offset readings called accurate | "Close to true, or just repeatable?" | Accuracy is closeness to true; precision is repeatability. |
| Trusting without calibration | Readings used with no reference | "Against what standard?" | Calibrate against a traceable standard. |
| Sensitivity confused with range | Slope and interval mixed up | "Slope, or width of inputs?" | Sensitivity is the slope; range is the input interval. |
| Assuming linearity | Extrapolating a curved response | "Is the calibration actually straight?" | Check the calibration curve before assuming a line. |
Practice ladder
An instrument reads 48.5 when the true value is 50.0. Find the error and relative error.
Show answer
e = 48.5 − 50.0 = −1.5; relative = −1.5 / 50 = −3%.
A pressure sensor gives 0 to 10 V over 0 to 200 psi. Find its sensitivity and output at 120 psi.
Show answer
K = 10 / 200 = 0.05 V/psi; output = 0.05 × 120 = 6 V.
A displacement sensor has a static sensitivity of 4 mA/mm over a 0 to 25 mm range. Find the output span.
Show answer
Span = K × range = 4 mA/mm × 25 mm = 100 mA.
Design a calibration for a bathroom scale: list the standards, the input points, and the checks you would run.
What good work looks like
Use certified reference masses spanning the range (for example 0, 20, 40, 60, 80, 100 kg), record the scale reading at each, plot output against known mass to find sensitivity and linearity, repeat to assess precision, and compute the error at each point against the standard. A good answer names traceable standards and separates accuracy from repeatability.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Calibrate one real instrument against a reference: build a calibration curve, report the sensitivity, and state the error band.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Write the error.
e = reading − true value.
2. Accuracy versus precision?
Accuracy is closeness to the true value; precision is repeatability.
3. What is static sensitivity?
The slope of the calibration curve, output per unit input.
4. Why calibrate against a standard?
To make readings traceable and correct systematic error.
5. Systematic versus random error?
Systematic is a consistent offset; random is scatter.
Textbook mapping
This module follows Figliola and Beasley, Theory and Design for Mechanical Measurements, 5th edition. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| The general measurement system | Figliola and Beasley, Section 1.2, General Measurement System |
| Calibration and sensitivity | Figliola and Beasley, Section 1.4, Calibration |
| Standards and the test plan | Figliola and Beasley, Sections 1.3 and 1.5 |
Section numbers refer to Figliola and Beasley, 5th edition. Any edition with the same chapter titles is equivalent for study.