PE2-4.1 Measurement Characteristics & Error Analysis

What this is and why it exists

A measurement without an error estimate is not a measurement. It is a number.

This topic is what stands behind every figure the rest of the course produces. It is also short on glamour and long on consequence, because the same vocabulary appears on every datasheet you will ever read.

The vocabulary

• Sensor — the element responding to the quantity being measured.
• Transducer — the device converting that response into another form, usually electrical.
• Range — the span of values a device can measure.
• Resolution — the smallest change it can distinguish.
• Sensitivity — how much its output changes for a given change in input.
• Repeatability — how closely repeated measurements of the same thing agree.
• Accuracy — how close a measurement is to the true value.
• Linearity — how closely the response follows a straight line.
• Dead band — a range of input over which the output does not change at all.
• Backlash — lost motion when a mechanism reverses direction.
• Systematic error — an error that repeats identically every time.
• Random error — an error that varies unpredictably between measurements.
• Loading effect — the measurement changing the quantity being measured.

The mental model

Two distinctions carry most of the weight here, and both are routinely muddled.

The first is sensitivity against resolution. Sensitivity is a slope: how much output you get per unit of input. Resolution is a step: the smallest change you can see. A device can be very sensitive and have poor resolution. A large output swing read by a coarse display tells you nothing finer than that display allows.

The second is accuracy against repeatability. Accuracy is about truth. Repeatability is about consistency. A device can be one without the other, and the case worth picturing is a device that gives the same wrong answer every time. It is perfectly repeatable and entirely inaccurate, and no amount of repeating will reveal the problem.

That leads to the most important idea in the topic. Errors divide into three kinds, and what you can do about each differs completely.

Gross errors are mistakes: a misread scale, a wrong connection. They are found by care and by cross-checking.

Random errors vary unpredictably from one measurement to the next, sometimes high and sometimes low. Averaging many measurements reduces them, because they partly cancel, and the improvement follows the square root of the number of readings.

Systematic errors are the dangerous ones. They repeat identically every time, so averaging does nothing at all. A thousand readings with a mis-calibrated instrument give a thousand identically wrong answers and a very confident-looking average. Systematic error can only be removed by calibration or by correction, never by repetition.

Error propagation is the practical skill. When a result is computed from several measurements, their uncertainties combine into the result's uncertainty, and how they combine depends on the calculation. Two useful patterns are worth carrying. For a sum or difference, the absolute uncertainties combine. For a product or quotient, the relative uncertainties do. And in most real chains one instrument dominates the total, so improving any of the others is wasted effort until that one is improved.

Loading effect is the error people forget entirely. Connecting a voltmeter draws a little current, which changes the voltage you came to measure. Putting a thermometer in a small sample warms or cools it. The act of measuring disturbs the thing measured. That is why input impedance appears among the specifications: a high input impedance is a promise to disturb the circuit only slightly.

The topic ends by making the vocabulary numerical. Take repeated readings, compute their mean and their spread, and the spread quantifies the random error. Quote a result as a value and an uncertainty, and the number becomes a measurement.

What you should now be able to explain or do

Distinguish sensor from transducer, sensitivity from resolution, and accuracy from repeatability. Say why averaging removes random error and does nothing to systematic error. Propagate uncertainty through a sum and through a product. Explain loading effect and connect it to input impedance. Quote a measurement with an uncertainty derived from repeated readings.

Check yourself