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Random Error and Drift by the Numbers
Noise, averaging, correlated noise, random walks, Allan deviation, thermal noise and drift for production measurement, worked through one checkweigher
Separate noise, random walk and drift in production measurements by numbers: averaging, correlated noise, Allan deviation, thermal noise, line speed, recalibration and uncertainty budgets.
Start readingAbout this book
Every measurement wobbles. Weigh the same pack twice and the readings differ in the last digit; leave a sensor running overnight and its zero wanders; run a line faster and its measurements get noisier. Most of the time those wobbles are treated as a nuisance to be filtered out by eye or by a setting nobody remembers choosing. They can instead be measured, separated into their kinds and designed for, and that is what this book does.
The mathematics comes from an unexpected place. The random jiggling of pollen and dust particles in water, Brownian motion, was explained in 1905 by Einstein as a random walk, and refined in 1930 by Ornstein and Uhlenbeck into a process with memory that reverts to its mean. The same two models, the random walk and the mean-reverting process, describe the two kinds of wandering found in almost every sensor, and the thermal agitation that drives Brownian motion also sets a floor under the noise of every resistor. The book works from those ideas to the practical decisions: how long to average, how often to zero, how fast to run, how often to check, and how to state the result with an honest uncertainty.
Prerequisites and how to use the book
It assumes algebra, the idea of a standard deviation and a willingness to use a spreadsheet. Chapters 1 and 2 cover random and systematic error and averaging; Chapters 3 to 5 cover correlated noise, random walks and the Allan deviation that separates them; Chapter 6 the thermal noise floor; Chapter 7 the trade between line speed and precision; and Chapter 8 drift, check intervals and the uncertainty budget. Inspecting to the Drawing covers measurement of parts against tolerances; this book covers the behaviour of the measurement itself.
The running case
The book follows CW-5, a fictional checkweigher at the end of a packing line that fills 500 g packs of dry product at 120 packs a minute. Each pack crosses a weigh conveyor on a single strain-gauge load cell of 1 kg capacity, rated output 2 mV/V, excited at 5 V, read by an analogue-to-digital converter at 1,000 samples a second. Each pack spends about 300 ms on the weigh conveyor; the first 100 ms are lost to the transient as it arrives, leaving a 200 ms averaging window. Between packs, the controller re-zeroes the scale on the empty conveyor. Packs below a set threshold are rejected.
The noise model used throughout is invented for teaching but realistic in form: broadband electrical and mechanical noise of 0.5 g standard deviation per sample; a correlated vibration component of 0.2 g with a correlation time of 20 ms; a slow random walk in the zero of about 0.05 g per square root of a minute; and a temperature-driven span drift that wanders with a standard deviation of 0.15 g and a time constant of 30 minutes. This is an illustration. None of it describes a KEVOS® client, supplier or product.
Source limits and professional responsibility
The supplied 1967 lecture notes develop the mathematics of Brownian motion; they informed the subject, and nothing from them is reproduced. This book is educational. Measuring instruments used for trade in Australia, including checkweighers that determine whether prepackaged goods are compliant, are regulated under the National Trade Measurement Regulations 2009 and National Measurement Institute requirements, and their verification, settings and records must follow those rules and the maker’s instructions. Measurements that affect safety or compliance should be designed and checked by a competent person.
Who this book is for
Quality, process and instrumentation engineers, machine builders and technicians who specify, tune or troubleshoot sensors, weighing systems and production measurements.
What you will learn
- Distinguish random error, systematic error, random walk and drift, and recognise each in measured data.
- Predict how averaging reduces noise, and calculate the effective number of samples when noise is correlated.
- Calculate how a random walk accumulates, and use Allan deviation to choose an averaging or zeroing time.
- Estimate the thermal noise floor of a strain-gauge bridge and compare it with other noise sources.
- Trade line speed against precision, set check intervals for drift, and combine the components into an uncertainty budget.
Inside this book
Chapters
- 01Random error, systematic error and driftSort CW-5's measurement errors into random scatter, fixed offsets and slow wander, recognise each in a run of readings, and see why adjusting after every reading makes the scatter worse.
- 02Averaging and the square-root ruleApply the square-root rule to CW-5's 200-sample window, see why its measured scatter is nearly three times what the rule predicts, and use window length to cancel conveyor vibration.
- 03Correlated noise and the effective number of samplesModel CW-5's vibration as an Ornstein–Uhlenbeck process, calculate how many independent samples its averaging windows really contain, and find why a short zeroing window added noise to every pack.
- 04Random walks: errors that accumulateModel CW-5's wandering zero as a random walk, see how random and systematic errors build up in totals, and find the zero-tracking window that balances noise against wander.
- 05Allan deviation: one chart for every time scaleCalculate the Allan deviation of CW-5's zero from a logged signal, read white noise, random walk and drift from its slopes, and use its minimum to choose averaging and zero-tracking times.
- 06The thermal noise floorCalculate the Johnson–Nyquist noise of CW-5's strain-gauge bridge and the thermal motion of its load cell, compare them with amplifier and mechanical noise, and see why more excitation backfired.
- 07Speed against precision on the lineFind how CW-5's measurement noise grows with line speed, how many packs it wrongly accepts and rejects at each speed, and what guard bands and an uncorrected offset do to those numbers.
- 08Drift, check intervals and the uncertainty budgetModel CW-5's span drift as a mean-reverting process, set the interval between check-weight tests, chart the checks, and combine every component into an uncertainty budget with an expanded uncertainty.
Apply This in Practice
Use these KEVOS® tools to try the method from this reading with your own numbers.
Applies this method
EWMA Control Chart Check (Weighted Moving Average, Control Limits, Small Shifts)
Builds the exponentially weighted moving average chart this chapter names as more sensitive to slow drift, for the averages of your check readings.
Related Tools and Examples
Useful alongside
CUSUM Control Chart Check (Tabular Upper and Lower Sums, Decision Interval, First Signal)
Applies a tabular CUSUM to a series of check results, another way to catch the slow drift this chapter charts before it reaches a limit.
Related to Drift, check intervals and the uncertainty budget
Useful alongside
Standard Deviation Chi-Square Check (Process Spread Against a Target Sigma, Confidence Interval)
Tests whether an observed scatter of check readings, like this chapter’s twenty-reading checks, is larger than the standard deviation a noise model predicts.
Related to Averaging and the square-root rule
Useful alongside
Two-Variance F Test (Compare the Spread of Two Machines, Batches or Methods)
Tests whether two spreads, such as the twenty-reading checks before and after this chapter’s converter upgrade, differ by more than sampling scatter.
Related to Averaging and the square-root rule
References and source notes
The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 27 October 2026.
- Edward Nelson, Princeton University Press. Dynamical Theories of Brownian Motion (1967). Supplied source, read as a subject map of the Wiener process, Gaussian processes and the Ornstein–Uhlenbeck theory. The book is in copyright; none of its text, derivations, figures or exercises is reproduced. This book applies the same mathematics to measurement noise with its own case, figures and exercises.
- Albert Einstein, Annalen der Physik 322(8), 549–560. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen (1905). Origin of the square-root-of-time growth of a random walk, used in Chapter 4.
- G. E. Uhlenbeck and L. S. Ornstein, Physical Review 36, 823–841. On the theory of the Brownian motion (1930). Origin of the mean-reverting process used in Chapters 3 and 8 to model correlated noise and slow drift.
- David W. Allan, Proceedings of the IEEE 54(2), 221–230. Statistics of atomic frequency standards (1966). Origin of the two-sample (Allan) variance used in Chapter 5.
- J. B. Johnson and H. Nyquist, Physical Review 32, 97–109 and 110–113. Thermal agitation of electricity in conductors; and Thermal agitation of electric charge in conductors (1928). Measurement and theory of thermal noise in resistors, used in Chapter 6 for the noise floor of a strain-gauge bridge.
- Joint Committee for Guides in Metrology (BIPM and partner organisations). JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement (2008, corrected 2010). Framework for the uncertainty budget in Chapter 8: standard uncertainties, combination by root-sum-square and expanded uncertainty with a coverage factor.
- National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods (current). Background for the averaging, normal-distribution and control-chart methods used in Chapters 2, 7 and 8.
- NIST, CODATA values of the fundamental physical constants. Boltzmann constant (current). Exact value 1.380 649 × 10⁻²³ J/K used for thermal noise in Chapter 6.
- Commonwealth of Australia; administered by the National Measurement Institute. National Trade Measurement Regulations 2009 (2009, as amended). Named in Chapters 1 and 7 as governing the measurement of prepackaged goods for trade; no requirements are quoted. Real trade-use checkweighing must follow them and NMI guidance.