Uncertainty in measure
The Error Function
Reading the propagation equation term by term — where it comes from, why it is squared, and what it means
Building Intuition Before Equations · July 2026
By Joseph P. McFadden Sr.
Engineering Fellow, Zebra Technologies | Professor of Mechanical Engineering, Fairfield University
Companion materials
This reader stands on its own. If you would rather watch or listen, the same material exists in two other forms:
• The slide deck, as a PDF — twenty-two slides, with the diagrams referred to here.
• The recorded walkthrough — the narrated version, about forty minutes.
This began as a lecture for my mechanics laboratory students at Fairfield. I have written it out because the argument turns out to matter well beyond a teaching lab, and because one line of a course manual deserved more than the four minutes it usually gets.
One Line
Somewhere in every laboratory manual there is a line that students copy into a spreadsheet and never look at again.
In ours it sits in Section II.C.3, under the heading Estimation of Error, and it is introduced like this:
"The data analysis for the ME experiments assumes the same probability for all errors in measurement, therefore it can be shown that the total error for a function of independent variables is given by…"
And then the equation:
e = √[ ( ∂R/∂X₁ · e₁ )² + ( ∂R/∂X₂ · e₂ )² + … + ( ∂R/∂Xₙ · eₙ )² ]
where e₁, e₂ … eₙ are the uncertainties in X₁, X₂ … Xₙ, and R = f (X₁, X₂ … Xₙ)
That is the whole thing. One line of algebra, a definition of terms, and then the manual moves on to a worked example about copper wire.
I want to spend this reader taking it apart, because in more than forty years of engineering, teaching, and failure investigation I have met a great many people — good ones, careful ones, people I would trust with a design — who could use that equation flawlessly and could not tell you what it meant.
A correct number, handed to someone who does not understand it, is worth very little. In a failure investigation it can be worth less than nothing, because it carries a confidence it has not earned.
So the goal here is not to teach you to use the equation. If you have taken a laboratory course you can already use it. The goal is to teach you to read it: to know where every symbol came from, why the squaring is there and is not negotiable, and where the whole apparatus quietly stops being true.
Three questions organize what follows.
1. Where does the equation come from?
2. Why is it squared?
3. When does it stop being true?
And before we start, notice a phrase in the manual's own sentence that almost everybody reads straight past: a function of independent variables. That is not a description. That is an assumption, stated out loud, in the middle of the setup — and it is the assumption most likely to be false in an undergraduate laboratory. We will come back to it.
Chapter One
The Situation You Are Actually In
Here is the fundamental problem of measurement, stated plainly:
You almost never measure the thing you care about. You measure several other things, and then you calculate the thing you care about.
You do not measure elastic modulus. You measure a force, a gauge length, a diameter, and an extension, and then you calculate a modulus.
You do not measure impact energy. You measure a mass, an arm length, a release angle and a swing-through angle, and you calculate energy.
You do not measure the resistance of a copper wire at thirty degrees Celsius. You measure its resistance at twenty degrees, you measure the temperature, you look up a coefficient, and you calculate.
Every one of those inputs arrives with a wobble in it. It came through an instrument, was read by a person, in a room with a temperature, on a day when somebody may have been tired. And the question the error function answers — the only question it answers — is this:
If each of my inputs wobbles a little, how much does my answer wobble?
Think of it as a bridge with three spans. On the left, what you measured: your X's, each carrying its own plus-or-minus. In the middle, the model: R = f(X), which is your claim about how the system behaves. On the right, what you report: R, plus or minus e.
The error function is the middle span. And it is worth being blunt about this: there is no way around it. There is no path from four measured uncertainties to one reported uncertainty that does not go through some version of this equation. You can evaluate it by hand, you can let a spreadsheet do it, you can do it numerically without any calculus at all — but you cannot skip it and still write a plus-or-minus you are able to defend.
Chapter Two
Anatomy: Every Symbol Has a Job
Before deriving anything, let us name the parts.
Symbol What it is What it does
e the answer The combined uncertainty in R. One number, in the units of R. This is what goes after the plus-or-minus sign.
∂R/∂Xᵢ the sensitivity How hard R responds to a nudge in Xᵢ, with everything else held still.
eᵢ the uncertainty How well you know Xᵢ.
(…)² the squaring Turns each contribution into a variance so that they can legitimately be added.
√ the un-squaring Returns the total to the units of R. Ohms, not ohms squared.
The two middle rows are the ones worth dwelling on, because the distinction between them is the single most useful idea in this reader.
The sensitivity is a pure property of the model. It has nothing whatever to do with your instruments — it would be the same number whether you measured with a bathroom scale or a national standard.
The uncertainty is a pure property of your instrument and your technique. It has nothing whatever to do with the model.
They come from completely different worlds — the physics on one side, the equipment on the other — and the equation multiplies them together.
That product is the only thing that matters. Neither factor alone tells you anything, and Chapter Eight contains a worked example in which the variable with by far the largest sensitivity turns out to be almost entirely irrelevant.
Chapter Three
Where It Comes From
The equation is not handed down. It falls out of a Taylor expansion in three steps, and only the first involves any calculus.
Step One: Expand
Suppose every input is off by a little. Write down what the model produces at the wrong inputs, expanded about the right ones:
R (X₁+e₁, X₂+e₂, …) = R (X₁, X₂, …) + Σ (∂R/∂Xᵢ) eᵢ + ½ ΣΣ (∂²R/∂Xᵢ∂Xⱼ) eᵢeⱼ + …
Three pieces. The first is the true value — what you would have gotten with perfect inputs. The second is a weighted sum of the input errors, where the weights are the first derivatives. The third is everything else: second derivatives and beyond.
Subtract the true value from both sides, and what remains on the right is the error in your output:
ΔR ≈ Σ (∂R/∂Xᵢ) eᵢ
Read that literally, because it is the entire physical content of the error function: the output error is a weighted sum of the input errors, and the weights are the partial derivatives. Everything after this is bookkeeping on that one sentence.
Step Two: Drop the Tail
We threw away the second-order terms and beyond. Here is the justification.
Each successive term in the expansion carries a higher power of the error. If the errors are small compared with the values, those powers collapse fast.
Term Scales as If e is 1 % of X Relative size
1st order e 1 % 0.01
2nd order e² 0.01 % 0.0001
3rd order e³ 0.0001 % 0.000001
A one percent measurement error puts a distortion of about one hundredth of one percent into your answer. That is why the linear term is allowed to stand alone.
But hear what that actually says. The collapse depends on the errors being small compared with the values. This is a choice we are making, not a law of nature — an excellent choice for a well-behaved experiment, and a bad one near a turning point. Chapter Thirteen is about the cases where it fails.
Step Three: From Errors to Variances
This is the step that produces the squaring, and it is the intellectual centre of the whole subject.
ΔR is a sum of random quantities. You cannot add random quantities by adding their typical sizes, because they are as likely to cancel one another as to reinforce one another. What you add is their variance.
Two ordinary rules do the work:
• The variance of a sum of independent quantities is the sum of their variances.
• When a constant multiplies a random quantity, the constant comes out of the variance squared.
Apply both to ΔR = Σ (∂R/∂Xᵢ) eᵢ and you get:
Var (ΔR) = Σ (∂R/∂Xᵢ)² Var (eᵢ)
Write each standard uncertainty as the square root of its variance, and take the root of both sides:
e = √ Σ (∂R/∂Xᵢ)² eᵢ²
That is the manual's equation. Not approximately, not a version of it — that is the line in Section II.C.3, arrived at from first principles.
So Why Is It Second Order?
Because variance is a second moment.
Spread has always been measured by squaring deviations — that is what a second moment is. Errors combine as squares because that is the only way random quantities are permitted to combine.
The square is not a mathematical convenience that somebody chose for tidiness. It is the definition of how spread works. Once you see that, the equation stops looking like a formula and starts looking like an inevitability.
Chapter Four
What a Partial Derivative Actually Is
Students disengage at the partial derivatives, decide the topic is calculus rather than engineering, and copy the formula without asking what it does. So let us be concrete.
A partial derivative answers exactly one question: if I nudge this one input while holding everything else still, how far does the output move?
It is a slope. And a slope is a conversion rate.
So many ohms per degree. So many square centimetres per centimetre. So many gigapascals per micron. That is all it is.
The Rectangle
Suppose you are computing the area of a rectangle: area equals length times width. Your length measurement is off by one centimetre. How wrong is your area?
You cannot answer that without knowing the width.
If the plate is ten centimetres wide, being off by one centimetre in length costs you ten square centimetres. If the plate is a hundred centimetres wide, the identical one-centimetre slip costs you a hundred square centimetres.
Same error. Ten times the damage.
That factor — the width — is ∂A/∂L. It is a magnification factor, and nothing more mysterious than that. The damage is set by the geometry, not by the measurement.
The Tangent, and What Sits Just Beyond It
Picture the model as a curve: output on the vertical axis, one input on the horizontal. At your operating point, draw the tangent. Nudge the input sideways by e, and the tangent tells you how far the output moves.
Steep curve, and the same measurement error costs you more. Flat curve, and the same error barely registers.
Now look further along that picture, out at the edges, where the tangent line and the curve pull apart. That gap is the second-order term. Right now it is small, which is exactly why we were allowed to throw it away in Chapter Three. In Chapter Thirteen I will show you two situations where it is not small at all.
Chapter Five
The Geometry of the Equation
If you square two things, add them, and take the square root, you have done something you have known how to do since you were about fourteen. You have found a hypotenuse.
Two Inputs
Draw a right triangle. One leg is the damage from the first input — its sensitivity times its uncertainty. The other leg is the damage from the second. The hypotenuse is your answer.
Suppose those two contributions happened to be 3 and 4.
Method Arithmetic Result
In quadrature √(3² + 4²) 5
Added straight 3 + 4 7
Forty percent larger, and it would essentially never happen.
Adding the legs means walking around two sides of the triangle. Nobody crosses a room that way. The errors point in different directions — that is what independent means — so they combine like perpendicular vectors, and the result is the direct path.
Pythagoras is not an analogy here. For two inputs, a² + b² = c² is the error function. Same equation, same reason.
Three Inputs, and Then More
Add a third input and the triangle becomes a box. The answer is the body diagonal — the direct distance from one corner to the opposite corner. Three perpendicular contributions, combined once.
With n inputs you are working in n dimensions, which you cannot draw. Do not let that trouble you. Nothing in the algebra changed; only the number of terms in the sum. The picture runs out before the mathematics does, and that is a limitation of paper, not of the idea.
Quadrature Versus the Plain Sum
Most textbooks tell you that quadrature is correct and the plain sum is naive. I want to offer a better version of that.
Using the copper-wire contributions we are about to compute in full:
Method Result What it assumes
Quadrature 0.0305 Ω The errors are independent
Plain sum 0.0451 Ω All three go wrong the same way, at once
The plain sum is not wrong. It is the correct answer for perfectly correlated errors. Quadrature is the correct answer for perfectly independent ones. Those are the two ends of a range, and every real experiment sits somewhere between them.
Which means the equation you were handed assumes one particular end of that range. Hold that thought for Chapter Eleven.
And the Square Root
One more piece of structure, quickly. When you add the squares you are holding a variance, and its units are the units of R, squared. Ohms squared. Newtons squared. Gigapascals squared. Nothing physical has those units, and you cannot write it after a plus-or-minus sign.
Take the root and it becomes a standard deviation, in the units of R, and it is the only form you can legally report.
Notice that this is the same move you already make with standard deviation, every time, without thinking about it: square the deviations so they cannot cancel, average them, take the root to get back to something quotable. The error function is that identical operation, run across variables instead of across repeated readings.
Squaring to combine, rooting to report. Every statistic you will ever use is built that way.
Chapter Six
The Worked Example, Every Term Shown
The manual's own example is a genuinely good one, because the answer contains a surprise. Every figure below is printed in Section II.C.3; none of it is my reinterpretation.
The Situation
You want the resistance of a copper wire at thirty degrees Celsius, and you cannot measure it there — your bench is at room temperature and you have no oven. So you measure the resistance at twenty degrees, you measure the actual temperature, you look up the temperature coefficient of resistance for copper, and you calculate.
R = R₀ [ 1 + α (T − 20°) ]
Input Value Uncertainty Relative
R₀ 6.000 Ω ± 0.018 Ω 0.30 %
α 0.004000 /°C ± 0.000040 /°C 1.00 %
T 30.0 °C ± 1.0 °C —
A word about that temperature uncertainty, because it is the one students query. One degree is not a percentage of anything. It is what a typical bench thermocouple and readout actually deliver once you account for probe placement, thermal contact, and drift. Taking one percent of thirty degrees would give three tenths of a degree, which for most laboratory setups is optimistic to the point of dishonesty. Choose the number your equipment has earned, not the number that makes the arithmetic pretty.
The Nominal Value
R = 6 × [1 + 0.004 × 10] = 6 × 1.04 = 6.24 Ω
The Three Sensitivities
• ∂R/∂R₀ = 1 + α(T − 20) = 1.04. Every ohm of error in R₀ becomes 1.04 ohms of error in R.
• ∂R/∂α = R₀(T − 20) = 60. An error of one thousandth in α becomes six hundredths of an ohm in R — a sixtyfold magnification.
• ∂R/∂T = R₀α = 0.024 Ω/°C. A full degree of temperature error costs twenty-four thousandths of an ohm.
Stop on those three numbers before reading on: 1.04, 60, and 0.024.
If you were asked right now which input to go and measure more carefully, almost everyone points at α. Sixty is an alarming number sitting next to 0.024.
Hold that thought.
The Contributions
Input Sensitivity Uncertainty Product Squared Share of variance
R₀ 1.04 0.018 Ω 0.01872 0.00035044 37.6 %
α 60 0.000040 0.00240 0.00000576 0.6 %
T 0.024 1.0 °C 0.02400 0.00057600 61.8 %
Total 0.00093220 100 %
e = √0.00093220 = 0.0305 Ω, which is 0.49 % of the nominal value.
R = 6.24 ± 0.03 Ω (0.49 %)
The Surprise
Look at the last column.
α, with the terrifying sensitivity of sixty, contributes six tenths of one percent of the total uncertainty. It is irrelevant.
T, with the trivial sensitivity of 0.024, contributes sixty-two percent. It dominates.
Sensitivity alone tells you nothing. What matters is sensitivity multiplied by uncertainty. α is a knife-edge variable that happens to be known to four decimal places from a century of metallurgical measurement, so its knife edge never gets tested. Temperature is a forgiving variable that you happen to measure badly, and the forgiveness is not quite enough to save you.
This is the practical payoff of the whole exercise, and it is why I make students build the contribution table rather than just quote the final number. The table is a shopping list. It tells you, in rank order, where your next hour and your next dollar should go: buy a better thermometer, improve the resistance measurement second, and do absolutely nothing about α.
Seeing It as Area
There is a way to draw this that makes it stick.
Draw each contribution as a square whose side is that contribution. R₀ gives a square of side 0.0187. T gives a square of side 0.0240. And α gives a square of side 0.0024 — a speck you can barely see.
The three areas add up exactly to a fourth square, and the side of that square, 0.0305, is your answer.
That is the equation, drawn. Variances are areas. Areas add. And what you report is the side of the square you end up with — not the sum of the sides.
Chapter Seven
Reading the Result
"6.24 ± 0.03 Ω" is a probability statement. Here is what it actually claims:
If nothing systematic is left in the measurement, then roughly sixty-eight times out of a hundred the true resistance lies between 6.21 and 6.27 ohms.
Sixty-eight. Not ninety-five.
The equation returns one standard uncertainty — k = 1 — because it inherits that convention from variance. Almost nobody states it, which is where a good deal of the confusion in laboratory reports begins.
Convention Multiplier Coverage For the copper wire
Standard uncertainty k = 1 ≈ 68 % 6.24 ± 0.03 Ω
Expanded uncertainty k = 2 ≈ 95 % 6.24 ± 0.06 Ω
Those two differ by a factor of two, and a reader cannot tell them apart by looking. So say which one you are quoting. If you are handing a number to somebody who will act on it, quote the expanded form and label it.
And What It Does Not Say
The interval says nothing whatever about accuracy.
If your thermocouple reads two degrees high on every single reading, the error function never finds out. It only ever sees the uncertainties you handed it.
The equation propagates what you tell it. Feed it optimistic inputs and it will hand you back an optimistic answer with a perfectly straight face. It has no way to audit the numbers you put in.
That is not a defect in the method. It is a reminder that the method sits downstream of your judgement, and cannot substitute for it.
Chapter Eight
Two Collapses Worth Recognizing on Sight
The general equation simplifies dramatically in two very common cases. Recognizing them will save you a great deal of time.
Sums and Differences
If R = X + Y or R = X − Y, then both partial derivatives are ±1, and squaring kills the sign. So:
e = √( eₓ² + e_y² ) — identical whether you add or subtract
Absolute uncertainties combine in quadrature.
Worked, and then the trap
A deflection computed from two dial readings:
52.40 ± 0.05 mm − 50.10 ± 0.05 mm = 2.30 ± 0.07 mm
The absolute uncertainty barely moved — 0.05 became 0.071. But watch the relative uncertainty:
Quantity Value Uncertainty Relative
First reading 52.40 mm 0.05 mm 0.10 %
Second reading 50.10 mm 0.05 mm 0.10 %
The difference 2.30 mm 0.07 mm 3.1 %
Subtracting two nearly equal numbers threw away roughly thirty times your precision. No instrument failed. Nobody did anything wrong.
This is the single most common way a well-run experiment produces a worthless number. Watch for it anywhere you compute a deflection from two positions, or a temperature rise from two absolute temperatures. If the setup lets you zero the indicator and read the difference directly, do that instead.
Products, Quotients and Powers
If R = XY, then ∂R/∂X = Y and ∂R/∂Y = X. Substitute those in, divide through by R² = X²Y², and every value cancels:
e / R = √[ ( eₓ / X )² + ( e_y / Y )² ]
No derivatives left. For any pure product, quotient or power you can work entirely in percentages — and each one is scaled by its exponent.
Relationship Exponent Effect of a 0.3 % error in d
R = X Y 1 adds 1 × the relative uncertainty
R = X / Y 1 division behaves exactly like multiplication
A = πd²/4 2 0.3 % in d becomes 0.6 % in area
V = πd³/6 3 0.3 % in d becomes 0.9 % in volume
Which is why diameter is usually the sneakiest measurement in a laboratory. It almost never enters a formula to the first power.
A worked case: modulus from a tension test
Elastic modulus is a pure product-quotient — E = FL / (Aδ), with area going as d² — so it needs no calculus at all.
Input Value Uncertainty Relative Exponent Scaled Share of variance
Force F 5000 N ± 25 N 0.500 % 1 0.500 % 4.3 %
Gauge length L 50.00 mm ± 0.05 mm 0.100 % 1 0.100 % 0.2 %
Diameter d 6.00 mm ± 0.02 mm 0.333 % 2 0.667 % 7.6 %
Extension δ 0.088 mm ± 0.002 mm 2.273 % 1 2.273 % 88.0 %
Combined: 2.42 % relative. The nominal modulus works out to 100.5 GPa, which is right for brass.
E = 100.5 ± 2.4 GPa (standard uncertainty, k = 1, one specimen)
And the share column carries the whole engineering lesson: extension is eighty-eight percent of the problem. Force — the measurement everyone watches, because the load frame has an impressive digital display — is four percent.
Buy the good extensometer. Do not buy the better load cell. And if you are using crosshead travel instead of an extensometer, your modulus is not a material property at all — it is a measurement of your load frame's compliance, with a little brass mixed in.
Chapter Nine
The Three Assumptions Hiding Inside One Line
Every time you use that equation you agree to three things. All three fail in ordinary undergraduate experiments, and the equation gives you no warning when they do. It returns a number either way, formatted just as confidently.
1. Independence. The inputs' errors are unrelated to one another. The manual says so out loud — a function of independent variables — and it is the assumption most often false in a teaching laboratory.
2. Small errors. The model is close enough to straight, across the span of your uncertainty, that the tangent line can stand in for the curve.
3. Standard uncertainties in, standard uncertainty out. Every value you feed the equation must be one sigma.
That third one is the quiet killer, so let me be specific about it. Manufacturers quote all sorts of things. A specification that reads accuracy ± 0.5 % is almost always a limit, not a standard uncertainty. Before it goes anywhere near this equation, convert it.
Source What you are given Standard uncertainty
Digital display, resolution d last-digit rounding d / √12 ≈ 0.29 d
Analog scale, division D, readable to half a division half-width 0.5 D 0.5 D / √3 ≈ 0.29 D
Manufacturer's limit, no distribution stated half-width a a / √3 ≈ 0.58 a
Calibration certificate at k = 2 expanded uncertainty U U / 2
The old workshop rule — estimate instrument uncertainty at one fifth to one tenth of a scale division — lands in the same territory as the formal calculation above. It was never wrong. It simply never showed its work.
Chapters Ten and Eleven take the first two assumptions apart.
Chapter Ten
When Independence Fails
The full equation has terms the manual's version leaves out — one for every pair of inputs:
e² = Σ (∂R/∂Xᵢ)² eᵢ² + 2 ΣΣ (∂R/∂Xᵢ)(∂R/∂Xⱼ) · cov (Xᵢ, Xⱼ)
Those second terms are covariances. They measure how much two inputs' errors move together. The manual's version assumes every one of them is zero.
How They Stop Being Zero
• One caliper, slightly mis-zeroed, used to measure both the length and the width.
• One operator with one parallax habit, reading every gauge on the bench.
• One thermocouple setting the reference temperature for every run.
• One calibration standard sitting behind two different instruments.
Every one of those is ordinary. Most of them are what a well-organized laboratory looks like.
What It Does to You
Errors that move together stop cancelling. The covariance term is positive, quadrature understates the real uncertainty, and you report a tighter number than you have earned.
Which is exactly the wrong direction to be wrong in.
Recall the two ends of the range from Chapter Five. Perfect correlation is the plain sum; independence is quadrature. Sharing one instrument across two inputs slides you toward the wrong end of that range — quietly, without changing a single number on your data sheet.
The fix is upstream, not in the arithmetic. Calibrate the shared instrument, so that its bias is removed rather than propagated. Or use two instruments. Or, if neither is possible, estimate the covariance and carry it.
Chapter Eleven
When Linearization Fails — the Second-Order Term Returns
Now we pick up the term we dropped in Chapter Three.
For a single input, the leading discarded term is:
½ (∂²R/∂X²) e²
Normally it is negligible, and we showed why: it scales with the square of a small quantity. But there is one situation in which it is not negligible at all — it is everything.
The Case That Breaks the Equation
Suppose the first derivative is zero at your operating point.
The first-order equation then predicts that this input contributes nothing whatever to your uncertainty. And that is simply false.
What it actually contributes is two things:
Effect Magnitude Note
A bias ½ R″ e² The answer shifts — this is not just a widening
A spread R″ e² / √2 Scales with e², not with e
Both scale with the square of the uncertainty rather than the first power. And the first one is a bias — it moves your answer, which no amount of repetition will reveal.
Where You Will Meet It
In an impact test, pendulum energy goes as the cosine of the angle. So the derivative of energy with respect to angle goes as the sine of the angle. As the swing-through angle approaches the bottom of the arc, the sine goes to zero — and the first-order equation announces that your angle reading has stopped mattering.
It has not. The entire contribution has moved into the second-order term, where the equation cannot see it.
More generally, you are exposed anywhere you are sitting at a maximum, a minimum, or a turning point in a calibration curve. A zero derivative is not a gift. It is a warning.
The Practical Rule
If a sensitivity comes out at or near zero, or if the model curves visibly across the span of your uncertainty, stop trusting the closed-form equation and perturb the calculation numerically instead.
The numerical method requires no calculus and takes five spreadsheet columns:
1. Compute your answer with all inputs at their nominal values. That is the baseline.
2. Increase one input by exactly its uncertainty. Leave everything else alone. Recompute.
3. The change in the answer is that input's contribution. Record it.
4. Restore the input, and repeat for every other input in turn.
5. Square the contributions, add them, take the square root.
It gives the identical answer for a linear model — I have checked it against the copper wire, and it reproduces 0.0305 Ω exactly. For a nonlinear one it is arguably the more honest result, because it reflects what your calculation actually does rather than what its tangent line does. And unlike a hand-derived partial derivative, anyone can audit it by rerunning it. There is nowhere for a sign error to hide.
Chapter Twelve
What the Error Function Actually Says
Let me put the equation back up whole, and read it the way you can now read it.
e = √ Σ ( ∂R/∂Xᵢ · eᵢ )²
• Each input's damage is its sensitivity times its uncertainty — the model's part, times your part.
• Those damages are squared because variance is a second moment, and that is the only way random quantities are permitted to combine.
• Squares add. The answer is the side of the resulting square — a hypotenuse, not a sum.
• It returns one sigma. It assumes independence. And it assumes the tangent stands in for the curve.
• When a sensitivity vanishes, or when inputs share an instrument, it quietly stops being true and does not tell you.
That is one line of a laboratory manual. If you have read this far, I hope it reads differently than it did an hour ago.
A Closing Thought
We poke systems in order to understand them. We load them, heat them, strike them and twist them, and then we listen to the answer. But the answer never arrives clean. It arrives through an instrument, read by a person, in a room with a temperature.
The error function is how we carry that imprecision through the arithmetic without lying about it at the far end.
Being honest about what you do not know, and about exactly how much you do not know it, is not a weakness in an engineering result. It is the strongest thing about it.
Appendix
Quick Reference
The equation, in its forms
• General: e = √ Σ (∂R/∂Xᵢ · eᵢ)²
• Sums and differences: absolute uncertainties add in quadrature.
• Products, quotients and powers: relative uncertainties add in quadrature, each scaled by its exponent.
• A quantity raised to the power n: its relative uncertainty is multiplied by |n|.
• No derivatives available: perturb each input by its uncertainty, record the change in the output, combine those changes in quadrature.
• Correlated inputs: add 2 ΣΣ (∂R/∂Xᵢ)(∂R/∂Xⱼ) · cov(Xᵢ, Xⱼ).
Reporting
• Round the uncertainty first — one significant figure, or two if the leading digit is 1 or 2. Then round the value to match its decimal place.
• State the coverage. k = 1 is about 68 %; k = 2 is about 95 %; they differ by a factor of two.
• Name the dominant contributor. "Uncertainty is dominated by the extension measurement, which contributes 88 % of the total variance" is the sentence that proves you understand your own experiment.
The one-page procedure
1. Write down what you are calculating and every input it depends on.
2. For each input, decide where its uncertainty comes from, and convert it to a standard uncertainty.
3. Guess which input will dominate. Write the guess down.
4. Build the contribution table: input, sensitivity, uncertainty, product, square, share of variance.
5. Take the square root of the total.
6. Compare the share column against your guess. You will be wrong often enough to learn something.
7. Check the assumptions: any shared instruments? Any sensitivity near zero?
8. Round, state the coverage, and name the dominant term.
A Note on Sources
The equation, the copper-wire example and its numbers are from Section II.C.3 of the MEEG 2206L Mechanics Laboratory Manual at Fairfield University. The derivation, the geometry, the worked tension-test case and the failure modes in Chapters Ten and Eleven are my own additions, developed over a good many semesters of watching where students — and practising engineers — actually come unstuck.
The thinking and the opinions here are mine, tested against forty-four years of arguing with broken parts. I worked with AI collaborators to organize and sharpen the writing; the clarity, where you find it, benefited from a tireless partner who never tired of the seventh revision.
If any of this is useful, or if you would like to argue with it, I would genuinely like to hear about it.
Joseph P. McFadden Sr.
