Chapter 3: Propagation of Uncertainties

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So imagine you were a college student, right?

You are sitting there staring down your very first major physics lab report.

The panic is real.

It really is.

You have all this data and you suddenly have to figure out how to calculate your margins of error,

which, you know, can feel completely overwhelming.

Absolutely.

But that is exactly our mission today.

We're going to make that process feel, well, intuitive.

Yeah, we are doing a deep dive into chapter three of the classic text, Introduction to Error Analysis.

The study of uncertainties in physical measurements.

Right.

And today's focus is on the propagation of uncertainties, which honestly sounds like a terrifying phrase.

It does sound intimidating, but the core concept is actually pretty straightforward.

In science, you basically never measure the exact thing you want directly.

Like velocity.

Right.

You don't just measure velocity.

Exactly.

You measure the distance a car traveled and you measure the time it took and then you calculate the velocity.

So the big question is, how does the the built -in flaw of your tape measure?

And maybe the slight delay in your stopwatch.

How do those combine and propagate into your final answer?

Right.

How do they spread into that final calculated velocity?

And understanding this isn't just dry math.

It is, I mean, it's arguably the most vital toolkit for critical thinking you can develop.

Because by the end of this deep dive, you'll know how to confidently report not just your answer, but how trustworthy that answer actually is.

Yes.

Because knowing the exact boundaries of your own ignorance is what separates a wild guess from a rigorous scientific claim.

I love that.

So let's build this from the ground up.

Before we see how errors spread, we kind of need to know where they start.

Which is with our direct physical measurements.

Right.

Let's talk about the instruments we use.

Yeah.

Because there's this weird psychological trap when it comes to analog versus digital.

Oh, this is a huge one.

Yeah.

So when you read an analog scale, like a standard plastic ruler or say an old mercury thermometer, you are physically estimating the gaps.

You're using your eye.

Exactly.

If the red line on the thermometer is somewhere between 20 and 21 degrees,

you estimate it.

You might say, okay, it's 20 .5 degrees plus or minus 0 .5 degrees.

And because you can literally see that gap, you naturally trust your eyes to assign a reasonable uncertainty.

You know it's not perfect.

But digital displays, which, you know, we use for almost everything now, they completely strip that visual context away.

They do.

And a digital readout just looks so authoritative.

It really does.

If a digital voltmeter flashes exactly 81 microvolts, your overwhelming temptation is just to write down 81 .000.

Because you assume it's perfectly 81, but that is a really dangerous assumption.

Unless the manufacturer's manual explicitly tells you otherwise, the rule of thumb is that your uncertainty is at least plus or minus 1 in that final least significant digit.

Because the meter is just rounding away invisible decimals behind the scenes.

Exactly.

It might actually be 81 .4 or 81 .6.

You just don't know.

So you have to record it as 81 plus or minus 1.

And the source text has this brilliant,

highly relatable example of this false digital confidence.

The Atwood machine.

Yes.

So imagine you're in a lab using an Atwood machine.

That's the classic setup with a string over a pulley,

weights hanging off both sides.

Right.

A staple of freshman physics.

You release the weights and you use a digital timer to measure how long it takes the heavier one to hit the floor.

And the timer flashes like a beautiful 8 .01 seconds.

And you think, wow, hundredth of a second precision.

Exactly.

But then you reset the exact same pulley, drop the exact same weights from the exact same height, and the timer flashes 8 .41 seconds.

Which is a massive 0 .4 second discrepancy.

Right.

And it perfectly highlights why you always, always have to repeat your measurements.

It's non -negotiable.

Because if you just trusted that very first digital readout, your claimed accuracy would be completely inflated.

The physical reality of the setup, like the friction in the pulley, or your personal reaction time hitting the button, or even just a slight swing of the weights, all of that creates a spread of values.

And that spread tells you your true uncertainty way better than whatever tiny fractions of a second the digital screen claims it can measure.

But you know, sometimes the instrument is actually totally fine.

Sometimes the problem is just the universe.

Ah, yes.

What the textbook calls the problem of definition.

This concept completely blew my mind.

It basically means the thing you're trying to measure is inherently fuzzy.

The Optics Bench experiment is the perfect illustration of this.

So imagine you're measuring the focal length of a thick glass lens.

Okay.

You've got a light source on one end of a track, the lens in the middle, and a little paper screen on the other end.

And you're sliding that screen back and forth until the image is perfectly sharp.

And my track has a brass ruler glued to it that's precise down to the millimeter.

Right.

So you'd naturally think your measurement uncertainty is just one millimeter.

Because that's what the ruler says.

Exactly.

But when you actually look at the projected image on the paper screen, you realize there's a depth of field.

The image might look perfectly indistinguishably in focus over a sliding range of a whole centimeter.

Oh, wow.

And on top of that, your lens is this thick, curved piece of glass.

Where exactly is the literal center of that lens that you're supposed to be measuring from?

So if I just blindly write down plus or minus one millimeter because of the ruler, I am completely ignoring the physical reality.

You'd be badly underestimating your total uncertainty.

Sometimes by a factor of 10,

the fuzziness of the focus is your real limit, not the markings on the track.

That makes so much sense.

And this idea of inherent fuzziness brings us to a totally different kind of direct measurement, which is counting random events.

Right, like babies born in a hospital or the clicks of a Geiger counter next to a radioactive rock.

The text introduces this thing called the square root rule for counting.

And I have to be honest, it feels a little bit like magic.

It really does.

If a demographer counts 14 births in a two -week period at a local hospital, the uncertainty of that average rate is literally just the square root of 14.

It feels like a magic trick, but it's deeply rooted in the mathematics of probability,

specifically the Poisson distribution.

OK, unpack that for us.

Think about radioactive decay.

An unstable atom doesn't have like a tiny clock inside it ticking down to a specific second.

It decays randomly, but a massive chunk of those atoms will decay at a definite average rate.

Because the underlying mechanism is truly random,

any sample you take is going to naturally fluctuate around that true average.

It's sort of like slipping a coin a hundred times.

You know the true average should be 50 heads, but if you actually do the physical experiment, you might get 48 or 53.

Exactly.

There is a predictable wobble to the universe.

A predictable wobble.

I like that.

So for events that happen randomly, but at a definite average rate, your best estimate of the true average is simply the raw number you just counted.

Let's call that number nu.

Right.

And the statistical wobble, the uncertainty, is wonderfully simple.

It's just the square root of nu.

So back to the hospital.

The demographer counted 14 births.

The square root of 14 is roughly 3 .7, which we can round up to 4.

So their final rigorous scientific conclusion is just 14 plus or minus 4 births.

That's it.

And if you put a Geiger counter next to a sample and you count 907 decays in 50 minutes, your uncertainty is the square root of 907.

Which is what, about 30?

Roughly 30.

So your measurement is 907 plus or minus 30.

It is a wildly powerful shortcut for any rate -based counting experiment.

OK.

So we've got our starting line established.

We know how to find the uncertainties of our direct measurements,

whether that's reading a ruler,

recognizing the fuzziness of a lens, or just taking the square root of a random count.

Now we have to start mixing them together.

Right.

What happens when we plug these flawed numbers into an actual equation?

Let's start simple, with basic addition and subtraction.

Say you're a chemist, and you need the total mass of liquids poured from two different flasks.

For addition and subtraction, the provisional rule is really straightforward.

You just stack the absolute uncertainties.

So if quantity X could be off by 2 grams, and quantity Y could be off by 3 grams, the absolute worst -case scenario is that both errors happen in the exact same direction.

So the uncertainty of your total calculated mass is simply 2 plus 3.

Your total could be off by 5 grams.

I follow that.

Absolute errors just stack.

So moving to multiplication and division, I assume we just multiply the absolute errors together like 2 grams times 3 grams?

You'd think so, but that would actually lead to absolute mathematical chaos.

Wait, really?

Why?

Think about the units.

If you multiply an error of 2 meters by an error of 3 seconds, you get 6 meter seconds of error.

Oh.

Right.

That doesn't mean anything.

Exactly.

When we multiply or divide, we have to completely abandon absolute numbers and pivot to percentages.

What the text calls fractional uncertainties.

Let's define fractional uncertainty real quick so we don't lose anyone.

If you measure a length as 100 centimeters and your absolute uncertainty is 2 centimeters, the fractional uncertainty is just 2 divided by 100.

So it's 0 .02, or 2%.

Spot on.

So let's apply that.

Imagine a surveying team trying to calculate the height of a massive redwood tree.

They use a formula that involves multiplying two measured lengths and dividing by a third.

Because they are multiplying and dividing, they simply add the percentage errors of those So if length 1 has a 1 % error, length 2 has a 2 % error, and length 3 has a 4 % error,

you just add them up?

Yep.

1 plus 2 plus 4 equals 7.

The fractional uncertainty of the final calculated tree height is 7%.

That is surprisingly clean.

You add absolute errors for sums, and you add percentage errors for products.

It is an elegant symmetry.

And actually, this fractional rule is the secret behind a really cool trick for measuring Oh yes, I teased this at the very beginning of the deep dive, the impossible measurement.

Exactly.

Because if you try to measure one sheet of printer paper with a millimeter ruler, your error is larger than the paper itself.

But if you stack 200 identical sheets into a thick block, you can easily measure that block.

Right.

Let's say the block measures 1 .3 centimeters thick, plus or minus 0 .1 centimeters.

That 0 .1 centimeter error on a 1 .3 centimeter block represents a fractional uncertainty of roughly 8%.

Now to find the thickness of a single sheet, you simply divide the block's total thickness by 200.

But wait, 200 is just a counted number, it's an exact integer, there's literally no uncertainty in the number 200.

Which is the magic of the trick.

Because the denominator has zero uncertainty, the fractional uncertainty of your final answer remains exactly the same as the numerator.

The total block had an 8 % error.

Therefore your calculated thickness for a single sheet also has exactly an 8 % error.

You've measured a microscopic thickness with an 8 % margin of error, using nothing but a cheap plastic ruler and a little bit of experimental design.

That is just deeply satisfying.

It's a great lab hack.

And there's one more special case for multiplication we should mention, which is dealing with powers.

Like if you're calculating kinetic energy,

your formula includes velocity squared.

Right, and velocity squared is just velocity times velocity.

Since it's multiplication, you add the fractional uncertainties.

So the fractional uncertainty of V plus the fractional uncertainty of V is just 2 times the fractional uncertainty of V.

Exactly.

So if a quantity is raised to a power of n, you simply multiply its percentage error by n.

If your velocity measurement is off by 3%, your calculated V squared will be off by 6%.

Okay, so these provisional rules, stacking absolutes for addition, stacking percentages for multiplication, they're highly logical.

But here is where the text drops a massive plot twist.

Yes, the plot twist.

It turns out those simple addition rules, they actually mathematically overestimate the error.

They are way too pessimistic.

But they represent the absolute upper bounds, because when we simply added our 2 and 3 gram errors earlier, we were assuming the universe hated us.

We assume that if you overestimated the first mass, you also simultaneously overestimated the second mass by the maximum possible amount.

To be fair, sometimes the universe does hate you.

The text uses this great analogy of a steel tape measure.

Oh, the temperature example.

If you are doing construction outside on a freezing cold winter day,

that steel tape measure is literally going to shrink.

So it will yield readings that are systematically too long.

Every single thing you measure with it will be overestimated.

And that is a classic systematic error.

Because the exact same underlying physical flaw is pushing every single measurement in the exact same direction,

there is zero chance for the errors to cancel out.

So in that specific scenario, you really must use the simple addition rules just to be safe.

But what if your errors aren't systematic?

What if they are independent and random?

Say I measure the length of a lab table and you measure the width.

My slight visual slip of the hand has absolutely nothing to do with you, like slightly misreading your dial.

Exactly.

And this is where probability steps in to save our data.

If errors are truly independent and random, there is a 50 % chance your length measurement is slightly too high.

And there's a 50 % chance my width measurement is slightly too low.

If one is high and the other is low, they naturally partially cancel each other out when we combine them in a calculation.

Assuming they both go completely wrong in the exact same direction is just mathematically unfair to our own experiment.

So we need a new mathematical tool that accounts for this cancellation.

And the text calls this adding in quadrature.

Which honestly sounds intensely complex.

It does, but it's really just the Pythagorean Theorem.

Just a triangle.

Yep.

It's the exact same math you use to find the long side of a right triangle.

Visually it makes perfect sense.

If you have two independent absolute uncertainties, you don't stack them end to end like a straight line.

You put them at a right angle to each other.

Okay, so you square the first error, you square the second error, add those squares together, and take the square root of the whole thing.

Precisely.

Because the hypotenuse of a right triangle is always shorter than the sum of the two separate sides, the error calculated by quadrature is always realistically smaller than the error calculated by straight addition.

The text walks through this amazing aha moment with an electric motor efficiency problem.

That's a great example.

You're calculating efficiency by multiplying and dividing five completely distinct variables.

There's the mass of a lifted weight,

the height it was lifted, the voltage of the motor, the current, and the time it took.

And each one has its own fractional uncertainty.

Say 1 % here, 5 % there.

Right.

And if you use the old provisional rule and just straight up add all those percentages together, you get this bloated total error of 9%.

But the voltage of the wall supply has absolutely nothing to do with the physical mass of the lead weight being lifted.

They are totally independent.

Exactly.

We are fully justified in adding their fractional uncertainties in quadrature.

So you square the 1%, you square the 5%, add them all up, take the square root, and your total margin of error drops from 9 % down to just 5%.

It's amazing.

You practically double your claimed precision just by using the correct statistical geometry.

And this reveals what we can call the golden rule of quadrature.

It's almost a diagnostic superpower.

It really is.

Look closely at what happens when you square numbers.

You drastically exaggerate the big ones, and you mathematically diminish the small ones.

Let's break that down for the listener.

In that motor example, the uncertainty in the time measurement was our biggest flaw.

It was 5%.

Right.

But the mass and the voltage only had tiny 1 % errors.

Well, when you square a 5 % error, it blooms into 25 units of badness.

25 units of badness.

I love that.

But when you square a 1 % error, 1 times 1 is just 1.

So under that square root bracket, the stopwatch error is contributing 25 parts to the final sum while the highly precise scale is only contributing one part.

So the tiny errors are basically mathematically swallowed whole by the dominant error.

Exactly.

This is huge for anyone actually running an experiment.

If you know one of your measurements is 5 times sloppier than the rest, you can essentially ignore all the other tiny errors.

Right.

Do not waste three hours trying to calibrate a slightly better scale for your mass when your stopwatch is terrible.

Put all your energy into finding a more precise stopwatch.

Quadrature tells you exactly where your experiment is bleeding precision.

It tells you how to allocate your most precious resource, which is your time.

That is just so practical.

Okay, so we've conquered basic arithmetic.

But physics formulas aren't always just flat lines of addition and multiplication.

Physical phenomena curve.

Yes, they do.

What happens when your formula involves a complex function like a sine wave or an exponential decay or a logarithm?

How does a tiny error in your input behave when you drop it onto a slope that is constantly changing?

Well, the book starts by having you visualize a graph.

Let's say your function is just q of x equals sine of x.

You find your best measured input value on the horizontal x -axis, go straight up to the curving line, and find the resulting output on the vertical y -axis.

Simple enough.

Then you look at your margin of error on the x -axis, your maximum and minimum probable values.

You trace those boundaries up to the curve and over to the axis.

And the spread on the a -axis is your final uncertainty.

It makes complete physical sense.

You are literally just tracing the physical boundaries of your error through the shape of the curve.

Right.

But drawing graphs by hand every single time is exhausting.

It is.

So the true master shortcut relies on calculus.

If your input uncertainty is relatively small, the curve over that tiny interval is basically a straight line.

And in calculus, the steepness of a line is its derivative.

Exactly.

The derivative tells you exactly how sensitive the output is to tiny changes in the input.

So the shortcut formula is actually quite beautiful.

The uncertainty in your final answer is just the absolute value of the derivative multiplied by the uncertainty of your input.

Yes.

Think about what that means physically.

If the derivative is a large number, the function is practically a vertical cliff.

Even a microscopic input error gets blown up into a massive output error.

But if the derivative is near zero, the curve is flat.

It just squashes your error down to almost nothing.

And here we must issue a vital public service announcement directly from the text.

Oh yes.

Because this traps almost everyone at least once.

The trigonometry trap.

If you are calculating the uncertainty of a trig function, like a cosine, a sine, a tangent,

you absolutely must convert your input uncertainty from degrees into radians before you multiply it by the derivative.

You have to.

The calculus rules you memorize, like the derivative of sine being cosine, are only mathematically true if the angles are measured in radians.

Because radians are based on the physical geometry of a circle, right?

Well degrees are just this arbitrary historical number.

Exactly.

If you measure an angle as 20 degrees plus or minus 3 degrees, you cannot just plug the integer 3 into the derivative formula.

It will work.

You have to convert that 3 degrees into roughly 0 .05 radians first.

Otherwise your calculation will be completely detached from physical reality.

Consider yourselves warned.

So armed with this calculus trick and our quadrature rules, we can tackle nasty equations step by step.

We can.

The text shows an example measuring gravity with a swinging pendulum.

The formula involves length divided by time squared.

To solve it step by step, you handle the time squared part first.

It's a power, so the fractional error is twice the fractional error of time.

Then you handle the division.

You take the fractional error of the length and the fractional error of the time block and you add them in quadrature because the length of the string and the swing of the pendulum are independent measurements.

You just snap the mathematical legos together.

That's a great way to put it.

It's incredibly satisfying.

But then the book throws a massive wrench into the gears.

It introduces the cart on the incline problem.

Ah, yes.

The cart.

You are calculating the acceleration of a cart rolling down a ramp.

The formula involves subtracting an initial velocity squared from a final velocity squared and then dividing by distance.

Now the initial velocity is length over time 1 and the final velocity is that exact same length over time 2.

And if you try to calculate the error for this step by step calculating the error for velocity 1, then velocity 2, then their swears, then subtracting them, you will end up estimating an uncertainty of about 9%.

Which sounds totally fine until you realize there is a hidden flaw.

A very big flaw.

Because velocity 1 and velocity 2 both rely on the exact same underlying length measurement,

they are inherently not independent.

And this is the fatal danger of stepwise propagation.

Think about the physical reality.

If you overestimate the length of the track, you will calculate a velocity 1 that is too high.

Right.

But you will also calculate a velocity 2 that is too high.

Because the acceleration formula subtracts velocity 1 from velocity 2, those two overestimates actually partially cancel each other out.

They are compensating errors.

But the stepwise method is completely blind to that.

Completely blind.

It calculated the velocity 1 error and the velocity 2 error as if they happened in parallel dimensions.

It missed the cancellation completely and artificially inflated our final error estimate to 9 % when the reality was much tighter.

So faced with a function where the same variable appears multiple times, the stepwise method just shatters.

We need a more powerful tool that evaluates the entire architecture of the equation all at once so it can detect those internal cancellations.

And that brings us to the final boss of Chapter 3, the general formula for error propagation.

The master key.

It's the key that unlocks any physics equation no matter how entangled it is.

Now it looks deeply intimidating on the page.

It really does.

But conceptually, it is just an extension of what we already did with calculus.

The general formula relies on partial derivatives.

Okay.

For those of us who break into a cold sweat at the phrase multi -variable calculus, let's break down how this actually works.

Sure.

Imagine an equation with three variables, x, y, and z.

To take the partial derivative with respect to x, you simply pretend that y and z are fixed constants.

You act like they are just regular numbers, like a 5 or a 10, and you take the normal basic derivative just for x.

This isolates the sensitivity of the final answer to changes in x alone.

It's like trying to figure out how much each individual ingredient changes the flavor of a soup.

Exactly.

You isolate the salt, you pretend the pepper and the garlic are totally fixed, and you measure just the salt's impact.

Then you isolate the pepper and so on.

That's a perfect analogy.

You find the partial derivative for each variable.

You multiply that partial derivative by that variable's uncertainty.

This gives you the isolated chunk of error contributed by that one specific ingredient.

And then you combine them.

Right.

If the variables are independent and random, you combine all those isolated chunks in quadrature.

But wait, I understand the soup analogy.

But mathematically,

how does the partial derivative actually know that my velocities in the card experiment are canceling each other out?

Because of how the calculus resolves the signs.

When you take the partial derivative of that acceleration formula with respect to the shared length variable, the algebra naturally produces a positive term from the final velocity and a negative term from the initial velocity being subtracted.

The compensating errors literally cancel themselves out in the algebra before you even plug your numbers in.

The math basically does the thinking for you.

That is genuinely beautiful.

This single general formula is really the foundation of everything else in the chapter.

It is.

If you apply this general partial derivative formula to the simple equation, q equals x plus y, out pops the simple addition rule.

If you apply it to q equals x times y, out pops the fractional addition rule.

It's the underlying source code of error propagation.

It's the ultimate diagnostic tool.

And to circle back to where we started, that feeling of staring down a daunting lab report, this chapter really proves that having uncertainty isn't a failure of your experiment.

Not at all.

Whether you are building an aerospace component, coding a machine learning algorithm, or just passing a freshman physics lab,

you are always going to have imperfect data.

Perfection is a myth.

The true goal of science is not absolute certainty.

It is rigorously quantified uncertainty.

Knowing the precise boundaries of what you don't know is what makes a measurement truly useful.

I couldn't agree more.

I want to leave you with a final thought to mull over.

We live in a modern society that demands absolute certainties.

We have AI chatbots that generate answers with unshakable confidence.

We certainly do.

We have news headlines that present data without a shadow of a doubt.

What if we demanded to see the error bars on everyday information?

That's a fascinating idea.

How would it change our online arguments, our policies, or just our understanding of the world if everyone fundamentally grasped that truth usually exists within a rigorously calculated margin of error?

That is a critical thinking exercise worth carrying with you long after the lab equipment is put away.

On behalf of the Last Minute Lecture team, thank you for learning with us and good luck with your analysis.