Measurement errors and error propagation

End of September, Year 1. At a liceo (academic high school), in physics, after the first lessons on quantities and the International System of Units comes the chapter on measurement errors — usually together with the first graphs and proportionality. At the technical-technological institute, with physics at 3 hours and the lab, errors come in right away, already in October, with the first lab reports to write: there the first test arrives in four to seven weeks, so between the end of October and the beginning of November. A detail for anyone starting the technical track this year: under the new curriculum (ministerial decree DM 29/2026) first-year physics and chemistry merge into “Scienze sperimentali” (experimental sciences) — the programme of measurements and errors stays the same, only the name of the subject on the report card changes.

Error propagation: the formulas for the physics test

This chapter is almost always assessed with the same scheme: a measurement (or a series of measurements) from which to obtain the mean value and the uncertainty, and then a calculation — an area, a density, a speed — to do with those uncertain numbers, writing the final result with its error. Those who lose marks usually do not get the calculations themselves wrong: they get wrong how many digits to write in the error, or apply the wrong rule (the one for sums where the one for products is needed). The rest of this post is the method for not doing that.

If your programme already covers significant figures and conversions and you are looking for that part, you will find it in physics in Year 3: measurements and significant figures: I do not repeat it here, because the added value of this post is repeated measurements and propagation.

Systematic and random errors

Every real measurement is affected by an error: there is no perfect instrument. Errors divide into two families:

• Systematic errors: they repeat always in the same direction (a badly calibrated scale that always weighs 2 g too much, a stopwatch that starts with a fixed delay). They are not eliminated by repeating the measurement — they are eliminated only by calibrating the instrument or correcting the method.
• Random (or accidental) errors: they change unpredictably from one measurement to the next (the reaction time of the person starting and stopping a stopwatch, small vibrations). They are treated statistically: by repeating the measurement several times, they partly cancel out.

The chapter on propagation always works on the random error, estimated with the half-range spread (semidispersione): the systematic part is assumed to be corrected or negligible, unless the exercise says otherwise.

A single measurement: the error is the instrument’s sensitivity

If you measure a quantity only once, the absolute error you assign to it is the sensitivity of the instrument: the smallest variation the instrument can distinguish. A millimetre ruler has a sensitivity of $0.1\mathrm{cm}$ (or, to be cautious, half the smallest division: $0.05\mathrm{cm}$ — it depends on how your teacher defined it, but the millimetre ruler is the case most cited in tests). A measurement of $l=12.3\mathrm{cm}$ with that instrument is written:

Repeated measurements: mean value and half-range spread

When you can repeat the measurement several times (the typical lab case: the period of a pendulum, the fall time of an object), the most reliable value is not a single reading but the arithmetic mean of all the measurements, and the error is estimated with the maximum half-range spread: half the difference between the largest and the smallest value obtained.

The half-range spread is not the standard deviation you will see later in statistics: it is a simpler and more cautious estimate (in the worst case it overestimates the true error), but it is the one required in Year 1 because it needs nothing beyond a mean and a difference.

Relative error, percentage error and how to write a measurement

The absolute error $\Delta x$ has the same unit as the quantity and says little on its own: an error of $1\mathrm{cm}$ is enormous if you are measuring a pencil, negligible if you are measuring a football pitch. To compare how “precise” a datum is you need the relative error:

and the percentage error, which is the same number multiplied by 100:

A complete measurement is always written as value ± error, with the unit of measurement, and with a rule on digits to respect: the error is rounded to one significant figure (rarely two, if the teacher explicitly asks for it), and the central value is rounded to the same order of magnitude as the error. Writing $l=(12.347\pm 0.1)\mathrm{cm}$ is wrong: those digits after the decimal point in the central value make no sense, because the error tells you that you do not know the value beyond the first decimal place. The correct form is $l=(12.3\pm 0.1)\mathrm{cm}$.

Error propagation: the three rules

When you calculate a quantity from two or more uncertain measurements, the error propagates into the result. The rules change according to the operation, and this is where the most common mistake in tests is concentrated: using the rule for sums where the one for products is needed, or vice versa.

Sum and difference → the absolute errors add. If $z=x+y$ or $z=x-y$:

It holds for both operations, sum and difference: the uncertainty is never subtracted, even when you are taking a difference of measurements — in the worst case the two errors add.

Product and quotient → the relative errors add. If $z=x\cdot y$ or $z=x/y$:

So to find the absolute error on the result, you first add the relative ones and then multiply by the value of $z$: $\Delta z=z\cdot \left(\frac{\Delta x}{x}+\frac{\Delta y}{y}\right)$.

Power → the exponent multiplies the relative error. If $z=x^n$:

This is why, when you calculate a volume as $V=l^3$, the relative error on the volume is three times that on the side, not the same: cubing amplifies the uncertainty.

Typical mistakes

• Using the sum rule for a product (adding the absolute errors of two measurements that should be multiplied): it gives a final error with the wrong unit of measurement, and almost always an absurd value.
• Forgetting the exponent in the power: calculating the relative error of the volume as if it were equal to that of the side, without multiplying it by 3.
• Writing the error with too many significant figures ($\pm 0.347$ instead of $\pm 0.3$): the error must be rounded to one significant figure, always.
• Rounding the central value before calculating the error, instead of at the end: it introduces a discrepancy that sometimes changes the last digit of the result.

Three worked examples

1. Period of a pendulum: mean and half-range spread

Five measurements of the oscillation period of a pendulum, in seconds: $2.04$; $2.02$; $2.06$; $2.03$; $2.05$.

The mean:

The half-range spread, with $T_{max}=2.06$ and $T_{min}=2.02$:

The period is written $T=(2.04\pm 0.02)\mathrm{s}$. The relative error is $0.02/2.04\approx 0.0098$, that is about $1\%$: a very precise measurement, with a discrepancy of one hundredth of a second out of a total of two seconds.

If you want to check for yourself whether you have understood the method before going on, in the Year 1 physics test — measurements, significant figures and errors you will find more exercises of this kind among the 15 questions: the correct answer is free, the guided step-by-step solution unlocks with the first lesson.

2. Area of a rectangle: propagation on a product

A rectangle has base $l=(12.3\pm 0.1)\mathrm{cm}$ and height $h=(5.4\pm 0.1)\mathrm{cm}$. The area:

Product of two measurements → the relative errors add:

about $2.7\%$. The absolute error:

which is rounded to one significant figure, $\Delta A\approx 2{\mathrm{cm}}^2$, and the central value to the same order of magnitude: $A=(66\pm 2){\mathrm{cm}}^2$.

3. Density: propagation on a quotient

A small block has mass $m=(145.2\pm 0.1)\mathrm{g}$ and volume $V=(62.0\pm 0.5){\mathrm{cm}}^3$. The density:

Quotient of two measurements → the relative errors add again:

The relative error on the volume ($0.0081$) almost single-handedly dominates the result: the error on the mass is almost negligible by comparison, and that is normal — in a product/quotient what matters more is the measurement with the larger relative error, not the one with the larger absolute error in numerical value. The absolute error:

Final result: $\rho =(2.34\pm 0.02)g/c{\mathrm{m}}^3$.

An error worth checking by yourself

Try redoing exercise 2 changing only the height to $h=(5.4\pm 0.3)\mathrm{cm}$ (larger error) and see how $\Delta A$ changes: it is a good way to check that you have understood that the larger relative error, not the absolute one, decides how much each measurement weighs in the final result — the same principle as exercise 3 on density.

The same rules, with the English vocabulary (uncertainty, significant figures), come back in international exams: I talk about it in IGCSE and A-level Physics for an Italian student.

If the chapter after errors in your programme is already graphs and proportionality, you will find the sequel in direct and inverse proportionality in physics graphs. For the general picture of the subject there is physics at academic and technical high school, and if you want to arrive at the test with the method already automatic instead of right before the date, you can start with a free introductory call.