Measurement Uncertainty and Error Analysis

Concise answer

Laboratory-methods questions ask what a number is worth, not just what it is. Random scatter shrinks when you average, systematic offsets do not; absolute uncertainties combine for sums and differences while relative uncertainties combine for products, quotients, and powers; averaging $N$ independent readings tightens the mean only as $1/\sqrt{N}$ ; and a real ammeter or voltmeter perturbs the circuit unless its resistance is negligible or enormous.

Definitions

Random error: Scatter that varies unpredictably from trial to trial (reading noise, thermal fluctuation, judgment of a needle position). It averages toward zero, so repeating the measurement narrows it.
Systematic error: A repeatable offset that shifts every reading the same way (a miscalibrated scale, a zero error, a meter that loads the circuit). Averaging does not remove it; only calibration or a changed method does.
Accuracy versus precision: Accuracy is closeness to the true value; precision is closeness of repeated readings to each other. A tightly clustered set of readings around the wrong value is precise but inaccurate.
Absolute and relative uncertainty: The absolute uncertainty $\delta A$ carries the units of $A$ ; the relative (fractional) uncertainty $\delta A/A$ is dimensionless, and multiplied by 100 it is the percent uncertainty.
Standard error of the mean: The uncertainty in an average of $N$ independent readings, $\sigma_{\bar{x}} = \sigma/\sqrt{N}$ , where $\sigma$ is the spread of a single reading. It shrinks with $N$ , but only as a square root.
Counting statistics: For events arriving independently at a steady average rate — decays in a detector, photons at a photomultiplier — the number recorded is Poisson distributed, so $N$ counts carry an uncertainty $\sqrt{N}$ and a relative uncertainty $1/\sqrt{N}$ .

Intuition

Every measurement is a claim plus a margin, and the margin is the part the question is really testing. If you cannot say whether a difference between two numbers is bigger than the uncertainty in that difference, you have not answered the physics question yet.

Averaging fights noise, not bias. Ten repeats of a reading on a scale that is 2 grams heavy give you a beautifully precise wrong answer; the scatter falls as $1/\sqrt{N}$ while the 2-gram offset sits there untouched. That single distinction decides most laboratory-methods items.

Uncertainty propagation follows the shape of the formula. Add quantities, and their absolute uncertainties combine, because the units must match. Multiply them, and their percent uncertainties combine, because a fractional wobble in either factor is a fractional wobble in the product. A power just repeats the multiplication, which is why an exponent multiplies the percent.

Concept walkthrough

Start by naming the error. Random error scatters readings around a center and is what a standard deviation describes; systematic error moves the center itself. Because averaging only cancels contributions that change sign from trial to trial, repeating a measurement improves precision without touching accuracy. A GRE-style prompt that describes a stopwatch started slightly late every time, a ruler with a worn end, or an ammeter with non-negligible resistance is describing a systematic effect, and the correct response is to change or calibrate the apparatus, not to take more data.

Then propagate. For a sum or difference $Q = a \pm b$ , absolute uncertainties combine: the conservative bound is $\delta Q = \delta a + \delta b$ , and for independent random errors the standard combination is $\delta Q = \sqrt{(\delta a)^2 + (\delta b)^2}$ . For a product, quotient, or power $Q = a^m b^n$ , the relative uncertainties combine instead: the conservative bound is $\delta Q/Q = |m|\,\delta a/a + |n|\,\delta b/b$ , and the independent-error version is $\delta Q/Q = \sqrt{(m\,\delta a/a)^2 + (n\,\delta b/b)^2}$ . Two consequences are worth memorizing: an exponent multiplies the percent uncertainty, so a radius known to 1% makes a volume $\propto r^3$ known only to 3%; and a difference of two nearly equal numbers is uncertainty poison, because the absolute uncertainty stays fixed while the value itself shrinks.

Next, decide what more data buys. The mean of $N$ independent readings has uncertainty $\sigma/\sqrt{N}$ , so cutting the uncertainty in half costs four times the data, and cutting it by ten costs a hundred times the data. Counting experiments are the same statement with the spread supplied by the physics rather than measured: events arriving independently at a steady rate are Poisson distributed, so a total of $N$ counts carries an uncertainty $\sqrt{N}$ and a relative uncertainty $1/\sqrt{N}$ . A run that records 100 counts is a 10% measurement; 10,000 counts is a 1% measurement.

Finally, respect the apparatus and the digits. A real ammeter goes in series and must have a very low resistance, and a real voltmeter goes in parallel and must have a very large resistance; when either condition fails, the meter draws or diverts current and the reading is biased low or high in a predictable direction — a systematic error created by the act of measuring. On reporting, a product or quotient carries as many significant figures as its least precise factor, a sum or difference carries as many decimal places as its least precise term, and the uncertainty itself is normally quoted to one significant figure, with the value rounded to the same decimal place. Writing $R = 6.00 \pm 0.25\,\Omega$ as $R = 6.00483 \pm 0.25\,\Omega$ claims a precision the measurement does not have.

After this page, you should be able to

Classify an error as random or systematic and say which one repeating the measurement can fix.
Propagate uncertainty through a sum, a product, a quotient, and a power without mixing absolute and relative forms.
Use $\sigma/\sqrt{N}$ and $1/\sqrt{N}$ to predict how much averaging or counting longer actually buys.
Report a result with the number of digits its uncertainty supports.
Explain why an ammeter must have small resistance and a voltmeter large resistance, and what happens when they do not.

Formulas and assumptions

Percent uncertainty

\text{\% uncertainty} = \frac{\delta A}{A}\times 100\%

Variables

A: measured value
delta A: absolute uncertainty in A, in the same units as A

Assumptions

A is nonzero; percent uncertainty is meaningless for a quantity consistent with zero.
The uncertainty describes random scatter, not a known calibration offset.

Sum or difference: absolute uncertainties combine

Q = a \pm b \;\Rightarrow\; \delta Q = \delta a + \delta b \;\text{ or }\; \sqrt{(\delta a)^2 + (\delta b)^2}

Variables

Q: derived quantity
a, b: measured quantities with the same units
delta a, delta b: absolute uncertainties

Assumptions

The linear form is an upper bound; the quadrature form assumes the two errors are independent and random.
Subtraction does not reduce the uncertainty — the absolute uncertainties still add.

Product, quotient, or power: relative uncertainties combine

Q = a^{m}b^{n} \;\Rightarrow\; \frac{\delta Q}{Q} = |m|\frac{\delta a}{a} + |n|\frac{\delta b}{b}

Variables

m, n: exponents (negative for a quantity in the denominator)
delta a / a, delta b / b: relative uncertainties

Assumptions

Uncertainties are small fractions of the values, so first-order propagation applies.
For independent random errors, replace the sum with the quadrature sum of the same terms.
The sign of an exponent does not matter; a quantity in the denominator contributes just as much as one in the numerator.

Standard error of the mean

\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{N}}

Variables

sigma: standard deviation of a single reading
N: number of independent readings averaged
sigma_mean: uncertainty in the average

Assumptions

Readings are independent and drawn from the same distribution.
Only random scatter shrinks; a systematic offset is untouched by N.

Counting statistics

\sigma_N = \sqrt{N}, \qquad \frac{\sigma_N}{N} = \frac{1}{\sqrt{N}}

Variables

N: total number of counts recorded
sigma_N: standard deviation of that total

Assumptions

Events arrive independently at a steady average rate (Poisson statistics).
Background and detector dead time are negligible or already subtracted.

Worked example

Uncertainty in a resistance measured from V and I

A student measures $V = 12.0 \pm 0.2\,\text{V}$ across a resistor and $I = 2.00 \pm 0.05\,\text{A}$ through it. Report $R = V/I$ with its uncertainty.

1Compute the central value: $R = V/I = 12.0/2.00 = 6.00\,\Omega$ (three significant figures, since both inputs carry three).
2Convert each uncertainty to a relative form, because $R$ is a quotient: $\delta V/V = 0.2/12.0 = 0.0167 \approx 1.7\%$ and $\delta I/I = 0.05/2.00 = 0.025 = 2.5\%$ .
3Combine the relative uncertainties. The conservative bound adds them: $\delta R/R = 1.7\% + 2.5\% = 4.2\%$ .
4Convert back to an absolute uncertainty: $\delta R = 0.042 \times 6.00 = 0.25\,\Omega$ .
5Round the uncertainty to one significant figure and the value to the same decimal place: $R = 6.0 \pm 0.3\,\Omega$ .
6If the two errors are treated as independent and random, the quadrature form gives $\sqrt{(1.7\%)^2 + (2.5\%)^2} = 3.0\%$ , or $\delta R = 0.18\,\Omega$ — a smaller, less conservative estimate. State which convention you used.

$R = 6.0 \pm 0.3\,\Omega$ using the conservative rule that percent uncertainties add ( $4.2\%$ ); treating the two errors as independent gives $6.00 \pm 0.18\,\Omega$ ( $3.0\%$ ). The current dominates the error budget either way, so a better ammeter buys more than a better voltmeter.

Common traps

Adding absolute uncertainties for a product or quotient. Only sums and differences combine absolute uncertainties; products combine relative ones, and the two have different units.
Adding relative uncertainties for a sum. A $2\%$ length plus a $2\%$ length is not a $2\%$ total unless the two lengths happen to be equal; convert to absolute first.
Dropping the exponent multiplier. If $V \propto r^3$ and $r$ is known to $1\%$ , then $V$ is known to $3\%$ , not $1\%$ .
Expecting more repeats to fix a calibration offset. Averaging shrinks random scatter as $1/\sqrt{N}$ and leaves systematic error exactly where it was.
Treating $1/\sqrt{N}$ as if it were $1/N$ : halving a counting uncertainty takes four times the counts, not twice.
Swapping the meters: an ammeter in parallel is a near short circuit, and a voltmeter in series nearly opens the branch.
Quoting a result to more digits than the uncertainty supports, which reports precision the apparatus never delivered.

GRE Physics overview DC Circuits Interference and Diffraction Sign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
University Physics Volume 1, Section 1.6: Significant Figures — OpenStax. Accessed 2026-08-15. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 2, Section 10.4: Electrical Measuring Instruments — OpenStax. Accessed 2026-08-15. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 4.6: Poisson Distribution — OpenStax. Accessed 2026-08-15. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 2.7: Measures of the Spread of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

Sources and corrections

Sources last checked 2026-08-15

Every source cited on this page was checked on the date shown, and we update the page when a source changes. If something looks wrong, tell us and we'll recheck it.

Report an issue with this page

Measurement Uncertainty and Error Analysis

Concise answer

Definitions

Intuition

Concept walkthrough

After this page, you should be able to

Formulas and assumptions

Percent uncertainty

Sum or difference: absolute uncertainties combine

Product, quotient, or power: relative uncertainties combine

Standard error of the mean

Counting statistics

Worked example

Uncertainty in a resistance measured from V and I

Common traps

Related pages and practice

Sources

Sources and corrections