Measurement Uncertainty: Read ± and Compare a Reference

Nablio · 6 questionsPublished October 11, 2026

A result such as $g=(9.6\pm0.3)\,\mathrm{m/s^2}$ contains two numbers with different jobs. The first is an estimate. The second describes its uncertainty under a stated convention. Before deciding whether the result agrees with a reference, find out what that convention is.

Before you start: use units, percentages, square roots and simple substitutions. The core lesson needs no calculus; an optional derivation explains the propagation formula.

Goal: explain a reported measurement, distinguish uncertainty from a difference with a reference, and propagate uncertainty using an appropriate model. All data and exercises here are original teaching examples, not an assessed lab report.

1. Read the meaning of ±

Let $x$ denote a measured estimate. An absolute uncertainty has the same unit as $x$ . “Absolute” describes the units; it does not say the uncertainty is a guaranteed bound.

Standard uncertainty $u(x)$ : an uncertainty expressed as an estimated standard deviation.
Expanded uncertainty $U$ : $U=k\,u_c$ , where $u_c$ is the combined standard uncertainty and $k$ is the stated coverage factor. A coverage probability needs further assumptions.
A stated bound $b$ : the exercise may instead assume the value is within $[x-b,x+b]$ . Such a bound is not automatically a standard uncertainty.

The symbol ± alone does not identify which meaning applies. Check the lab instructions, instrument specification or uncertainty statement. A confidence level cannot be read from that symbol alone. Even when ± means one standard uncertainty, “about 68%” needs an appropriate approximately normal model and a reliable uncertainty estimate.

For a nonzero estimate, relative standard uncertainty compares the uncertainty with the magnitude of the estimate:

$u_r(x)=\frac{u(x)}{|x|}.$

Multiply by 100 to express it as a percentage. The ratio is dimensionless. At $x=0$ , this relative form is undefined; use an absolute uncertainty.

2. Compare a measurement with a reference

A gravity measurement

For this example, define $g_m=9.6\,\mathrm{m/s^2}$ and $u(g_m)=0.3\,\mathrm{m/s^2}$ . Thus $g=(9.6\pm0.3)\,\mathrm{m/s^2}$ uses a standard uncertainty. Use the exercise's reference $g_{\mathrm{ref}}=9.8\,\mathrm{m/s^2}$ , treating its uncertainty as negligible here. This reference is a supplied comparison value, not a claim that gravity is identical everywhere.

Signed difference: $g_m-g_{\mathrm{ref}}=-0.2\,\mathrm{m/s^2}$ . The measured estimate is lower.
Absolute discrepancy: $|g_m-g_{\mathrm{ref}}|=0.2\,\mathrm{m/s^2}$ .
Percentage error relative to this reference: $100|g_m-g_{\mathrm{ref}}|/|g_{\mathrm{ref}}|\approx2.04\%$ . The denominator is the reference.
Relative standard uncertainty: $100u(g_m)/|g_m|=3.125\%$ , or about $3.1\%$ . The denominator is the measured estimate.

The one-standard-uncertainty interval is $[9.3,9.9]\,\mathrm{m/s^2}$ , which contains 9.8. The discrepancy is about $0.67$ standard uncertainties:

$\frac{|g_m-g_{\mathrm{ref}}|}{u(g_m)}=\frac{0.2}{0.3}\approx0.67.$

Interpretation: the reference lies within the stated ± interval. This observation does not prove that the measurement is correct, that all systematic effects were included, or that a particular confidence level applies. A formal agreement test requires a defined model and decision rule.

In introductory exercises, “percentage error” often names the reference-relative calculation above. It is a comparison with the supplied reference, not a measurement of the unknown true error. It is undefined for a zero reference. A small percentage error does not demonstrate a small measurement uncertainty; an estimate can land near the reference by chance.

Optional extension: an uncertain reference

Define the signed difference $\Delta g=g_m-g_{\mathrm{ref}}$ . If the reference itself has standard uncertainty $u(g_{\mathrm{ref}})$ , and the two estimates are independent, the standard uncertainty of their difference is:

$u(\Delta g)=\sqrt{u(g_m)^2+u(g_{\mathrm{ref}})^2}.$

Compare $|\Delta g|$ with $u(\Delta g)$ , rather than ignoring the reference uncertainty. Correlation changes this expression. Overlapping error bars alone are not a universal statistical test.

3. Identify sources of uncertainty

Begin with how the measurement was made. Repeated readings reveal scatter, but do not expose every effect. A ruler with a shifted zero can produce tightly grouped readings that all share an offset. Correct a known offset where appropriate and account for uncertainty in the correction.

For independent repeated measurements under stable conditions, the sample mean estimates the quantity. The sample standard deviation $s$ describes the scatter of individual readings. The standard uncertainty contribution to the mean from that scatter is $s/\sqrt n$ , where $n$ is the number of readings. This is not automatically the total uncertainty. Repetition does not remove a common calibration offset.

An instrument's display step is useful information, but is not automatically its total uncertainty. For example, if rounding is the only effect and the rounding error is modeled as uniform from $-q/2$ to $+q/2$ , a display step $q$ contributes standard uncertainty $q/\sqrt{12}$ . A different specification or model can give a different result.

Reporting: keep extra digits during calculations, then round the uncertainty and estimate together. In this lesson, calculated uncertainties are reported with two significant digits, and the estimate is rounded to the same decimal place. Follow a different convention if your lab specifies one. Rounding communicates the calculation; it does not establish the instrument's accuracy.

4. Propagate standard uncertainties

Suppose a cart travels a measured distance $d=1.20\,\mathrm m$ in measured time $t=0.80\,\mathrm s$ . Standard uncertainties are $u(d)=0.02\,\mathrm m$ and $u(t)=0.01\,\mathrm s$ . These inputs are independent, their relative uncertainties are small, and the measurement model is $v=d/t$ . Here $v$ is average speed over that distance. Assume other effects are negligible in this teaching example.

Estimate first, uncertainty second

$v=\frac{1.20}{0.80}=1.50\,\mathrm{m/s}.$

For independent inputs, first-order propagation gives:

$\begin{gathered}\frac{u(v)}{|v|}\approx\\ \sqrt{\left(\frac{u(d)}d\right)^2+\left(\frac{u(t)}t\right)^2}.\end{gathered}$

$\begin{gathered}\frac{u(v)}{|v|}\approx\\ \sqrt{\left(\frac{0.02}{1.20}\right)^2+\left(\frac{0.01}{0.80}\right)^2}\\ =0.020833\ldots\end{gathered}$

$\begin{gathered}u(v)\approx1.50(0.020833\ldots)\\ =0.03125\,\mathrm{m/s}.\end{gathered}$

With the reporting convention above, write $v=(1.500\pm0.031)\,\mathrm{m/s}$ , where ± denotes the combined standard uncertainty from this first-order model. The displayed value 1.500 aligns with the uncertainty's decimal place; it does not imply an exact measurement.

Why do the relative terms add in quadrature?

For $v=d/t$ , the sensitivity coefficients are $\partial v/\partial d=1/t$ and $\partial v/\partial t=-d/t^2$ . With zero covariance:

$\begin{aligned}u(v)^2&\approx\left(\frac1t\right)^2u(d)^2\\ &\quad+\left(\frac d{t^2}\right)^2u(t)^2.\end{aligned}$

Dividing by $v^2=d^2/t^2$ gives the relative formula. The minus sign disappears on squaring. If the inputs are correlated, the variance expression also contains $2(1/t)(-d/t^2)\operatorname{cov}(d,t)$ ; do not discard it without justification.

Optional extension: powers and products

For an exact constant $C$ and independent inputs in $y=Cx^az^b$ , first-order propagation gives $u(y)/|y|\approx\sqrt{a^2[u(x)/x]^2+b^2[u(z)/z]^2}$ , where the inputs and output are nonzero and the model is smooth over their relevant range. Exponents multiply relative contributions. If a constant is measured rather than exact, include its uncertainty too. Large uncertainties, strong nonlinearity or a denominator near zero need more care than this approximation.

5. Use bounds when bounds are supplied

Now change the assumptions: suppose the same numerical ± values are bounds, with $d\in[1.18,1.22]\,\mathrm m$ and $t\in[0.79,0.81]\,\mathrm s$ . Assume every combination in these intervals is allowed. Because both quantities are positive, speed is smallest at the smallest distance and largest time, and largest at the largest distance and smallest time:

$v_{\min}=\frac{1.18}{0.81}\approx1.45679\,\mathrm{m/s}.$

$v_{\max}=\frac{1.22}{0.79}\approx1.54430\,\mathrm{m/s}.$

An outward-rounded enclosing interval is $[1.456,1.545]\,\mathrm{m/s}$ . It is not the ±0.031 standard-uncertainty result. The input meanings changed, so the calculation changed. A first-order sum of absolute contributions can approximate a worst-case bound, but the endpoint calculation here gives the actual range under the stated interval model.

6. Try six questions

Identify the uncertainty convention before calculating. Each exercise adds one decision.

1. What is missing?

A label reads $L=(20.0\pm0.2)\,\mathrm{cm}$ with no other explanation. Find the endpoints and identify what you cannot infer.

Show a hint

Arithmetic determines endpoints. The symbol does not determine coverage.

Show the explained solution

The displayed interval is 19.8 to 20.2 cm. We do not know whether 0.2 cm is a standard uncertainty, expanded uncertainty or bound. We cannot assign a confidence level. Ask for the uncertainty definition, method and any coverage factor.

2. Choose the denominator

An estimate is $4.80\,\mathrm s$ , its standard uncertainty is $0.12\,\mathrm s$ , and a supplied reference is $5.00\,\mathrm s$ . Calculate absolute discrepancy, percentage error relative to the reference and relative standard uncertainty.

Show a hint

The two percentage calculations use different denominators.

Show the explained solution

The absolute discrepancy is $|4.80-5.00|=0.20\,\mathrm s$ . Percentage error is $100(0.20/5.00)=4.0\%$ . Relative standard uncertainty is $100(0.12/4.80)=2.5\%$ . None of these values substitutes for the others.

3. Include the reference uncertainty

Use the gravity example, but now let the independent reference have standard uncertainty $0.4\,\mathrm{m/s^2}$ . Find the standard uncertainty of the difference and the discrepancy divided by that uncertainty.

Show a hint

Combine standard uncertainties in quadrature before forming the ratio.

Show the explained solution

$u(\Delta g)=\sqrt{0.3^2+0.4^2}=0.50\,\mathrm{m/s^2}$ . The ratio is $0.2/0.5=0.4$ . The discrepancy is smaller than one standard uncertainty of the difference. No confidence level or formal acceptance rule has been supplied.

4. Does repetition remove every uncertainty?

For 16 independent readings under stable conditions, $s=0.08\,\mathrm s$ . Find the standard uncertainty contribution to the mean from scatter. Would taking more readings remove an unknown common timing offset?

Show a hint

Separate the spread of individual readings from uncertainty in their mean.

Show the explained solution

$u_{\mathrm{scatter}}(\bar t)=0.08/\sqrt{16}=0.020\,\mathrm s$ . A common offset is not reduced by averaging. Investigate the timing method or calibration and include the relevant uncertainty contribution; 0.020 s alone may not be the total.

5. A power in the measurement model

A square's side is $L=0.500\,\mathrm m$ , with standard uncertainty $u(L)=0.005\,\mathrm m$ . Use first-order propagation for $A=L^2$ , assuming this is the only uncertainty source. Report the result with two significant digits in the uncertainty.

Show a hint

Both appearances of $L$ are the same measured input, not two independent measurements.

Show the explained solution

$A=0.250\,\mathrm{m^2}$ . The relative standard uncertainty is $2u(L)/L=0.020$ , so $u(A)\approx0.0050\,\mathrm{m^2}$ . Report $A=(0.2500\pm0.0050)\,\mathrm{m^2}$ , with ± denoting standard uncertainty. Treating $L\times L$ as independent inputs would incorrectly give a factor $\sqrt2$ instead of 2.

6. Bounds on a difference

Assume $A\in[9.8,10.2]\,\mathrm{cm}$ and $B\in[6.9,7.1]\,\mathrm{cm}$ , with every combination allowed. Find the exact interval for $A-B$ . Why is quadrature inappropriate for these supplied bounds?

Show a hint

Subtract the largest B from the smallest A to find the minimum.

Show the explained solution

The minimum is $9.8-7.1=2.7\,\mathrm{cm}$ ; the maximum is $10.2-6.9=3.3\,\mathrm{cm}$ . Thus $A-B\in[2.7,3.3]\,\mathrm{cm}$ , equivalently a centre of 3.0 cm with bound ±0.3 cm. The supplied half-widths are bounds, not standard uncertainties with an independent-input probability model.

7. Choose your next step

Before reporting a result, check: What quantity did I estimate? What does ± mean? Are all quantities in compatible units? What assumptions justify my propagation rule? What comparison does my conclusion actually support?

For a mechanics application, revisit motion graphs and kinematics. To practise interpreting a straight-line slope, try points, slope and straight-line graphs. A slope from measured data can also have uncertainty; this lesson does not cover line-fitting uncertainty.

For physics tutoring, bring the uncertainty convention you were given and a step you have tried. We can work through the choice of model, units and interpretation using a fresh practice example. Follow your course's academic-integrity rules for assessed work; your submitted analysis and report should remain your own.

8. References and course scope

This independent, topic-first lesson is for introductory physics measurement practice. Use it when measurement uncertainty appears in your current course or lab instructions. It does not establish the syllabus or assessment requirements of any current course section.

The examples and practice questions were written for this lesson. The references support the uncertainty conventions and their assumptions.

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Measurement Uncertainty: Read ± and Compare a Reference

1. Read the meaning of ±

2. Compare a measurement with a reference

A gravity measurement

3. Identify sources of uncertainty

4. Propagate standard uncertainties

Estimate first, uncertainty second

5. Use bounds when bounds are supplied

6. Try six questions

1. What is missing?

2. Choose the denominator

3. Include the reference uncertainty

4. Does repetition remove every uncertainty?

5. A power in the measurement model

6. Bounds on a difference

7. Choose your next step

8. References and course scope

Useful for

Work on the step you’re missing.

Free study guides and practice

Motion Graphs: From the Story to the Equation

Points, Slope and Straight-Line Graphs