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Propagate uncertainty through a ratio with a worked example

A concrete guide to propagating uncertainty through a ratio, with a synthetic worked example, unit checks, assumptions, and common mistakes.

November 5, 2026 · 8 min read · ResearchForge

A notebook gives you two measured quantities and asks for a ratio. For example, mass is 125.4 g and volume is 15.20 cm^3. Dividing them is easy: 125.4 / 15.20 = 8.25 g/cm^3. The harder part is the next question: what uncertainty should be attached to that ratio, and how can you tell the result is internally consistent?

A careful answer starts with the measurement model, not with a memorized percentage rule. NIST’s guidance says a measurement result is only complete when it includes a quantitative uncertainty statement, and it organizes uncertainty components as standard uncertainties that are later combined into a combined standard uncertainty using the law of propagation of uncertainty or an equivalent documented method NIST uncertainty components, NIST combined standard uncertainty.

This article shows a practical way to propagate uncertainty through a ratio, with a fully worked synthetic example, unit checks, assumptions, and a reusable procedure you can apply to other engineering or mathematics problems.

Define the ratio and the uncertainty inputs

Write the derived quantity as

y = A / B

where A is the numerator quantity and B is the denominator quantity.

That step matters because uncertainty propagation depends on the mathematical model. If you skip straight to “add the percentages in quadrature,” you can miss hidden assumptions such as independence, corrections, or a denominator that is too close to zero for a simple linear approximation to behave well.

For each input, record:

• the measured value,
• its unit,
• its standard uncertainty,
• how that standard uncertainty was obtained.

NIST separates uncertainty components by evaluation method: Type A components come from statistical analysis of repeated observations, while Type B components come from other information sources and are still represented as standard uncertainties NIST uncertainty components. NIST also notes that correlations between components are described using covariances or correlation coefficients, so independence is an assumption to state, not something to leave implicit NIST uncertainty components.

A useful working layout is:

A = value ± u(A)
B = value ± u(B)
y = A / B

Here u(A) and u(B) are standard uncertainties, not tolerances, not maximum possible errors, and not percentages unless you have explicitly converted them.

Derive the ratio formula instead of treating it as magic

The relative-uncertainty rule follows from the derivatives of the ratio. Derive it before using the shortcut.

For

y = A / B

the sensitivity coefficients are

∂y/∂A = 1/B
∂y/∂B = -A/B^2

Using the first-order law of propagation of uncertainty, which is the same framework NIST identifies for combining standard uncertainties NIST combined standard uncertainty, the propagated variance is

u(y)^2 ≈ (∂y/∂A)^2 u(A)^2 + (∂y/∂B)^2 u(B)^2

if A and B are treated as independent.

Substituting the derivatives gives

u(y)^2 ≈ u(A)^2 / B^2 + A^2 u(B)^2 / B^4

Since y = A/B, this can be rearranged into the familiar relative form:

(u(y)/y)^2 ≈ (u(A)/A)^2 + (u(B)/B)^2

Two distinctions are easy to lose and worth keeping explicit:

• u(y) is an absolute standard uncertainty. It has the same units as y.
• u(y)/|y| is a relative standard uncertainty for nonzero y. It is dimensionless.

That dimensionless-versus-dimensional split is not bookkeeping trivia. Use dimensional consistency as an algebra check.

If the inputs are not independent, include covariance:

u(y)^2 ≈ (∂y/∂A)^2 u(A)^2 + (∂y/∂B)^2 u(B)^2 + 2(∂y/∂A)(∂y/∂B) cov(A,B)

That covariance term is not optional when correlation is known to matter; it is part of the model.

Fully worked example: a synthetic density-style ratio

Hypothetical example using synthetic data.

Suppose a sample has:

• m = 125.4 g with standard uncertainty u(m) = 0.3 g
• V = 15.20 cm^3 with standard uncertainty u(V) = 0.08 cm^3

Define density-style ratio

ρ = m / V

This is a convenient example because the units make physical sense and are easy to check.

Step 1: Calculate the central value

ρ = 125.4 / 15.20 = 8.25 g/cm^3

Keep extra digits internally:

ρ = 8.2500 g/cm^3

Step 2: Convert the input uncertainties to relative form

u(m)/m = 0.3 / 125.4 = 0.002392
u(V)/V = 0.08 / 15.20 = 0.005263

These are dimensionless.

Step 3: Combine them, assuming independence

(u(ρ)/ρ)^2 ≈ (u(m)/m)^2 + (u(V)/V)^2

Substitute the numbers:

(u(ρ)/ρ)^2 ≈ (0.002392)^2 + (0.005263)^2
               ≈ 0.00000572 + 0.00002770
               ≈ 0.00003342

So

u(ρ)/ρ ≈ 0.00578136  (using the unrounded input ratios)

The relative standard uncertainty is therefore about 0.578%.

Step 4: Convert back to an absolute standard uncertainty

u(ρ) = 8.25 × 0.0057813617 ≈ 0.04769623 g/cm^3

Step 5: Report the result clearly

A clear report is:

ρ = 8.25 g/cm^3 with combined standard uncertainty u(ρ) = 0.05 g/cm^3

A compact equivalent is:

ρ = 8.25 ± 0.05 g/cm^3  (standard uncertainty)

That final label matters. NIST states that combined standard uncertainty represents an estimated standard deviation of the result, and under conditions such as an approximately normal distribution and a negligible uncertainty in u_c(y), the interval y ± u_c(y) corresponds to an approximate 68 percent level NIST combined standard uncertainty. So if you report standard uncertainty, say that you are doing so.

Check the result two different ways

A good ratio calculation should survive at least two checks: a derivative-form recalculation and a unit or dimensional check.

Independent numerical check using derivatives

Recompute the same propagated uncertainty without the relative shortcut.

∂ρ/∂m = 1/V = 1/15.20 = 0.06579 cm^-3
∂ρ/∂V = -m/V^2 = -125.4/(15.20)^2 = -0.54276 g·cm^-6

Now propagate:

u(ρ)^2 ≈ (0.06579)^2 (0.3)^2 + (0.54276)^2 (0.08)^2
       ≈ 0.0003895 + 0.00188539
       ≈ 0.00227493

Therefore

u(ρ) ≈ sqrt(0.00227493) ≈ 0.0477 g/cm^3

This matches the earlier result.

Unit and dimensional check

OpenStax describes volumetric mass density as having dimension M/L^3 OpenStax dimensional analysis. Since the numerator is mass and the denominator is volume, g/cm^3 is dimensionally consistent with the intended physical quantity.

NIST’s SI style guidance is also useful when writing the result. Unit symbols should be unambiguous, and repeated solidi should be avoided unless parentheses make the grouping clear NIST unit notation. The same chapter also says not to mix unit names and unit symbols in hybrid forms NIST unit notation. In practice:

• g/cm^3 is clear.
• grams/cm^3 is a mixed style and better avoided in symbolic expressions.
• u(ρ) should use the same unit as ρ.

A suggested heuristic: if the ratio unit looks awkward, rewrite it once using negative exponents and once using a fraction bar. If the two forms do not obviously mean the same thing, the notation may be too ambiguous for publication or grading.

Reusable checklist for ratio uncertainty problems

The following is a suggested heuristic, not a source-proven universal procedure.

A compact decision procedure is:

1. Require B != 0 and write y = A/B.
   The relative form also requires A != 0; otherwise use derivatives.
2. If A and B are independent:
      ry = sqrt((u(A)/A)^2 + (u(B)/B)^2)
   else:
      use the covariance form
3. u(y) = |y| ry
4. Check units and dimensions
5. Report y with u(y) and the uncertainty type

Assumptions, limits, and common mistakes

Independence is an assumption, not a default fact

If the same calibration constant, environmental correction, or instrument affects both numerator and denominator, the covariance can shift the result. NIST explicitly recognizes correlation as part of uncertainty evaluation NIST uncertainty components.

The simple ratio rule is a first-order approximation

The formulas above are a derived first-order linearization of the uncertainty model, not a direct quotation from the sources. A suggested heuristic: be cautious if the denominator has a large relative uncertainty or if plausible denominator values approach zero. In those cases, the ratio can behave badly and a first-order approximation may be a poor summary.

Corrections still carry uncertainty

NIST says recognized systematic effects should be corrected when appropriate, and the uncertainty attached to a correction is the standard uncertainty of that correction NIST combined standard uncertainty. So if either A or B was corrected before forming the ratio, do not drop the uncertainty of that correction.

Common mistakes to catch early

• Mixing absolute and relative uncertainty

Adding u(A) and u(B) directly for a ratio is not the same as propagating uncertainty.

• Forgetting units on uncertainty

If y is in g/cm^3, then u(y) must also be in g/cm^3.

• Reporting a bare ± value without saying what it is

A reader should know whether the number is a standard uncertainty.

• Ignoring correlation silently

If independence is assumed, say so.

• Skipping the dimensional check

A wrong unit can reveal a wrong model even when the arithmetic is flawless.

ResearchForge describes itself as a site for engineering and math calculators, articles, and locally reproduced research ResearchForge. For a reader working through homework-pattern or engineering-style derivations, the main practical lesson here is the same one such a workflow should reinforce: a ratio result is not finished when the division is done.

When you propagate uncertainty through a ratio, the reliable path is to write the model, keep standard uncertainties attached to the inputs, state the independence assumption, compute the combined standard uncertainty, and then verify the answer with units and an independent calculation path. That turns a one-line division into a result someone else can actually audit.