Measurement sensitivity and uncertainty
Dilution Sensitivity Calculator
Propagate small independent relative standard uncertainties through C2=C1V1/V2 and compare the RSS result with a separate first-order worst-case bound.
CURRENT MODEL
Enter the nominal dilution and relative standard uncertainties
Laboratory analysts and students screening which stock, aliquot, or final-volume uncertainty most influences a dilution result.
Decision supportedIdentify the largest entered uncertainty contribution and decide whether the nominal dilution needs better measurement evidence or a covariance-aware uncertainty budget.
LIVE DECISION VIEW
Relative uncertainty contributions and combined markers
Horizontal contribution bars share one percentage scale; separate markers show the RSS combination and the conservative first-order sum without conflating them.
| Quantity | Symbol | Nominal value | Unit | Relative uncertainty (percent) | Elasticity |
|---|
Subject illustration
Three measurement records converge on one concentration
The editorial scene shows evidence provenance; the live contribution chart above carries the current percentages and combined bounds.
How to use
Screen a dilution uncertainty budget without hiding its assumptions
- Enter the nominal stock concentration, delivered aliquot, and final solution volume.
- Enter one-standard-uncertainty relative percentages for C1, V1, and V2 from traceable evidence.
- Keep percentages nonnegative and distinguish uncertainty from known bias or correction.
- Read the contribution bars to identify which input dominates the current RSS combination.
- Compare the RSS result with the separately labeled first-order worst-case sum.
- Review independence, linearity, denominator, and coverage limitations before interpreting the interval.
Sensitivity fundamentals
Five ideas behind the first-order uncertainty result
- Nominal measurand
- The prepared concentration C2=C1V1/V2 evaluated at the entered best estimates.
- Relative standard uncertainty
- A standard uncertainty divided by its quantity value, entered here as a percentage.
- Elasticity
- The fractional response coefficient: +1 for C1 and V1, and -1 for V2.
- RSS combination
- The square root of summed squared independent relative contributions.
- Worst-case bound
- A separate first-order magnitude sum representing aligned adverse changes, not a probability statement.
Calculation method
Propagate the product-and-quotient sensitivities
Logarithmic differentiation gives dC2/C2=dC1/C1+dV1/V1-dV2/V2. For mutually independent standard uncertainties, the signed elasticities are squared, so their contribution magnitudes combine by root sum of squares.
The live view plots the three entered relative contributions on one percentage axis and overlays the combined RSS and arithmetic-sum markers. This makes dominance and the gap between statistical and conservative combinations visible.
Sign and magnitude answer different questions
The negative V2 elasticity means an upward final-volume change lowers concentration. RSS uses squared magnitudes, while directional scenario analysis must retain the sign.
Independence is a model claim
Using the same calibration reference, temperature correction, balance, or operator can correlate inputs. Covariance terms may increase or decrease the combined result and cannot be inferred from this simplified page.
Uncertainty is not tolerance
A manufacturer tolerance, resolution digit, observed repeatability, calibration certificate uncertainty, and process specification are different quantities. Convert evidence to standard uncertainties before entering percentages.
Detailed calculation process
Symbols, current substitution, intermediate quantities, and reconciliation
C2=C1V1/V2; u_r(C2)=sqrt(u_r(C1)^2+u_r(V1)^2+u_r(V2)^2)Percentages are converted once to decimals; nominal concentration, RSS, absolute uncertainty, interval, and worst-case sum use unrounded values before display formatting.
| Symbol | Meaning | Default | Unit |
|---|---|---|---|
| C1 | Nominal stock concentration | 1.00 | mol/L |
| V1 | Nominal stock aliquot | 10.0 | mL |
| V2 | Nominal final volume | 100 | mL |
| C2 | Nominal diluted concentration | calculated | mol/L |
| u_r(C1) | Relative standard uncertainty of stock concentration | 1.00 | % |
| u_r(V1) | Relative standard uncertainty of aliquot volume | 0.50 | % |
| u_r(V2) | Relative standard uncertainty of final volume | 0.20 | % |
| u_r(C2) | Combined relative standard uncertainty | calculated | % |
Waiting for valid inputs.
Result interpretation
Use the dominant contributor to choose the next measurement improvement
The largest entered percentage is the first screening target, but reducing it helps only if its estimate is credible and correlations are negligible. The displayed nominal plus/minus standard-uncertainty interval is not automatically a 95% interval, acceptance band, or worst-case range.
Evidence to retain
Make every uncertainty input traceable
Retain stock certificate and lot, concentration correction, pipette and flask calibration certificates, resolution, repeatability data, temperature and expansion correction, distribution assumption, divisor used to obtain standard uncertainty, degrees of freedom where relevant, covariance assessment, calculation version, reviewer, and measurement date.
Scope and limitations
Conditions outside this independent first-order screen
- Correlation or covariance among concentration and volume inputs
- Known bias, calibration correction, drift, evaporation, or transfer loss
- Large uncertainty, strong nonlinearity, or asymmetric distributions
- Final volume near zero or a model with discontinuities and clipping
- Coverage-factor, confidence-level, tolerance, or conformity decisions
- Additional contributors such as purity, density, temperature, mixing, and sampling
Positive nominal concentration and volumes; nonnegative relative standard uncertainties; small first-order changes; mutually independent inputs for the RSS result; no confidence-level claim.
Key terminology
Measurement-sensitivity glossary
- Measurand
- The quantity intended to be measured or calculated, here the diluted concentration C2.
- Standard uncertainty
- An uncertainty expressed as a standard deviation for one input or the result.
- Relative uncertainty
- Standard uncertainty divided by the magnitude of its associated quantity.
- Sensitivity coefficient
- The derivative that maps a small input change into a change in the measurand.
- Covariance
- A measure of joint variation that must be included when input errors are not independent.
- Coverage factor
- A multiplier used with combined standard uncertainty to form an expanded uncertainty under stated conditions.
Practical cases
Two budgets that lead to different actions
Stock certificate dominates
With 1.0% stock, 0.5% aliquot, and 0.2% flask relative standard uncertainties, the combined value is about 1.136%. Improving only the flask has limited impact; the stock evidence deserves first review.
Shared-temperature correlation
A pipette and flask used at the same off-reference temperature may share a systematic volume effect. The independent RSS number remains an arithmetic screen, but a defensible budget must add covariance and the temperature correction before use.
Important note
This screen does not certify measurement fitness
A complete uncertainty statement requires documented input distributions, correlations, corrections, degrees of freedom, coverage choice, and decision rule appropriate to the laboratory purpose.
Frequently asked questions
Why is the final-volume elasticity negative?
Final volume is in the denominator, so increasing V2 lowers C2. The sign matters for directional sensitivity, while RSS squares the contribution magnitude.
Is the RSS interval a confidence interval?
No. It is the nominal value plus or minus one combined standard uncertainty under the entered model. Coverage probability requires a distribution and coverage-factor analysis.
Why is the worst-case sum larger than RSS?
RSS treats independent standard uncertainties as orthogonal variance contributions. The summed magnitude assumes all first-order effects align adversely and is a separate conservative bound.
Can correlated volume errors be entered here?
Not correctly with this simplified RSS equation. Material correlations require covariance terms and a documented uncertainty budget.
What does zero uncertainty mean?
It is a valid arithmetic boundary that produces zero propagated uncertainty, but it should be entered only when supported; real measurement processes rarely have literally zero uncertainty.
When does first-order propagation become unreliable?
Large relative errors, nonlinear response, a denominator near zero, asymmetric distributions, clipping, or strong correlations require simulation or a fuller propagation method.
Authority and follow-on work
Reliable sources and related calculators
- NIST TN 1297 - Combined Standard UncertaintyDefines first-order uncertainty propagation and covariance-aware combination.
- NIST - Standard and Relative Standard UncertaintyDefines standard uncertainty and relative standard uncertainty.
- OpenStax Chemistry - Molarity and DilutionProvides the nominal C2=C1V1/V2 measurand relation.