Chemistry tools
Measurement uncertainty calculator

Measurement uncertainty calculator: concentration from weighing and volume

Calculate concentration and dilution uncertainty from weighing error, volume error, and related inputs directly in your browser.

Review combined standard uncertainty uc, expanded uncertainty U=k·uc, main contributors, and report-ready output in one place.

Calculate combined standard uncertainty uc and expanded uncertainty U=k·uc
Convert tolerance notation to standard uncertainty instantly
Highlight the main contributors so you know what to improve
Share URLs, copy output, and save drafts locally
Calculation template
Coverage factor k
k=2 is a common approximate 95% choice. Follow your standard or internal rule when required.

What this page covers

Combine weighing and volume uncertainty for concentration

Combine standard uncertainties for formulas such as C=m/V, C=(m·P)/V, and C=(m·P)/(M·V).

Convert tolerance notation to standard uncertainty

Handle certificate-style ±a(k=2), specification-style ±a, and triangular assumptions without a separate manual conversion.

See the dominant contributors

The contributor breakdown shows which factor dominates the variance, so you can target improvements efficiently.

Copy output ready for reports

Switch among plain text, Markdown, CSV, and JSON, then copy the exact format you need.

How to use the calculator

  1. Choose a template: concentration, dilution, ratio, or custom.
  2. Enter each factor and its uncertainty using SD or tolerance notation.
  3. Choose a distribution and coverage factor k when needed.
  4. Review the result, main contributors, and report-ready output before pasting it into a document.

Examples

mg/L uncertainty from weighing and a volumetric flask

Input

m=100.00 mg ±0.10 (rectangular), V=100.00 mL ±0.08 (normal, k=2)

Output

Shows concentration, uc, U, and the contribution split between m and V.

Dilution of a standard solution

Input

C1=1000 mg/L ±5, V1=10.00 mL ±0.02, V2=100.00 mL ±0.08

Output

Shows U for the diluted concentration and which volume error matters most.

Ratios and recovery

Input

A=98.0 ±0.5, B=100.0 ±0.2

Output

Shows uncertainty for A/B or Recovery(%).

Molarity with purity and molar mass

Input

Enter m, P, M, and V together

Output

Shows the contribution from purity and molar mass as well.

Glossary

Standard uncertainty u(x)

The standard-deviation-like uncertainty associated with an input quantity x.

Combined standard uncertainty uc

The standard uncertainty of result y after all input contributions are combined.

Expanded uncertainty U

The report-facing uncertainty calculated as U = k·uc.

Sensitivity coefficient c

A coefficient showing how strongly the result changes when one input changes.

Contribution ratio

The share of total variance attributable to one factor.

Calculation formulas

  • Standard uncertainty conversion: u=a/k, a/√3, or a/√6
  • Combined standard uncertainty: uc = √Σ(cᵢ·uᵢ)²
  • Expanded uncertainty: U = k·uc
  • Concentration: C = m/V, (m·P)/V, or (m·P)/(M·V)
  • Dilution: C2 = C1·V1/V2
  • Ratio: R = A/B

Frequently asked questions

I only have a ± value, not a standard deviation. What should I enter?

Yes. Choose a distribution assumption to convert tolerance notation into standard uncertainty: u=a/k for normal(k), u=a/√3 for rectangular, or u=a/√6 for triangular.

Which coverage factor k should I use?

k=2 is common, but follow your standard, internal rule, or customer requirement. The chosen k is always shown in the result.

Can I paste the output directly into a report?

Yes. Copy plain text, Markdown, CSV, or JSON; the output includes assumptions, inputs, results, and the main contributors.

Can I use it when inputs are correlated?

The first release assumes independent inputs. With correlated inputs, the result may be under- or over-estimated.

Is it reliable for large errors or strongly nonlinear formulas?

This calculator uses first-order propagation. For large relative uncertainties or strongly nonlinear formulas, verify the result with another method.

Notes

  • This calculator assumes independent inputs and uses first-order propagation.
  • With correlated inputs or a strongly nonlinear formula, the result may be under- or over-estimated.
  • Automatic conversion among arbitrary concentration units is not included. Enter the unit you intend to use.
  • Share URLs include free-form labels and formulas; do not enter confidential names.