How It Works
This calculator works in 4 steps.
Enter the stock uncertainty
Type the relative standard uncertainty of C₁ from the certificate.
Enter the stock volume uncertainty
Type the pipette's relative uncertainty at the volume used.
Enter the final volume uncertainty
Type the flask or cylinder relative uncertainty.
Read the combined result
The calculator returns combined relative uncertainty, absolute uncertainty, and expanded uncertainty at k = 2.
The Formula
The formula combines relative standard uncertainties in quadrature.
u(C₂)/C₂ = √[(u(C₁)/C₁)² + (u(V₁)/V₁)² + (u(V₂)/V₂)²]U = k × u(C₂), k = 2u = a / √3- u(x)/x:
- Relative standard uncertainty of input x
- k:
- Coverage factor
- a:
- Half-width of a stated ± tolerance
Stock 0.5%, pipette 0.8%, and flask 0.1% combine to √(0.25 + 0.64 + 0.01) = 0.949%. The expanded uncertainty is 1.9% at k = 2.
Conversion Table
The conversion table combines 6 sets of relative uncertainties into standard and expanded values.
| u(C₁) | u(V₁) | u(V₂) | Combined u(C₂) | Expanded (k = 2) |
|---|---|---|---|---|
| 0.5% | 0.8% | 0.1% | 0.949% | 1.9% |
| 1% | 1% | 0.2% | 1.43% | 2.86% |
| 0.2% | 0.3% | 0.05% | 0.364% | 0.728% |
| 2% | 1.5% | 0.5% | 2.55% | 5.1% |
| 0.1% | 5% | 0.1% | 5% | 10% |
| 0.5% | 0.5% | 0.5% | 0.866% | 1.73% |
What Is the Dilution Uncertainty Calculator?
The Dilution Uncertainty Calculator is a tool that propagates the uncertainty of the stock concentration, stock volume, and final volume into the combined uncertainty of the diluted concentration. For C₂ = C₁V₁ / V₂, relative uncertainties combine as a root sum of squares.
The method follows the Guide to the Expression of Uncertainty in Measurement (GUM). Inputs come from the certificate of analysis, pipette calibration data, and flask tolerance. The calculator returns the standard uncertainty and the expanded uncertainty at a coverage factor of k = 2, which corresponds to about 95% confidence.
Common Uses
There are 5 main uses of this calculator.
ISO/IEC 17025 budgets
Accredited labs include dilution steps in uncertainty budgets.
Standard certificates
Reference labs state uncertainty on prepared standards.
Method validation
Validation teams find the largest contributor to uncertainty.
Pipette choice
Analysts compare uncertainty with different pipettes.
Proficiency reporting
Labs report results with expanded uncertainty.