How It Works
This calculator works in 4 steps.
Enter the per-step random uncertainty
Type the relative uncertainty of one transfer-and-dilute step.
Enter the systematic bias
Type a constant per-step bias, or 0.
Enter the step count
Type the number of serial steps.
Read the per-tube results
The calculator returns random, systematic, and combined uncertainty for each tube.
The Formula
The formula multiplies the per-step random uncertainty by the square root of the step count and the per-step bias by the step count.
u_n = u_step × √nb_n ≈ n × b_stepU_n = 2 × u_n- u_step:
- Relative random uncertainty per step
- b_step:
- Relative bias per step
- n:
- Step number
At 1% random uncertainty per step, tube 6 carries 1% × √6 = 2.45%. A 1% bias per step adds up to about 6% by tube 6.
Conversion Table
The conversion table compares random and linear-bias growth for 6 step counts.
| Per-step uncertainty | Steps | Random (√n) | Bias (linear) | Random expanded (k = 2) |
|---|---|---|---|---|
| 1% | 1 | 1% | 1% | 2% |
| 1% | 3 | 1.73% | 3% | 3.46% |
| 1% | 6 | 2.45% | 6% | 4.9% |
| 2% | 6 | 4.9% | 12% | 9.8% |
| 0.5% | 8 | 1.41% | 4% | 2.83% |
| 1.5% | 10 | 4.74% | 15% | 9.49% |
What Is the Serial Dilution Error Calculator?
The Serial Dilution Error Calculator is a tool that tracks how pipetting uncertainty grows from tube to tube across a serial dilution. Random errors add in quadrature, so n steps of equal relative uncertainty u give a combined uncertainty of u × √n.
Random error grows with the square root of the step count, while a systematic bias, such as a pipette that delivers 1% high every time, grows linearly. After 6 steps at 1% bias, the last tube is about 6% off. The calculator reports both models side by side, so the user sees the best case and the worst case.
Common Uses
There are 5 main uses of this calculator.
Standard curve QC
Analysts estimate error at the lowest standard.
Design comparison
Users compare serial and parallel designs.
Plate count uncertainty
Microbiologists add dilution error to count uncertainty.
Pipetting training
Trainers show how small per-step errors grow.
Audit support
QA teams justify step limits in SOPs.