Serial Dilution Calculator
Serial Dilution Calculator
Find the concentration at the end of a serial dilution series, and see how quickly the total dilution — and the accumulated pipetting error — grows with each step.
Serial dilution
Start ÷ (factor to the power n)A 1,000 unit/mL stock, diluted 1 in 10 at each of 5 steps
Formula
total dilution = 1 in DFⁿ
- C₀
- starting concentration, in any unit; the answer is in the same unit
- DF
- the dilution factor applied at each step — 10 for a 1-in-10 dilution, 2 for a doubling dilution
- n
- the number of transfers, not the number of tubes: the neat starting tube is step 0
- DFⁿ
- the total dilution, which multiplies rather than adds — five 1-in-10 steps is 1 in 100,000, not 1 in 50
Worked example
A 1,000 unit/mL stock, diluted 1 in 10 at each of 5 steps
Total dilution = 10⁵ = 100,000-fold
1,000 ÷ 100,000 = 0.0100 unit/mL
Note that this is 1 in 100,000, not 1 in 50
A 1-in-10 series from 1,000 units/mL
| Tube | Steps completed | Total dilution | Concentration |
|---|---|---|---|
| Neat | 0 | 1 in 1 | 1,000 |
| 1 | 1 | 1 in 10 | 100 |
| 2 | 2 | 1 in 100 | 10 |
| 3 | 3 | 1 in 1,000 | 1 |
| 4 | 4 | 1 in 10,000 | 0.1 |
| 5 | 5 | 1 in 100,000 | 0.01 |
How a constant pipetting bias accumulates
| Steps completed | Compounded 5% bias | Error in the final concentration |
|---|---|---|
| 1 | 1.05 | 5% |
| 2 | 1.05² = 1.103 | 10% |
| 3 | 1.05³ = 1.158 | 16% |
| 4 | 1.05⁴ = 1.216 | 22% |
| 5 | 1.05⁵ = 1.276 | 28% |
Where a dilution series goes wrong
A serial dilution is a repeated fixed-ratio transfer, and the total dilution is the ratio raised to the power of the number of transfers. That geometry is the whole point of the technique and also the commonest source of confusion: five 1-in-10 steps is a 1 in 100,000 dilution, not 1 in 50. Count transfers rather than tubes, since the neat starting tube has had no dilution applied to it, and an off-by-one here is an off-by-tenfold in the answer.
Errors compound the same way the dilutions do. A pipette running 5% high, or a technique that consistently under-mixes before transferring, applies the same bias at every step, so by the fifth tube the accumulated factor is 1.05⁵, roughly 1.28 — a 28% error in a number the operator has no independent way of checking. Random error accumulates more slowly than that, but systematic error does not average out, which is why the calibration of the pipette used for the series matters more than its nominal precision suggests.
Technique at each step is what keeps the series honest. Mix thoroughly before drawing the next aliquot: an unmixed tube transfers whatever happens to be at the tip depth rather than a representative sample, and a dilution series is unforgiving of that because the mistake is carried down every subsequent tube. Change tips between steps as well, because the film of concentrated solution left on the outside of a reused tip is negligible against tube 1 and dominant against tube 6 — carryover is what sets the practical floor of the series, not the arithmetic.
Doubling dilutions, where the factor is 2, are the convention for antibody titres and for minimum inhibitory concentration testing, and the reason is resolution: a twofold series brackets an endpoint closely enough to be meaningful while remaining practical to set up across a microtitre plate. Titres are then reported as the reciprocal of the last dilution still giving a reaction, which is why a titre of 1 in 320 is a stronger result than 1 in 40. Because adjacent wells differ by only a factor of two, a single-well difference on repeat testing is within the expected variability of the method rather than a real change.
Frequently asked questions
How do I calculate the total dilution of a serial dilution?
Raise the dilution factor to the power of the number of steps. Five 1-in-10 steps gives 10 to the power 5, a 1 in 100,000 dilution — not 1 in 50. The dilutions multiply, they do not add.
Do I count the neat tube as a step?
No. Count transfers, not tubes. The undiluted starting material has had no dilution applied, so a row of six tubes represents five steps. This off-by-one is the most frequent error in the calculation.
Why do errors get worse further down the series?
A consistent bias is applied at every step and compounds geometrically. A pipette running 5% high produces about a 28% error by the fifth tube, and nothing in the series reveals it.
Do I really need to change tips between steps?
Yes. The film of concentrated solution carried on the outside of a tip is trivial compared with the first tube and dominant compared with the last. Carryover, not arithmetic, sets the practical limit at the dilute end.
Why are doubling dilutions used for titres and MICs?
A twofold series gives enough resolution to bracket an endpoint while remaining practical across a microtitre plate. It also means adjacent wells differ by only a factor of two, so a one-well shift on repeat testing is within normal method variability.
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References
- Clinical and Laboratory Standards Institute. Methods for Dilution Antimicrobial Susceptibility Tests for Bacteria That Grow Aerobically. CLSI standard M07.
- Sambrook J, Green MR. Molecular Cloning: A Laboratory Manual. 4th ed. Cold Spring Harbor Laboratory Press; 2012.