What a serial dilution does and why one big dilution will not do
A serial dilution reaches a very small concentration by repeating a modest dilution several times, instead of attempting one enormous one. Six ten-fold steps dilute a stock a million-fold using nothing but a 1 mL pipette and ten 10 mL tubes. Achieving the same million-fold dilution in a single step would mean measuring 1 µL into a litre — a volume you cannot deliver accurately and a vessel you cannot mix properly.
That is the whole argument for the technique. Accuracy is dominated by the smallest volume you have to measure, and a series keeps every measured volume in the range where your pipette is actually specified to perform. The trade is that errors accumulate across the steps rather than occurring once, which is why the number of steps should be the smallest that reaches your target.
The technique is the backbone of viable counting, where you need plates in the countable window and cannot know the titre in advance; of antibody and antigen titration, where two-fold steps produce the classic endpoint titre; of minimum inhibitory concentration testing, where two-fold antibiotic series are prescribed by the reference methods; and of every standard curve in an immunoassay or qPCR run. Once the series exists, converting plates back to a titre is handled by the CFU per mL calculator.
The formula, and the ratio that trips everyone up
The dilution factor of one step is the final volume divided by the sample volume: DF = (Vs + Vd) ÷ Vs. Transfer 1 mL into 9 mL and the final volume is 10 mL, so the factor is 10 — written 1:10, or 10⁻¹, or "a ten-fold dilution".
The mistake that survives into published protocols is reading 1:10 as one part sample to ten parts diluent. That combination gives a final volume of eleven parts and a factor of 11, not 10 — an error of 10% at every step, compounding to 77% after six steps (11⁶ ÷ 10⁶ = 1.77). Whenever the ratio matters, state the volumes rather than the ratio, and check that the numbers you write down sum to the final volume you intend.
Because each step multiplies the previous dilution, cumulative dilution after n identical steps is DFn, and the concentration in tube n is C0 ÷ DFn. Two properties follow. First, in logarithms the series is a straight line: log₁₀C falls by log₁₀DF at every step, which is why a ten-fold series drops exactly one log per tube and a two-fold series drops 0.301. Second, the arithmetic never depends on the tube volume you keep — only on the ratio you transfer. Doubling every volume in the protocol changes nothing about the dilutions, which is what makes a series scale freely from a 96-well plate to a rack of 50 mL tubes.
The formula also tells you what to do if the steps are not identical. Multiply the individual factors instead of raising one to a power: a 1:10 followed by a 1:5 followed by a 1:2 is a cumulative 1:100. This calculator assumes identical steps because that is what almost every protocol specifies; for a one-off mixed dilution use the C₁V₁ = C₂V₂ dilution calculator for each stage.
Worked example: a six-tube ten-fold series from 1 × 10⁶ cells/mL
You have a suspension at 1 × 10⁶ cells/mL and want plates in a countable range, so you build a decimal series: 1 mL transferred into 9 mL of diluent, six times.
- Dilution factor per step. (1 + 9) ÷ 1 = 10.
- Tube 1. 1 × 10⁶ ÷ 10 = 1 × 10⁵ per mL, a cumulative 1:10.
- Tube 2. 1 × 10⁶ ÷ 10² = 1 × 10⁴ per mL, cumulative 1:100.
- Tube 3. 1 × 10³ per mL, cumulative 1:1 000.
- Tube 4. 1 × 10² per mL, cumulative 1:10 000.
- Tube 5. 1 × 10¹ per mL, cumulative 1:100 000.
- Tube 6. 1 × 10⁶ ÷ 10⁶ = 1 cell per mL, cumulative 1:1 000 000.
The cumulative dilution is 10⁶ and log₁₀ of that is 6, so the series spans exactly six logs. Total diluent required is 6 × 9 mL = 54 mL, and each tube holds 10 mL after the transfer arrives but only 9 mL once you have taken the next transfer out of it.
Now pick the plating dilution. If you spread 0.1 mL, tube 4 delivers 100 per mL × 0.1 mL = 10 colonies, tube 3 delivers 100, and tube 2 delivers about 1 000. Tube 3 is the one that lands inside the 25–250 countable window, and tubes 2 and 4 bracket it — which is exactly why you plate three consecutive dilutions rather than betting on one.
Change the volumes and check the ratio still holds: 0.5 mL into 4.5 mL is also (0.5 + 4.5) ÷ 0.5 = 10, and consumes half the diluent. 100 µL into 900 µL is the same factor again in a microplate.
Choosing the step size and the number of steps
Match the step size to what you are measuring. Ten-fold steps cover ground quickly and are the standard for viable counts, where the titre may be unknown within several orders of magnitude. Two-fold steps give fine resolution and are prescribed for antimicrobial susceptibility testing and for antibody endpoint titres, where you need to distinguish adjacent concentrations rather than survey a wide range. Three- to five-fold steps are common for standard curves in immunoassays and dose-response work, where eight points should span the assay's dynamic range without wasting points outside it.
Choose the number of steps from the target, not from habit. If you need to fall from 10⁸ to 10² per mL, that is six logs, which is six ten-fold steps or twenty two-fold steps (2²⁰ = 1.05 × 10⁶). The two-fold route reaches the same place with far more pipetting and far more accumulated error, so use it only when you need the resolution.
Watch the volume you have left. Each tube gives up the transfer volume to the next tube, so a 10 mL tube in a 1-into-9 series has 9 mL available for assays — plenty. A 200 µL well in a two-fold plate series that transfers 100 µL leaves only 100 µL to read, which is why plate titrations are usually set up with the final volume in mind first and the transfer volume derived from it.
One structural point that catches people out: the last tube in the series never gives up a transfer, so it retains its full volume. If your protocol says to discard 100 µL from the final well to equalise volumes across the plate, that instruction exists for photometry, not for the dilution math — the concentration in that well is correct either way.
Common transfer volumes and the factors they give
| Transfer | Into diluent | DF per step | Cumulative after 6 steps | Cumulative after 10 steps |
|---|---|---|---|---|
| 1 mL | 1 mL | 2 | 64 | 1 024 |
| 1 mL | 2 mL | 3 | 729 | 59 049 |
| 1 mL | 3 mL | 4 | 4 096 | 1 048 576 |
| 1 mL | 4 mL | 5 | 15 625 | 9 765 625 |
| 1 mL | 9 mL | 10 | 1 000 000 | 10¹⁰ |
| 0.5 mL | 4.5 mL | 10 | 1 000 000 | 10¹⁰ |
| 100 µL | 900 µL | 10 | 1 000 000 | 10¹⁰ |
| 1 mL | 10 mL | 11 | 1 771 561 | 2.59 × 10¹⁰ |
The last row is the classic error: one part into ten parts of diluent is a 1:11 dilution, and after six steps it is 77% away from the 1:1 000 000 you intended.
What ruins a dilution series
- Not changing the tip between steps. Carry-over on the outside of a tip is a fixed volume of concentrated material added to a progressively more dilute tube, so it dominates the far end of the series. Change tips at every transfer.
- Failing to mix before the next transfer. Vortex or pipette-mix each tube thoroughly. A transfer taken from an unmixed tube samples whatever stratified at the top, and no arithmetic recovers from it.
- Reading 1:10 as one part in ten parts of diluent. That is a factor of 11. Over six steps it compounds to a 77% error.
- Letting cells settle in the stock. A suspension that sat on the bench for ten minutes is no longer at the concentration you measured. Mix the stock immediately before the first transfer.
- Using a pipette below its calibrated range. Delivering 2 µL with a 200 µL pipette is far outside its accuracy specification. ISO 8655 sets tolerances by volume, and error grows sharply at the bottom of a pipette's range.
- Assuming the diluent is inert. Some organisms lose viability rapidly in plain water or unbuffered saline. Use the diluent the method specifies, and plate within the holding time it allows.
- Adding transfers you do not need. Every step adds error. Reach the target in the fewest steps the volumes allow.
How error builds along the series
Two kinds of error behave very differently down a series, and knowing which you have determines whether a long series is acceptable.
Random error — the ordinary scatter of pipetting — partially cancels, because a step that runs slightly high is as likely as one that runs slightly low. For independent errors of similar relative size at each step, the relative uncertainty of the cumulative dilution grows roughly with the square root of the number of steps. Ten steps carry about three times the relative uncertainty of one step, not ten times.
Systematic error does not cancel at all. A pipette that consistently delivers 2% low applies that bias at every step, so the cumulative dilution is out by 1.02n: 13% after six steps, 22% after ten. This is the argument for calibrating the pipette you use for a standard curve, and for using the same pipette for the whole series so that any residual bias is at least consistent.
There is also a floor you cannot pipette your way past. Carry-over on the outside of a tip, aerosol during vortexing, and adsorption of the analyte to tube walls all add material to tubes that should be nearly empty. In practice this sets a limit on how deep a manual series is meaningful, and it is why an extremely dilute tube should be verified against a direct measurement rather than trusted from the arithmetic alone. For cells, that verification is a plate count or a chamber count with the hemocytometer cell count calculator; for growth experiments, the two counts feed straight into the cell doubling time calculator.
Set the whole rack out before you start
Pre-dispense the diluent into every tube, label them with their cumulative dilution rather than their step number, and lay out one tip per step. Labelling tubes 10⁻¹ to 10⁻⁶ instead of 1 to 6 removes the most common transcription error at the plating bench, because the label then matches the number you will type into a calculation. The total diluent figure on this page tells you how much to pour before you begin — 54 mL for a six-step 1-into-9 series.
Key terms
- Dilution factor
- Final volume divided by sample volume for one step. A 1:10 dilution has a factor of 10 and reduces concentration to one tenth.
- Cumulative (total) dilution
- The product of every step's factor. For n identical steps it is DF raised to the power n.
- Decimal dilution
- A ten-fold step, the standard in microbiology because it moves the count by exactly one log₁₀ per tube.
- Two-fold (doubling) dilution
- A 1:2 step, used for antibody titres and MIC panels because it gives fine resolution across a narrow range.
- Carry-over
- Concentrated material transferred unintentionally, usually as a film on the outside of a tip. It sets the practical depth limit of a manual series.
- Endpoint titre
- The reciprocal of the highest dilution still giving a positive result — read directly off the cumulative dilution column of a two-fold series.