Serial dilutions

14 min

Calculate a cumulative dilution factor
Calculate the destination concentration in a serial dilution step
Predict how a transfer or diluent volume error changes a serial dilution

Try first

A serum is diluted 1:4, and then the mixed 1:4 tube is diluted 1:5. Which factor converts a reading on the second tube back to the serum?

Right. The next section explains why.

The next section explains it.

I don't know yet

The next section explains it.

Get the idea

Each tube comes from the one before

In a serial dilution each tube is made from the tube before it, after that tube has been mixed. Every step therefore divides a concentration that earlier steps have already divided.

Each step has its own factor, taken from the volumes put into that tube:

step factor = (transfer volume + diluent volume) ÷ transfer volume

The tube's concentration comes from its source, the tube just before it:

destination concentration = source concentration ÷ step factor

Multiply the step factors

The total factor for any tube, compared with the original, is the product of every step factor up to and including that tube.^1,2 Adding the factors describes nothing that happened in the tubes. Two 1:5 steps leave one twenty-fifth of the original, so correcting a result by 5 + 5 = 10 would give a value 2.5 times too low.

A series that starts at 240 mg/L shows how the numbers move:

Tube	Transfer + diluent (µL)	Step factor	Total factor	Concentration (mg/L)
1	500 + 500	2	2	120
2	250 + 750	4	8	30
3	200 + 800	5	40	6

Tube 3 is 30 mg/L ÷ 5, using tube 2 as its source. Dividing the original 240 mg/L by 5 would give 48 mg/L, which belongs to no tube in this series.

When one volume is off

The factors must describe the volumes that were pipetted. If one transfer or diluent volume is off:

That tube's step factor changes.
Every later total factor is a product that includes it, so the error travels down the whole series even when the later steps are done perfectly.²
Calculating with the intended factors then misstates every concentration after the slip.

Remake the affected tube and every tube after it. Keep the tubes made before the slip.

References

Bishop ML, Fody EP, Van Siclen C, Mistler JM, Moy M. Clinical Chemistry: Principles, Techniques, and Correlations. 9th ed. Jones & Bartlett Learning; 2023.
Kingston University. Concentration of solutions: serial dilution: overview. Reviewed March 31, 2021. Accessed September 23, 2026.

Watch one

A tenfold series is planned from a solution at 80 mg/L: three tubes, each 100 µL of the previous tube plus 900 µL of diluent.

Tube 1 is found to hold 100 µL of the solution in only 400 µL of diluent.
Tubes 2 and 3 were made as planned.

What does tube 3 hold, and what do you do?

Take tube 1's actual volumes: 100 µL + 400 µL = 500 µL, so its step factor is 500 ÷ 100 = 5, half the planned 10.
The factor has to describe what went into the tube.
Keep the later step factors at 10 and 10.
Tubes 2 and 3 were pipetted as planned, so their own step factors stand.
Multiply the actual step factors: 5 × 10 × 10 = 500. The plan assumed 1,000.
Each later tube was made from the one before, so the first step's factor carries forward.
Tube 3 holds 80 mg/L ÷ 500 = 0.16 mg/L, twice the intended 0.08 mg/L.
The concentration follows the total factor of the tubes as made.
Remake tube 1 and then tubes 2 and 3 from it.
The slip carries into every tube made after it.

Tube 3 holds 0.16 mg/L at a total factor of 500. Remake tube 1 and every tube after it.

Your turn

Problem 1 of 3

A specimen is diluted 1:5, and that diluted sample is diluted 1:5 again. Which dilution factor converts a result measured on the second tube into the original specimen concentration?

Incorrect. Factor 5 reverses only the second dilution, so it gives the concentration in the first tube. Multiply by another factor of 5 to recover the original concentration.

Incorrect. Adding the factors gives 10, but the second dilution acts on a mixture that is already one fifth of the original concentration. One fifth of one fifth is one twenty-fifth, so the factors multiply: 5 × 5 = 25.

Correct. Each 1:5 step leaves one fifth of the previous concentration, and one fifth of one fifth is one twenty-fifth. The result measured on the second tube multiplied by 25 gives the original specimen concentration.

Hint

The second tube is made from the first tube, which is already weaker than the specimen.
After the first step the tube holds one fifth of the specimen's concentration; the second step takes one fifth of that.
Combine the two step factors the way the fractions combine.

Review Serial factors multiply

Problem 2 of 3

A solution at 180 mg/L is diluted 1:5 into tube 1, which is mixed. You then transfer 200 µL of tube 1 into 400 µL of diluent to make tube 2. What is the concentration in tube 2?

Hint

Tube 2's source is tube 1. Find tube 1's concentration first.
Tube 1 is 180 ÷ 5 = 36 mg/L. Now find tube 2's step factor from its volumes.
The step factor is the whole volume in tube 2 divided by the part transferred.

Show the answer

12 mg/L

Tube 1 is 180 ÷ 5 = 36 mg/L. Tube 2 holds 200 + 400 = 600 µL, so its step factor is 600 ÷ 200 = 3, and 36 mg/L ÷ 3 = 12 mg/L.

Review Serial factors multiply

Problem 3 of 3

A twofold series of three tubes is planned, each 500 µL of the previous tube plus 500 µL of diluent. Tube 2 is made with 500 µL of tube 1 and 1,000 µL of diluent; tubes 1 and 3 are made as planned. Which total factor describes tube 3?

Eight is 2 × 2 × 2, the plan. Tube 2's step factor is 1,500 ÷ 500 = 3, and tube 3 was made from tube 2, so the plan's factor no longer applies.

Used the intended factor after preparing different volumes

A transfer of 200 µL into 900 µL of diluent gives a factor of 1,100 ÷ 200 = 5.5, short of the intended 10, and the error carries into every later tube. After two more tenfold steps the total factor is 550, where the plan called for 1,000. Using the intended factor misstates every downstream concentration.

Seven is 2 + 3 + 2. Each tube divides the concentration of the tube before it, so the step factors multiply.

Added the dilution factors

Each step acts on the concentration the step before it produced, so two 1:5 steps leave one twenty-fifth of the original. Adding the factors gives 10, and a result corrected by 10 is 2.5 times too low.

The step factors are 2, then 1,500 ÷ 500 = 3, then 2, and 2 × 3 × 2 = 12.

Four leaves out tube 2 altogether. Tube 3 was made from tube 2, so tube 2's step counts.

Review Unequal steps and volume errors

Use it

Try this at the Dilution stationPendingOriginal · mg/L100Source concentrationUnequal steps: 2, 5, 10Dilute specimen

Keep

Sources checked September 23, 2026

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