🔬 Dilution Calculator

Plan Serial Dilutions and Back-Calculate Titers from Plate Count Data

DOI: 10.5281/zenodo.19411018

by Stephen T. Abedon Ph.D. (abedon.1@osu.edu)

phage.org | phage-therapy.org | biologyaspoetry.org | abedon.phage.org | google scholar

Jump to:   🧪 Dilution Planner  |  🦠 Titer from Plate Count  |  📐 Exponents  |  🔢 Scientific Notation  |  🎯 Significant Figures  |  🧮 More Calculators

What this tool does: The Dilution Planner helps you design a serial dilution series — either to reach a target final titer in a tube, or to deliver a target number of pfu/cfu to a plate. The Titer from Plate Count tab back-calculates the titer of your original sample from plaque or colony counts. The Exponents and Scientific Notation tabs explain the underlying math, pitched for microbiology students but useful as a quick reference for experienced researchers too.

To cite this tool: Abedon, S.T. (2026). Dilution Calculator. dilution.phage.org. DOI: 10.5281/zenodo.19411018

dilution.phage.org  ·  Abedon’s Books  ·  DOI: 10.5281/zenodo.19411018

How can I improve this page?  contact: dilution@phage.org

🧪 Dilution Planner

Follow the steps below. Enter your starting titer, where you want to end up, and how you want to get there — the calculator will lay out every dilution step.

Step 1 of 5 — Goal

What is the purpose of your dilution series?

Choose whether the final diluted volume will be plated directly (the goal is countable plaques or colonies on a plate) or whether you simply need a tube containing a lower titer for some other use.

Reduce liquid titer mode: The goal is simply to have a tube containing a lower concentration — for example, preparing a working stock or a dilution standard. No plating volume is involved.
Step 2 of 5 — Starting titer

What is the titer of your starting stock?

Enter the concentration of your undiluted sample in pfu/mL (phage) or cfu/mL (bacteria). Scientific notation is fine — for example, type 3.2e9 to mean 3.2 × 109. If you do not know your starting titer and need to determine it from plate counts, see titering.phage.org.

pfu/mL or cfu/mL
Step 3 of 5 — Target titer

What final titer do you want in the last tube?

Enter the titer you want to achieve after all dilution steps are complete. For example, type 2e4 to mean 2 × 104 pfu/mL. The calculator will use whole-number serial steps to get close, then add a final fractional step to hit the target exactly.

pfu/mL or cfu/mL in the final tube
Step 4 of 5 — Dilution scheme

How do you want to dilute?

Choose the combination of dilution steps. Each step is performed in its own tube: transfer a small volume of sample into a larger volume of diluent (buffer, broth, or saline).

100-fold + 10-fold: Transfer 1 part sample into 99 parts diluent for each 100-fold step (e.g., 100 µL into 9,900 µL), or 1 part into 9 parts for each 10-fold step (e.g., 100 µL into 900 µL). This scheme reaches large dilution factors with fewer tubes.
Step 5 of 5 — Volumes (optional)

What transfer and diluent volumes will you use?

Enter the total final volume you want in each dilution tube (e.g., 1000 µL = 1 mL). The calculator will work out how much to transfer and how much diluent to add for each step. A microliter (µL) is 1/1000 of a milliliter (mL), so 1000 µL = 1 mL, 100 µL = 0.1 mL, etc.

µL per tube (e.g. 1000 µL = 1 mL)

🦠 Titer from Plate Count

Enter your plate count and your dilution information to back-calculate the titer of your original sample. You can enter either the total dilution factor directly, or specify each dilution step individually. This tab is intended for quick ballpark titer estimates. For rigorous titer determination — including trimmed means, replicate handling, efficiency of plating, and statistical guidance — see titering.phage.org. If instead you need to determine phage concentrations from bacterial survival or killing data, see the Phage Killing Titer Calculator.

Step 1 — Dilution

How do you want to enter your dilution?

Enter the product of all dilution steps, including any plating volume factor. For example, if you did three 10-fold serial steps (total 10−3) and then plated 0.1 mL of the final dilution (another 10-fold factor), the total dilution factor is 10−3 × 0.1 = 10−4 = 1e-4.

must be < 1 (e.g., 1e-8)
Step 2 — Plate data

Count and volume

total plaques or colonies on one plate
mL actually spread or poured
Is your count in a reliable range? Counts that are very low (below about 30–50) are subject to higher statistical uncertainty — even a count of 10 typically introduces around 30% error, though the effect compounds with replicate variability. Counts that are very high risk overlap and undercounting. The right upper limit depends on plaque (or colony) size: small plaques may be reliably counted at higher densities than large ones. Rather than fixed universal cutoffs, the principle is to identify what works consistently for your system and stick with it. For a more careful treatment of what "countable range" means statistically, see Abedon & Katsaounis (2021) and titering.phage.org. Low counts are not necessarily invalid and should not automatically be discarded — see that reference for guidance.

📐 How Exponents Work

Exponents appear throughout microbiology — in phage and bacterial titers, dilution factors, and population sizes. This tab walks through the key rules with examples relevant to lab work.

What is an exponent?

An exponent (also called a power) tells you how many times to multiply a number by itself. The number being multiplied is called the base.

General form: baseexponent = base × base × base × … (exponent times)
103 = 10 × 10 × 10 = 1,000
106 = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000
100 = 1    (any number raised to the power zero equals 1)
101 = 10

Negative exponents — the key to dilutions

A negative exponent means "1 divided by the positive version." This is exactly how dilution factors work.

10−1 = 1/10 = 0.1    → a 10-fold dilution
10−2 = 1/100 = 0.01    → a 100-fold dilution
10−6 = 1/1,000,000 = 0.000001    → a million-fold dilution
10−8 = 0.00000001    → a hundred-million-fold dilution
Laboratory insight: When we say a phage stock is 1010 pfu/mL, we need to dilute it to a range where plates show countable plaques. That means diluting roughly 108-fold so that 1010 × 10−8 = 102 = 100 pfu/mL in the final diluted suspension. We achieve this with a series of smaller dilution steps.

Multiplying with exponents (same base)

When you multiply two powers with the same base, you add the exponents. This is why combining dilution steps is equivalent to adding their log-scale numbers.

10a × 10b = 10a + b

Example — three serial dilution steps:
10−2 × 10−3 × 10−3 = 10(−2)+(−3)+(−3) = 10−8
(a 100-fold step, then two 1,000-fold steps → 100,000,000-fold total)
Common student mistake: Many students try to add the dilution factors (e.g., 0.01 + 0.001 + 0.001) instead of multiplying them. Adding gives the wrong answer. You must either multiply the factors (0.01 × 0.001 × 0.001 = 10−8) or, equivalently, add the exponents (−2 + −3 + −3 = −8).

Dividing with exponents — how much dilution do I need?

When you divide, you subtract the exponent of the denominator from the exponent of the numerator. This is the calculation for figuring out the required total dilution factor.

10a ÷ 10b = 10a − b

Example:
Starting titer: 1010 pfu/mL
Target titer: 102 pfu/mL
Required dilution = 102 ÷ 1010 = 102−10 = 10−8

Exponents and logarithms

The logarithm (base 10) is the inverse of the exponent. If 10x = n, then log10(n) = x.

log10(1,000,000) = log10(106) = 6
log10(0.000001) = log10(10−6) = −6
log10(500) ≈ 2.699    (because 102.699 ≈ 500)

This is why we talk about "log reductions." A 3-log reduction means a 1,000-fold reduction (103); a 6-log reduction is a million-fold reduction.

🧮 Interactive Exponent Explorer

Calculate baseexponent

Enter any base and exponent. For the base, you can use a number or the mathematical constant e ≈ 2.71828 (click the button). Non-integer exponents such as −6.5 are allowed.

What exponent (dilution) do I need?

Given a starting value and a target ending value, this calculates the log10 ratio — i.e., the exponent you would need if diluting base-10.

🔢 Scientific Notation

Phage titers routinely span 10 or more orders of magnitude. Scientific notation is the standard way to write very large or very small numbers compactly and unambiguously.

The basic form

A number in scientific notation is written as:

Scientific notation: M × 10n

where M (the coefficient, or mantissa) is a number ≥ 1 and < 10, and n is an integer exponent.

1,000,000,000 = 1 × 109
3,200,000,000 = 3.2 × 109
0.00000047 = 4.7 × 10−7
100 = 1 × 102 = 102

Converting a regular number to scientific notation

  1. Move the decimal point until you have a number between 1 and 10.
  2. Count how many places you moved it — that count becomes the exponent.
  3. If you moved the decimal left (the number was large), the exponent is positive.
  4. If you moved the decimal right (the number was small), the exponent is negative.
4,700,000 → move decimal 6 places left → 4.7 × 106
0.0000047 → move decimal 6 places right → 4.7 × 10−6

Dilution factors in scientific notation

A dilution factor of 10−8 is scientific notation with an implied coefficient of 1 (i.e., 1 × 10−8). Non-integer dilutions such as 3.33 × 10−7 follow the same rules.

Calculators and spreadsheets: When you type 1e-8, it means 1 × 10−8. The "e" here stands for "exponent of 10" — it is not the mathematical constant e ≈ 2.718. So 4.7e6 = 4.7 × 106 = 4,700,000.

Multiplying numbers in scientific notation

Multiply the coefficients, then add the exponents.

(3 × 104) × (2 × 105)
= (3 × 2) × 104+5
= 6 × 109
Titer example:
Starting titer: 3 × 109 pfu/mL
Dilution factor: 3 × 10−8
Result: (3 × 3) × 109+(−8) = 9 × 101 = 90 pfu/mL
Adjusting the coefficient: If multiplying coefficients gives a result ≥ 10, shift the decimal and adjust the exponent. For example, 12 × 103 → 1.2 × 104.

Dividing numbers in scientific notation

Divide the coefficients, then subtract the exponents.

(6 × 109) ÷ (3 × 104)
= (6 ÷ 3) × 109−4
= 2 × 105

🧮 Scientific Notation Converter

Decimal → Scientific Notation

Scientific Notation → Decimal

🎯 Significant Figures

Significant figures (often abbreviated sig figs) communicate the precision of a measurement or calculation. In microbiology, titer values and dilution factors almost always have limited precision, so knowing how many figures are meaningful prevents the illusion of false accuracy.

What counts as a significant figure?

A digit is significant if it carries real information about the measured quantity. The rules:

  1. All non-zero digits are significant.   472 has 3 sig figs.
  2. Zeros between non-zero digits are significant.   4,072 has 4 sig figs.
  3. Leading zeros are never significant.   0.0047 has 2 sig figs (the 4 and 7).
  4. Trailing zeros after a decimal point are significant.   4.70 has 3 sig figs — the final zero tells you the measurement was precise to the hundredths place.
  5. Trailing zeros in a whole number are ambiguous without a decimal point.   4,700 could have 2, 3, or 4 sig figs. Scientific notation eliminates this ambiguity: 4.7 × 103 has exactly 2 sig figs.
0.00470 → 3 sig figs (leading zeros don't count; trailing zero after decimal does)
4.70 × 106 → 3 sig figs
1.000 × 10−8 → 4 sig figs
47 → 2 sig figs

Sig figs in multiplication and division

The result should have no more sig figs than the least precise input.

47 pfu × (1 / 0.1 mL) = 470 pfu/mL → report as 4.7 × 102 pfu/mL  (2 sig figs)

Titer calculation example:
Count = 47 (2 sig figs)
Volume = 0.10 mL (2 sig figs)
Dilution = 1.0 × 10−8 (2 sig figs)
Titer = 47 ÷ (1.0 × 10−8 × 0.10) = 4.7 × 1010 pfu/mL  (2 sig figs)
Common mistake: A calculator display of 47,000,000,000 implies 11 significant figures. That precision is false — your count of 47 plaques, read by eye, is good to at most 2 sig figs. Always round to match your least precise measurement.

Sig figs in addition and subtraction

The result should be rounded to the least precise decimal place (not the fewest sig figs).

1,247.3 + 0.06 = 1,247.36 → round to 1,247.4  (tenths place, matching 1,247.3)

This rule matters less often in dilution calculations, which are dominated by multiplication and division.

Why sig figs matter for phage titers

A plaque count of, say, 43 plaques on a plate is probably good to ±2–3 plaques — so at best 2 sig figs. A titer reported as 4.3 × 109 pfu/mL honestly conveys that precision. Reporting the same value as 4,300,000,000 pfu/mL implies a precision that no plaque assay can deliver.

Practical convention: Phage and bacterial titers are almost always reported to 1–2 significant figures. If you report 3 sig figs, you are implying your counting method and dilution volumes are accurate to better than 1% — a high bar. Two sig figs (e.g., 4.3 × 109) is almost always appropriate for plaque assay data.

Scientific notation makes sig figs explicit

This is one of the main practical reasons scientists use scientific notation: it removes ambiguity about which zeros are significant.

4,300,000,000 pfu/mL  → ambiguous (2? 3? 10 sig figs?)
4.3 × 109 pfu/mL  → unambiguously 2 sig figs
4.30 × 109 pfu/mL  → unambiguously 3 sig figs

🧮 Significant Figure Counter

Enter a number and this tool will identify how many significant figures it has, and show it in unambiguous scientific notation.

Count significant figures

Round to a specified number of significant figures

🧮 Phage Biology, Phage Therapy, and Teaching Calculators

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Dilution Calculator — DOI: 10.5281/zenodo.19411018 — phage.org — Version 2026.05.11