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Define the population and averages
Chapters 1–2. Keep molecule counts, mass weighting and the sample boundary distinct.
Construct matching summaries
Chapters 3–5. Work two distributions and verify both molar-mass averages.
Test the missing information
Chapters 6–7. Compare threshold fractions and identify what the moments establish.
Read distributions as evidence
Chapters 8–10. Build a family of counterexamples and check measurement limits.
Practise and continue
Chapters 11–12. Change mass scale and sample selection before further reading.
Full chapter index · Practice and worked checks · Polymer Science subject root
Two polymer samples can have the same number-average molar mass, the same mass-average molar mass and the same dispersity, yet contain different proportions of short and long molecules. Matching the summaries does not uniquely determine the population.
We will construct two explicit distributions, check every reported average, and then examine a high-molar-mass fraction that the shared averages fail to identify. All populations are invented and use exact representative molar masses for teaching.
Chapter index
Define the population and averages · Chapters 1–2
1. Define the molecular population
2. Distinguish the two averaging rules
Construct matching summaries · Chapters 3–5
3. Build two different distributions
4. Calculate both number averages
5. Calculate both mass averages and dispersities
Test the missing information · Chapters 6–7
6. Ask a question the summaries do not answer
7. Identify what the shared moments do establish
Read distributions as evidence · Chapters 8–10
8. Construct a whole family of matching samples
9. Choose measurements for the remaining question
10. Questions readers often ask
Practise and continue · Chapters 11–12
11. Practise with changed scales and boundaries
12. Choose a connected reading route
CHAPTER 1 OF 12 · Define the population and averages
1. Define the molecular population
Consider molecules of one stipulated polymer chemistry, counted across the entire stated sample. The counts below are relative count units: multiply all counts by any common large factor without changing the calculated averages.
We are not measuring six isolated peaks from a real chromatogram or claiming that actual polymer distributions have only a few exact masses. Discrete populations make the information loss visible with manageable arithmetic.
For every comparison, keep the population boundary fixed. Removing an insoluble fraction, excluding low-mass species or changing an integration limit creates a different analysed population. Matching results across changed boundaries needs a separate explanation.
CHAPTER 2 OF 12 · Define the population and averages
2. Distinguish the two averaging rules
If ni molecules have molar mass Mi, number-average molar mass is Mn = ΣniMi/Σni. Every molecule receives equal count weight. Mass-average molar mass is Mw = ΣniMi²/ΣniMi, giving more weight to molecules that contribute more mass.
Dispersity is Đ = Mw/Mn. IUPAC’s polymer-characterization report uses the ratio of mass-average to number-average molar mass; the mass average is also written Mm in that terminology. We use the familiar Mw notation here.
The ratio is dimensionless. Reporting Mn, Mw and Đ does not provide three independent distribution constraints: once the first two are fixed, their ratio is fixed automatically.
CHAPTER 3 OF 12 · Construct matching summaries
3. Build two different distributions
Let one mass unit in the arithmetic below mean 1,000 g/mol. Population A contains molecules at 10,000, 20,000 and 40,000 g/mol. Population B instead contains molecules at 5,000, 25,000 and 35,000 g/mol.
| Molar mass | Relative count in A | Relative count in B |
|---|---|---|
| 5,000 g/mol | 0 | 7 |
| 10,000 g/mol | 8 | 0 |
| 20,000 g/mol | 12 | 0 |
| 25,000 g/mol | 0 | 15 |
| 35,000 g/mol | 0 | 2 |
| 40,000 g/mol | 4 | 0 |
Each population has 24 relative count units. Their support points and proportions differ visibly. The next step is to check whether the summary averages notice that difference.
Working in the 1,000 g/mol mass unit, A’s count-weighted mass sum is 8 × 10 + 12 × 20 + 4 × 40 = 480. Dividing by 24 gives Mn = 20 units, or 20,000 g/mol.
B’s corresponding sum is 7 × 5 + 15 × 25 + 2 × 35 = 35 + 375 + 70 = 480. Its Mn is also 20,000 g/mol.
Equal total count and equal weighted mass sum establish equal Mn. They do not establish equal counts at any particular mass. The same overall mass can be distributed among the molecules in more than one way.
CHAPTER 5 OF 12 · Construct matching summaries
5. Calculate both mass averages and dispersities
For A, the squared-mass sum is 8 × 10² + 12 × 20² + 4 × 40² = 800 + 4,800 + 6,400 = 12,000. Divide by its mass sum 480 to obtain Mw = 25 units, or 25,000 g/mol.
For B, the squared-mass sum is 7 × 5² + 15 × 25² + 2 × 35² = 175 + 9,375 + 2,450 = 12,000. Its Mw is likewise 25,000 g/mol.
Both dispersities are 25,000/20,000 = 1.25. The distributions match all three commonly quoted numbers, exactly within this model. This is a counterexample to the claim that those numbers uniquely identify the chain population.
CHAPTER 6 OF 12 · Test the missing information
6. Ask a question the summaries do not answer
Suppose the question concerns the mass fraction contributed by molecules above 30,000 g/mol. In A, the four molecules at 40,000 contribute 160 of the total 480 mass units, giving 1/3.
In B, the two molecules at 35,000 contribute 70 of the total 480 mass units, giving 7/48, or about 14.58%. Matching the averages has left a substantial difference in this explicitly defined upper fraction.
| Reported quantity | Population A | Population B |
|---|---|---|
| Mn | 20,000 g/mol | 20,000 g/mol |
| Mw | 25,000 g/mol | 25,000 g/mol |
| Dispersity | 1.25 | 1.25 |
| Number fraction below 15,000 g/mol | 1/3 | 7/24 |
| Mass fraction below 15,000 g/mol | 1/6 | 7/96 |
| Mass fraction above 30,000 g/mol | 1/3 | 7/48 |
The thresholds are stipulated reporting questions, not physical transition points or product acceptance criteria. They show what additional distribution information a particular question may need.
CHAPTER 7 OF 12 · Test the missing information
7. Identify what the shared moments do establish
View M as the molar mass of a molecule selected with equal probability by count. Then Mn = E[M] and Mw = E[M²]/E[M]. Consequently the count-weighted variance is Var(M) = Mn(Mw−Mn).
For both populations this is 20,000 × 5,000 = 100,000,000 (g/mol)². Their count-weighted standard deviation is 10,000 g/mol. Thus they match not only the mean but also this variance.
That does not make the summaries useless. They constrain the distribution. They simply do not specify its full shape, its modes or every threshold fraction. The conclusion should retain both parts: the agreement is real, and the remaining ambiguity is real.
CHAPTER 8 OF 12 · Read distributions as evidence
8. Construct a whole family of matching samples
Form a new population in which a fraction θ of the molecules is drawn from A’s count distribution and the remaining fraction 1−θ from B’s. Because both original distributions share E[M] and E[M²], every such mixture shares those moments too.
Therefore every θ between zero and one gives Mn = 20,000 g/mol, Mw = 25,000 g/mol and Đ = 1.25. Yet the mass fraction above 30,000 g/mol becomes θ/3 + (1−θ)7/48.
At θ = 1/2, that fraction is 23/96, or about 23.96%. The interval between 7/48 and 1/3 describes this constructed mixture family only. It is not a proven universal bound over every possible distribution sharing the two averages.
CHAPTER 9 OF 12 · Read distributions as evidence
9. Choose measurements for the remaining question
If the decision depends on a distribution shoulder, a low-mass fraction or an upper tail, request distribution evidence and a declared integration rule. Two averages alone do not reconstruct those features.
The size-exclusion chromatography guide explains separation, calibration, detector response and integration limits. A raw detector trace is not automatically a number-fraction distribution. Converting it into molar-mass evidence requires a suitable measurement model.
The IUPAC polymer-characterization report provides a wider account of structural quantities and methods. Our exact discrete populations are mathematical teaching cases, not measurements reported in that paper.
Does identical dispersity mean identical distribution width?
Dispersity fixes a particular ratio. With Mn also fixed, it fixes the count-weighted variance used above. It still does not uniquely determine distribution shape or tail fractions. State which meaning of width is intended.
Does Population A have to perform better?
No. We calculated population summaries, not material performance. Chemistry, architecture, morphology, processing, temperature and the tested property need their own evidence.
Would a more precise average solve the ambiguity?
These examples match the averages exactly. Reducing measurement noise cannot recover information that the selected summaries do not uniquely contain.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with changed scales and boundaries
Double every molar mass in both populations while keeping the relative counts. Both Mn values become 40,000 g/mol and both Mw values become 50,000 g/mol. Dispersity remains 1.25. The matching-summary ambiguity survives the rescaling.
Now remove all molecules below 15,000 g/mol from the original populations. A retains 16 count units and 400 mass units, so its new Mn is 25,000 g/mol. B retains 17 count units and 445 mass units, giving about 26,176 g/mol. Their retained populations no longer match Mn.
Explain why this does not contradict the original result. The operation selected different parts of different distributions. It changed the populations whose averages were being calculated.
For definitions and structure, return to Polymer Science and Polymer Chemistry. For changes in molecule count, read chain scission and number-average molar mass and step-growth imbalance and the chain-length ceiling. Those models describe how a count evolves; this article examines what selected summaries leave unresolved.
For evidence, follow size-exclusion chromatography and the IUPAC polymer-characterization report.
For connected measurement routes, read multi-angle light scattering and molar-mass evidence and field-flow fractionation and size separation. Each needs its own detector, recovery and interpretation checks.
Continue through How X Works and The Gold Standard of Education. Before calling two samples equivalent, ask which observations agree and which question those observations actually answer.
Sources checked 22 October 2026. Worked examples are educational models with the assumptions stated above.