A polymer sample contains chains of many different lengths, so its molecular weight must be averaged — and the two common averages weight the distribution differently.

Number average counts every molecule equally:

\(M_n = \frac{\sum N_i M_i}{\sum N_i}\).

Because it is a simple head count, a large population of short chains pulls it down strongly — so \(M_n\) is sensitive to low molecular weight species. It is measured by methods that count molecules, such as osmometry and end-group analysis.

Weight average weights each chain by its own mass:

\(M_w = \frac{\sum N_i M_i^{2}}{\sum N_i M_i}\).

The \(M_i^{2}\) term means a single heavy chain counts far more than a single light one, so \(M_w\) is sensitive to high molecular weight species. It is measured by techniques that respond to mass, such as light scattering.

So the correct statement pairs each average with the end of the distribution it emphasises, and the first two options simply reverse that pairing.

The fourth option states the polydispersity index the wrong way round. The index is defined as \(PDI = \frac{M_w}{M_n}\), not \(\frac{M_n}{M_w}\). Since \(M_w \ge M_n\) always, the PDI is at least 1, equalling exactly 1 only for a perfectly monodisperse sample where all chains are identical — as in a protein. Synthetic polymers typically give values from about 1.05 for living anionic polymerisation up to 20 or more for free-radical processes, so the ratio is a direct measure of how broad the chain-length distribution is.

Hence M_w is sensitive to high molecular weight and M_n to low molecular weight species.