Radius of gyration ($R_g^2=\frac{1}{2N^2}{\sum }_{m,n}⟨|{\mathbf{R}}_m-{\mathbf{R}}_n{|}^2⟩$)

For $N$ equally weighted positions, the squared radius of gyration is the average squared distance from the center of mass: $R_g^2=N^{-1}{\sum }_n⟨|{\mathbf{R}}_n-{\mathbf{R}}_{\mathrm{cm}}{|}^2⟩$, where ${\mathbf{R}}_{\mathrm{cm}}=N^{-1}{\sum }_n{\mathbf{R}}_n$. Expanding pair differences gives ${\sum }_{m,n}|{\mathbf{R}}_m-{\mathbf{R}}_n{|}^2=2N{\sum }_n|{\mathbf{R}}_n-{\mathbf{R}}_{\mathrm{cm}}{|}^2$, and thereforeFor a Gaussian chain or freely jointed chain with these $N$ positions and mean squared link length $b^2$, each pair variance is $b^2|m-n|$. Summing separations yields $R_g^2=b^2(N^2-1)/(6N)\sim N{b}^2/6$. Counting all endpoints of $N$ links instead gives $N+1$ positions and the corresponding finite-size formula.

For a long ideal chain of $N$ independent statistical steps of length $b$, $R_g^2=N{b}^2/6$. With fixed chain chemistry and $b$, the molar mass $M$ is proportional to $N$. Thus multiplying molar mass by $s$ multiplies the radius of gyration by $\sqrt{s}$; solvent quality can change both the statistical segment length and the size exponent.

For a uniform solid sphere of radius $R$, the squared radius of gyration about its centre is the mass-weighted mean of $r^2$:Thus its volume is $(4\pi /3)(5/3{)}^{3/2}{R}_g^3$, about $9.01R_g^3$. A nonuniform object or a different shape need not obey this conversion. This distinction matters when estimating molecular number density from scattering measurements of a radius of gyration.