What is R squared?

R squared (R²), also known as the correlation coefficient or Pearson’s coefficient, is a calculated parameter that is used to describe how well graphical data fit an applied model, e.g. linear, exponential, power, etc. For a linear model or fit, R² is calculated as shown below, where n is the number of X-Y data points in the set.

Values for R² range from 0 to 1, with a value of 1 indicating that there is zero deviation of the data from the model or best fit line (or curve). The figure below shows a couple of examples of different models applied to two sets of data, and the effects of improper model selection on the calculated R² value.

Figure 1: Linear and exponential data sets with different curve fitting models applied.

With regard to the use of R² in the area of static light scattering, R² can be used to help one decide whether or not the Debye or linear form of the Rayleigh equation is appropriate for the sample and experimental conditions. In static light scattering measurements, the Rayleigh equation is used to calculate the molecular weight and 2^nd virial coefficient from the concentration dependence of the light scattered from the sample. The linear form of the Rayleigh equation is shown below, where C is the weight concentration, M is the weight average molecular weight, A₂ is the 2nd virial coefficient (indicative of particle -solvent interactions), K is a constant that embodies all of the optical properties of the solution being measured, ñ_o is the solvent refractive index, R_q is the Rayleigh ratio of scattered to incident light, and dñ/dC is the specific refractive index increment for the particle – solvent pair.

Deviations from the linearity of KC/R_q with C can arise from various sources, including particle interactions and/or larger than typical 3^rd virial effects. If deviations from linearity are significant, as would be indicated by a low R² value, one should consider alternate forms of the Rayleigh expression, such as the Barry form shown below.

For additional questions or information regarding Malvern Instruments complete line of particle and materials characterization products, visit us at www.malvern.com.