[updated 12/25/2016: R^{2} is a useful measure of indexing]

The R-squared (R^{2}) statistic describes a pattern of plotted data with respect to a straight line. R-squared is called the coefficient of determination (ref 1,2).

The black dots in figure 1 represent investment returns that are poorly related to market returns. There is a random distribution of investment returns with respect to market returns. The blue line is an inadequate representation of the relationship simply because there is no relationship. The R^{2} score for this distribution is 0.03. Conversely, the black dots in figure 2 show the ‘herding’ of data around a straight line.

Figure 2’s investment returns are highly related to market returns with an R^{2} of 0.997.

### Significance

The R^{2} score represents the degree of alignment of data to a best-fit line as determined by regression analysis. The lowest possible score of 0 indicates a random pattern of data with absolutely no alignment. The highest possible score of 1 represents complete alignment.

The product of R^{2} X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R^{2} measures the relavance of the best-fit line to a set of data. Relavance increases as the R^{2} score varies from 0 to 1.

The lowest score of 0 defies any financial analyst to draw a meaningful line for investment returns as they relate to market returns. In figure 1, the incline (β) and Y-intercept (⍺) of the blue line are unreliable measurements of investment performance.

The highest R^{2} score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R^{2} above 0.75 identifies a straight line for making predictions of returns. Lower scores represent increasingly random events. In figure 2, the incline (β) and Y-intercept (⍺) are reliable measurements of investment performance.

R-squared is an excellent measure of index fund performance. Websites for index mutual funds and ETFs publish R^{2} as a measure of alignment between fund returns and the market index. Funds that have an R^{2} score of nearly 1.00 track the index very closely.

### References

1. Lain Pardoe, Laura Simon, and Derek Young. STAT 501, Regression Methods. 1.5- The coefficient of determination, r-squared. Pennsylvania State University, Eberly College of Science, Online courses. https://onlinecourses.science.psu.edu/stat501

2. R-squared. 2016, Investopedia http://www.investopedia.com/terms/r/r-squared.asp?lgl=no-infinite