Why does Logger Pro show only "correlation" when linear is selected, but shows "RMSE" for all other fits?

Logger Pro reports two different measures of the goodness of fit for all fits. Non-linear fits report the root mean square error (RMSE), and the linear fit can provide both the correlation coefficient and the RMSE. All fits can display the uncertainty of the fitted coefficients. For example, the slope of a velocity vs time graph might be expressed as 9.7 +/- 0.2 m/s^2.

Correlation coefficients are not as useful as RMSE in science since they do not provide any measure of the uncertainty in the fitted coefficients, nor can a correlation coefficient be interpreted in terms of the uncertainties of any of the raw measurements.

In the curve fit dialog we do show only correlation for the linear fit, while we show RMSE for other fits. This was done mostly for historical reasons going back to earlier versions of our software. Perhaps when this dialog is re-done sometime we'll show only RMSE for all fits.

The Root Mean Squared Error (RMSE) is a measure of how close a fitted line is to data points. The RMSE is in the same units of whatever is plotted on the vertical axis. For every data point, you take the distance vertically from the point to the corresponding y value on the curve fit (the error), and square the value. Then you add up all those values for all data points, and divide by the number of points, finally taking the square root. The squaring is done so negative values do not cancel positive values. The smaller the RMSE, the closer the fit is to the data.

Key point: The RMSE is thus the distance, on average, of a data point from the fitted line, measured along a vertical line.

The RMSE is directly interpretable in terms of measurement units, and so is a better measure of goodness of fit than a correlation coefficient. One can compare the RMSE to observed variation in measurements of a typical point. The two should be similar for a reasonable fit.

Since the correlation coefficient doesn't have a physical interpretation (unlike the RMSE) we encourage the use of RMSE to evaluate the goodness of fit.

An aside: the correlation coefficient by itself is terribly misleading. See:
http://en.wikipedia.org/wiki/Anscombe%27s_quartet
for a fascinating set of graphs with the same correlation coefficient.

The reported errors are all defined as the standard deviation of the fitted parameters. This makes the explicit assumption, found in all least-squares fitting routines, that all of the uncertainty lies in the y values, not the x values.

Related TILs:
TIL 1014: What are Mean Squared Error and Root Mean Squared Error?