Ways to show how well or badly predicted data reflect observed data

I have observed rainfall data and would like to know how well they represent the predicted data. My first idea was to show the data on a scatter plot. Plotting x =observed vs y = predicted I set the axis to equal length as well as that the plots are square sizes.

t  x   y
1  2.3 3
2  1   1.5
3  4.4 4.8


Then I thought I am showing the Pearson correlation coefficient; however I am not sure what are the best ways to show how good my data are.

What other ways exist to assess such data? What would be a sound way to represent the fit of the data and represent it to other? Any creative ideas?

• In practice Pearson correlation might be useful here but in principle it doesn't measure agreement. Consider as a knockdown case, predicted $=$ positive constant $\times$ observed. Here Pearson correlation is identically 1, always, but the larger the constant $>$1, the poorer the fit. Otherwise and more concisely put, Pearson correlation measures linearity, not agreement. Concordance correlation is a measure of agreement. Commented Nov 13, 2017 at 18:34
• Some edits of language, but it's not totally clear what your situation is. Observed versus measured: what's the distinction there? Or is it observed versus predicted, and how were predictions obtained? Commented Nov 13, 2017 at 18:50
• Thanks a lot to all who took their time and giving those good answers. It already made certain things clearer. I do want to show how good the agreement between those two data sets are. Commented Nov 14, 2017 at 7:37
• Commented Sep 3, 2019 at 12:51

Plotting the predicted and actual values is always a good idea to see if there is some systematic error in the predictions.

Often people will use a measure of accuracy or error (list copied from here. Caveat: I' am author of this page.):

• Minimum maximum accuracy

• Mean absolute percent error (MAPE)

• Root mean square error (RMSE)

• Normalized root mean square error (NRMSE)

Accuracy measures tend to be reported as a percent or proportion, and so are unit-less. With 1 being a perfect fit. In my experience, they tend to be high (closer to 1) compared with r-squared values, so you have to be cautious in interpretation.

MAPE is reported as a percent error, so unitless, with 0 being a perfect fit. RMSE retains the units of the measured variable; again, 0 would be a perfect fit.