What does it mean for a linear regression to be statistically significant but has very low r squared?

I understand it to mean that the model is bad at predicting individual data points but has established a firm trend (e.g. y goes up when x goes up).

Answer

It means that you can explain a small portion of the variance in the data. For instance, you can establish that a college degree impacts salaries, but at the same time it’s just a small factor. There are many other factors that impact your salary, and the contribution of the college degree is very small, but detectable.

In practical terms it could mean that in average the college degree increases the salary by $500 per year, while the standard deviation of salaries of people is $10K. So, many college educated people have lower salaries than non-educated, and the value of your model for prediction is low.

Attribution
Source : Link , Question Author : PaperRockBazooka , Answer Author : Aksakal

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