Why do we need MAE (Mean Absolute Error) when we have MSE (Mean Squared Error)

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I understand that in MAE, because many small errors will drown out rare, but very large errors. Due to Squaring, MSE will quickly point out large errors.

My question is then why do we need MAE? Are there problems where MAE is better to be watched for?

Any examples for type of real-life problem where MAE better than MSE (and the other way also). The more intuitive the example the better. I am almost ok on the classification error types just because of the medical diagnostics test example fits it so well.

I am not a math guy and this whole topic makes my head hurt.

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You said it yourself- they have different ways of dealing with large errors, so it depends on how much worse a large error is than a small one.

If you’re dealing with a money loss issue for example, it might make more sense to use MAE-losing $12,000 once and then losing $1,000 10 times is really not any worse than losing $2,000 eleven times.

On the other hand, if you’re looking at something like blood loss during surgery, MSE might make more sense given that the effects aren’t linear. If you have 10 surgeries where everyone loses a little blood, that might just be 10 successful surgeries, but 9 surgeries with no blood loss and 1 surgery with a lot of blood loss could mean 9 successful outcomes and 1 dead patient.