There is quite a gap between wanting to be rational and wanting to know how unbiased you are. Since the test is self-administered, pursuing the first desire could easily lead to a favourable, biased, seemingly rational test result. This result would be influenced by personal expectations, and it's reliability is null according to Löb's Theorem. The latter desire implies one being open to his biased state and states his purpose of assessing some sort of bias/rational balance. This endeavour is more profitable than the previous because, hopefully, it offers actionable information.
Perhaps one could have a good shot at finding out more about his biases by making quick judgements and later trying to contemplate various aspects and sequences of his or her judgement with accounting of seemingly absurd alternatives and attention paid to the smallest of details. The result should occur as a percentage of correct/faulty conclusions. Apart from discovering some sort of rational/biased ratio in a line of thought, this process should automatically bring one closer to being rational by the memorizing of judgement flaws, their sources and pattern, and by the development of a habit for righteous thinking from a rationality point of view.
This test could have a much more reliable result when performed on someone else by providing all necessary information for a right conclusion to be reached together with vague, inconclusive information for incorrect conclusions to be reached, and great incentives for reaching some of the wrong conclusions.
Speaking of incentives, I believe anyone trying to be as rational as possible within a group could be influenced by group values and beliefs. Therefore, trying to find out biases within the group's/group members' judgements could be correlated with one's affinity for that group. Rationality should be neutral, but neutrality is seldom a group value so chances are high that instinctive-rationalists will be outliers. The tendency to agree with beliefs is probably as wrong as the tendency of finding biases, the two depending on one's grade of sympathy for a specific group.
Identifying exterior biases will be an unreliable measure of one's rationality, because of the incentives which exist in interacting with others and also because there is usually little information on exterior thought processes which led to specific outcomes. Also, beliefs widely spread across a social system can have consequences that seemingly prove those beliefs even without their being rational, in which case, comparing one's judgement to facts would be an indicator of power rather than rationality.
So you think you want to be rational, to believe what is true even when sirens tempt you? Great, get to work; there's lots you can do. Do you want to justifiably believe that you are more rational than others, smugly knowing your beliefs are more accurate? Hold on; this is hard.
Humans nearly universally find excuses to believe that they are more correct that others, at least on the important things. They point to others' incredible beliefs, to biases afflicting others, and to estimation tasks where they are especially skilled. But they forget most everyone can point to such things.
But shouldn't you get more rationality credit if you spend more time studying common biases, statistical techniques, and the like? Well this would be good evidence of your rationality if you were in fact pretty rational about your rationality, i.e., if you knew that when you read or discussed such issues your mind would then systematically, broadly, and reasonably incorporate those insights into your reasoning processes.
But what if your mind is far from rational? What if your mind is likely to just go through the motions of studying rationality to allow itself to smugly believe it is more accurate, or to bond you more closely to your social allies?
It seems to me that if you are serious about actually being rational, rather than just believing in your rationality or joining a group that thinks itself rational, you should try hard and often to test your rationality. But how can you do that?
To test the rationality of your beliefs, you could sometimes declare beliefs, and later score those beliefs via tests where high scoring beliefs tend to be more rational. Better tests are those where scores are more tightly and reliably correlated with rationality. So, what are good rationality tests?