As mentioned before, fasfa uses a basic formula to calculate the parental contribution amount. Data taken into consideration is gathered from your Federal tax filing and your input of current assets. Be honest when completing the fasfa.
When you receive a scholarship for tuition, usually the funds are directly sent to the educational institution. This isn't normally considered "income" to you or your parents. Some gifts, may be considered income depending on the amount and how it is distributed and to whom it is paid to (you vs. institution) - check with your tax consultant on these types of funds received.
It is the "income" received that would adjust the level of parental contribution in the fasfa formula. The fasa's calculation of parental contribution is really completely separate from whether not an institution decides to adjust the amount of a financial award. Usually, the award/scholarship has some documentation and will explain the intent of the award and how it is distributed, etc. Read this documentation (maybe be in fine print on scholarship's website) for a better understanding.
Whether or not the institution would reduce the scholarship given to you would probably depend on if it was a merit or need-based scholarship and the intended use of the scholarship.
Think about it, if an institution awarded you a need-based scholarship for 4 years tuition , and then you no longer had a need because you received additional scholarships that met that need, then why wouldn't they reduce it and give it to someone with more of a need? If it was Merit based, then they may not reduce it but in both these scenarios, keep in mind that scholarships are designated for specific purposes. If the funds can't be used for it's specific purposes fully, then the institution really has a duty to redistribute those unused funds and use them for the intent they were designed.
If this is the case, consider yourself blessed and bless someone else by letting your institution know you no longer need their scholarship and why. By releasing these funds, you will allow the college to act quickly and give another student the opportunity to attend college that they may not have attended otherwise.