I've researched the average salary of what my position makes and I make well less than the mean .
"Average salary" means nothing . . . unless you're average.
In reality, you're attempting assessment of your value as an employee, or your potential value to other companies.
6 months in, I'd be dubious as to the accuracy of an employee's self-assessment FWIW. Does not mean your situation is not unique though. If you're assuming some entitled compensation in this economy, you're making a mistake.
Before you approach your employer, I'd recommend testing the market. See what actual
opportunities are out there for you in your field. If there are none. Now is the time to count blessings for your current job.
OTOH, if you find IAW the data you've cited that you're being undervalued in your present situation, go after the raise. If under those circumstances it is not forthcoming, change employers.