Thanks for the help June. I was thinking that there was something wierd with my version of access as the design view for a select query does not have a totals box. This has thrown me off for awhile because most every screen shot online has this box. Hitting the totals ribbon button fixes this. So, thank you for that. 
As far as this query goes, one of our programmers happened to swing by today and take a look at what I was trying to do. He claims that the general syntax was correct, there were just a few modifications to be done. First, setting the statement up as I had it would return how many fields met both the DCN and Pass criteria as one record, and then DCN and FYI as a second record. In order to fix it we had to run a subquery to then combine the two records as one. Then, we switched the "WHERE" clause to a "HAVING" clause (not really sure why). The code below is the correct SQL statement and returns the numbers as intended.
Code:
SELECT [%$##@_Alias].Audit_DCN, Sum([%$##@_Alias].count_x) AS Total_Score
FROM (SELECT COUNT(Field_Result) as count_x,DCN as Audit_DCN
FROM Audit_Scores
GROUP BY DCN, Field_Result
HAVING (Field_Result = "FYI" OR Field_Result = "Pass")) AS [%$##@_Alias]
GROUP BY [%$##@_Alias].Audit_DCN;