Maybe not as technical as Anurag's explanation, but since I knew this was an average problem, I knew I could find the average using the formula largest number + smallest number / 2. After plugging in 200 into the formula, though, and pulling out -200, I realized that would have to be the lowest. I looked back at the original pattern, and noticed every odd digit was positive, and every even digit was negative, so if -200 is the SMALLEST number, then by plugging in 199, you'd find the largest digit (which is just 199, after computation).
So then I simply plugged these two into the standard average formula:
[199 + (-200)]/2 = -0.5