To give some perspective:
I lived in a one-room dorm that cost anywhere from 15,000 to 20,000 yen a month depending on how much gas/water I used. I had the JASSO scholarship which gave me 80,00 a month. That left me with around 60,000 a month to spend on whatever I wanted. After you took out food cost and the occasional drinking, I still had some money left over to save. Granted, I wasn't buying a ton of clothing/accessories/toys, etc, but I was able to save every month by budgeting.
I'm not sure how that would translate to a family of three and you'll probably be paying at least 3 times as much for rent, but it gives you a rough idea. Also, where will you be living? Rent in Tokyo is considerably higher than say rent in Sendai.