Jr pass needed?

I will be in Japan for 15 days. I'm thinking of going from narita to Tokyo to Kyoto to Osaka and back to Tokyo and narita. Would I need to get the 14 day jr pass??

7 Day JR Pass price = Round trip between Tokyo and Osaka
14 Day JR Pass price = Round trip between Tokyo and Hiroshima

https://www.jreast.co.jp/e/suica-nex/index.html

It's not a question of "needing" the pass. It's will you save money by it. In your case, not with a 14 day pass. A 7 day pass with a side trip to Nara would be worth it though.
You also might look into some alternatives, like an air or bus pass.
See:
http://www.jal.co.jp/yokosojapan/
http://www.ana.co.jp/wws/us/e/wws_common/fare/special/airpass.html
http://www.ana.co.jp/wws/us/e/wws_common/fare/special/visit.html
http://www.ana.co.jp/wws/us/e/wws_common/promotion/experience_jp/
http://willerexpress.com

I have an itinerary now that goes from Narita to tokyo,to takayama, to Nagoya, Osaka,koyosan,Osaka, Hiroshima, and back to tokyo. We have priced out individual tickets. And have included one day passes in Kyoto, Osaka, tokyo and Hiroshima. I'm surprised to see its nit cheaper with the rail pass. We are going to take a couple of busses that are not included in the jr pass so maybe that's the difference.

Be sure that when you are pricing the fares you include BOTH the fare and the seat (reserved or not) supplement.

I'm going to re price it.