How to check competition for keyword?

Normally the competitiveness of a keyword is about how many people are bidding for it in adwords. However, I think you mean how many people are attempting to rank for that keyword/phrase.  

Without paid tools or using google keyword planner, you can simply google the keyword and see how many results are returned. That will give you a guide to how many sites are being listed for that keyword. So gives you an idea of how easy it will be to rank for it.  

I think there is not any specific formula for doing calculation . The only way is to type that keyword on google search engine and check the corresponding results that how many websites are ranking towards it and then compare at your level to see its competition.

Thanks.