I wrote a post this week about the importance of twitter engagement. The process by which we can measure the success of our engagement is called analysis. There are many free tools online you can use to analyze your twitter account. There is no subscription process or need to give anything more than your twitter @soandso handle. After a few seconds you will be able to view your analysis. Try the sites I recommend below. After a few weeks of checking up and re-analyzing your account you will start to be able to distinguish patterns that you can use to better manage you tweeting experience.
Love this one. Pure numbers and stats, no funny business. It tells you exactly what you need to know.
This site gives some interesting stats. Not super helpful when trying to analyze engagement, but overall a nice tool.
Thanks to renjith krishnan for the photo