Do students with Masters Degree complete their PhDs (in US, but I am curious about other countries too) sooner than the one's who have entered the PhD program after their undergrad?
I know it depends upon the individual, project, major and the supervisor but is there a general trend?
This varies by program. In the US, in mathematics, "par" for a master's is 2 years. In some PhD programs, this will cut a year off your time-to-degree. In others, no help at all. In rare cases, it will cut two years off.
It depends on the university and the program.
In the US, most PhD granting institutions have pre-requisites for getting your PhD. This may include coursework and will also require research.
For research, one could argue that if you did research in the course of your Master's degree then you would have a time advantage in that you may be able to hit the ground running and engage in publish-able research more quickly.
In institutions that require coursework, the amount of coursework that overlaps with a Master's program can vary. Often, even if you've taken similar coursework in another university, you would still need to fulfill the course requirements in the department where you're going for your PhD. For programs that offer a combined Master's/PhD program, the courses for getting your PhD may be covered under the requirements for getting a Master's, so by already having a Master's degree you may only be shaving off a few of your requirements.
Last, at least in Computer Science, a Master's degree and a PhD are fundamentally different in what they're trying to educate you for. A Master's degree is more like an advanced bachelor's degree where you're taking additional courses and gaining extra skills. A PhD is for research. Given that fundamental difference, it's difficult to say how much would transfer over and again, it would really depend on the university and the program's expectations of what you should have when you enter and what you requirements you need to fulfill once you get there.
Not really, for example if you make your master's is like 2 years, and then your PhD in 4 years, roughly is going to be 6 years that you spent until you got your terminal degree.
I some universities is not required to follow a Master's first. I know the case of one undergraduate student who was always interested in research (when he was in his last year of bachelor he published two good articles in an important ACM conference); he applied directly to a PhD program and he got it. Now he has finished his PhD in only 4 years, and the background that he was missing he got it by following specific courses. So at the end all depends in your background and the passion you got for research.
In my department, students who enter with a masters don't have to take much formal coursework, while those who enter without one have to take much more.
Students with a masters therefore (in theory) have more time and focus to spend on research in the first year or two.
But this doesn't seem to help them finish faster (maybe by a tiny imperceptible amount). Most students aren't working very much on what will become their thesis work in that first year anyways - it's hard to come in and hit the ground running. So while the students entering with a masters might have more time for research in year 1, it doesn't usually directly advance the thesis work.
It is true that it may not shave much time off of your Ph.d, however, it prepares you for the radically different type of education that is graduate school. Forget about time, an M.A will ultimately contribute to your doctoral success in so many varying ways from acclimation to research habits.
From my experience in Biochemistry and Neurobiology (and I'm assuming most Biology and Chemistry fields).
Unfortunately, in the US, no it does not help in completing a PhD sooner.
Sometimes a masters is necessary for a person just to get into a PhD program, depending on the person's history and the program.
It would help if the PhD had some requirement of a certain level of understanding or even amount of research work, but instead a PhD tends to just require an amount of time as an indentured servant to your PI - and that amount of time doesn't change if you have a masters. Although, if you're clever and a good salesman maybe you can convince your PI and committee that your masters time should count toward your sentence/years of labor.
