The Social Security Death Index (SSDI) states 1883 as birth year and grave stone says 1882.
Which should I believe?
My first question in response to this would be why you are settling for only two sources that give direct evidence about when your research subject was born, and trying to decide which of them is more likely to be correct, instead of looking for more sources that could give you this information. One record is not proof of a person's birth year -- you arrive at a proof by constructing a well-reasoned argument, following genealogy standards such as the Genealogical Proof Standard.
One method for evaluating sources and the information contained in them is described in Elizabeth Shown Mills' QuickLesson 17: The Evidence Analysis Process Map. You can work through the 3 x 3 grid as you consider your research question. For your example, a productive research question might be: When was my research subject born?
The Social Security Death Index is an index: it is a derivative record. You could get a record closer to the time the person was born by using the information in the index to order an original record, the SS-5 (the application for the Social Security number).
Consider that the Social Security Board was created on August 14, 1935. Someone applying for a Social Security number who was born in the 1880s would have been in their fifties (or older) when they made the application, so the SS-5 would have been created decades after the person was born.
Evaluating the headstone is trickier because in your question, we know absolutely nothing about the stone. We don't know the age the person died, whether the style of the stone is contemporaneous with the time of death, or who ordered the stone, all of which could give us clues about whether the birth date was given by someone who was likely to know the date. (Contrast a memorial for an infant, where the date on the stone could have been given by an adult who was present at the infant's birth, versus a memorial for an elderly person, where all the survivors are younger people and only know the birth year from hearsay. Though it doesn't apply in your case, a death date on a memorial inscription for an infant might be more reliable than the dates for the adults -- or not, depending on when the MI was made.)
Both your sources give direct evidence to answer your research question. However, you may be able to build a stronger case by using other records, even if those records only have indirect evidence. One record by itself is not proof.
Another thing to consider is that all sources need to be evaluated on their own merits -- otherwise we may make mistakes due to source snobbery. In his article "Perils of Source Snobbery" published in OnBoard 18 (May 2012), Dr. Thomas W. Jones, Ph.D., CG, CGL, FASG, FNGS, FUGA cautions us that "despite passing analytical tests, preferred materials may contain false information" and that:
Disdained sources sometimes show the only or most efficient path to reliable, informative records, or they may provide evidence critical to the researcher’s conclusion. They are safely consulted when the researcher subjects them to correlative tests of accuracy—the same tests they apply to preferred sources.
So the answer is -- "It depends." One of your two sources might be correct, or neither of them might be.
In her presentation "Problems and Pitfalls of a Reasonably Shallow Search", Elissa Scalise Powell, CG, CGL warns that shallow research can cause us problems, including creating create the wrong kinship ties and causing brick walls. Keep an open mind, and gather more evidence.
If two sources differ in relation to a date then I think you should seek a third, fourth, or fifth source, if at all possible, in order to decide which, if either is more likely to be correct.
For example, the gravestone of my 3rd great grandmother Mary Symons says that she died on 22 Jan 1911, and yet her death and funeral notices appeared in South Australian newspapers a year earlier on 24 Jan 1910. The Australia Death Index, 1787-1985 (of Ancestry.com) says she died 22 Jan 1910.
I have concluded that the year of 1911 on her gravestone is incorrect.