If I'm 5.1% Jewish, how far away are my Jewish ancestors and do I count as a person of the Jewish race ?
This is perhaps not the best place for the question, but I'll try my best. There are many questions in one here too.
Firstly, I would question where you got the figure from, as sometimes DNA tests can get things wrong, even the big ones like Ancestry. As they don't look at the entire genome, only small slices of the genome known as "markers" their results can be incorrect especially at this early stage where gene sequencing is in its infancy.
Secondly, how far away are your Jewish ancestors? It depends vastly. It would help to know what country you are from to try and piece things together. If it was only one ancestor that was Jewish, this would be either four or five generations back, see below % of autosomal DNA shared:
Roughly 25% = grandparent Roughly 12.5% = great grandparent Roughly 6.3% = great great grandparent Roughly 3.2% = thrice great grandparent
It may however have been a mix of say a half Ashkenazi Jew and a full Ashkenazi Jew, mixing generations and percentages; the only way to find out would be to trace your family tree and find out.
Lastly, do you count as a member of the "Jewish race"? Firstly, there are many types of Jew and no single Jewish race. Ashkenazi Jews are one of many ethnic Jews which testify an ancestry back to Israel/Jacob, the figure from the Talmud and Old Testament.
Quite simply, if the figure is correct, you are 5.1% Jewish, but you are not an Ashkenazi Jew. If an individual buys a 500ml can of Coke and pours it into a vat of water is that a vat of coke?