My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated?
The rule for this style of accounting is straightforward: a person is 50% what their mother was and 50% what their father was.
Sometimes one will want to simplify a result. For instance, if your mother is Russian and your father is half Russian and half Chinese, then you are 50% Russian plus 50% half-Russian-half-Chinese, which you can simplify like this:
25% = 50% x 50% Chinese (father) 25% = 50% x 50% Russian (father) 50% = 50% x 100% Russian (mother)
25% = 50% x 50% Chinese (father) 75% = 25% + 50% = (50%*50%+50%) Russian (father plus mother)
So under the presumption that "being Russian", "being Jewish", etc. is inherited in this way, if your great-grandmother was Jewish, then you are 50%x50%x50% = 12.5% Jewish because of her, plus whatever percentage Jewish you inherit from your other seven great-grandparents.
Biologically, it is not certain that you will inherit some DNA from each grandparent, great grandparent, etc. At each generation, the DNA is a random mix of the DNA from each parent, each of which are in turn a random mix of their parents. Given enough generations, randomly some DNA from early ancestors may be lost. In fact, there's a 1 in 64 chance that you've inherited nothing at all (or twice as much) from a particular great great grandparent (rather than the 1/16th you'd expect), and a 1 in 8 chance of losing DNA from the generation before that.
Apart from that problem, there is also no way to know the correct ancestry proportions of each ancestor. They may be described as German, but in fact have a Russian parent (so reducing their German proportion by 50%), or grandparent (reducing it by 25%). So you'd need to go back far enough so that the chances of their ancestry being different become insignificant - say 7 generations to bring it under 1%. At that point, the chances of some ancestry being lost or amplified (as described above) becomes quite likely.
This means that in practical terms there is no accurate way to tell what proportions you inherit from each distant ancestor, except maybe through genetic testing of yourself and very many distant cousins.
You're also mixing up ethnic origins and nationality (and both are very ill-defined terms, evolving over time). It's perfectly possible to be Jewish (or Arab, or Romany, or Scots) and have nationalities from one or more countries. For example, maybe that Jewish grandmother was somebody who was born in Germany, grew up in the same place that was now in Poland and later became American.
So in summary I don't think it's very practical to try to calculate an "ethnic mixture", as the source data is incomplete (not back far enough) and subjective (depends on cultural assumptions and biases) and the biology means the calculation can't be accurate.
In addition to the answers given above, sometimes it's important to know who is counting. What does it mean to be Jewish? According to Jewish tradition, to be Jewish, your mother had be Jewish. (Exercise in recursion left to the reader.) But (for example) in the Soviet Union, the government determined race (and put it in your passport so that everybody would know) based on the paternal line. So according to these rules, my father (whose father was Russian and whose mother was Jewish) is Russian by the old Soviet (and probably new Russian) standards, and Jewish according to Jewish custom.
In addition, there is the issue of intentional obfuscation. When someone self-identifies as Russian, Jewish, or something else, what does that really mean? Whose interpretation is being applied, and why? In some cases, Jews might have wanted to obfuscate their racial identity to avoid quotas, persecution, etc.
I grew up hearing all those statements as well and the best answer I can offer you is that, DNA testing is the best way to see the percentage of ethnicity your lineage contains. We can go back as many generations as we like and still not get the true I'm a third Cherokee, half Irish, one eighth Dutch, & one eight German, or a Heinz 57 (what my Father would call it).
here are some links below that will provide you scientific data and help you on your way to discovering exactly where your lineage originated.
Altough your genetics may say otherwise, most people when answering this question mean for it be calculated with equal distribution. Its not usually meant to be exact, its what you and your relatives and acestors use to identify as their nationality, Most of the time people self identify nationality whether its in their genes or not, so just use the formula for equal distribution or go through the huge uneccesary genetic testing just to find out what percentage of Chinese or Italian is in your blood. If you do the gentic testing there's much more useful info to obtain than race or nationality.
All these 50% calculations assume that whatever constitutes genetic ethnicity is equally divided at the point that eggs and sperm are formed (meiosis). In fact genes are more likely to be randomly assorted. My son with red hair and beard and a fair skin is clearly "more celtic" than his tanned, black-haired sibling but these calculations would assign them the same percentage of Scot-ness.
The only way in which these measures of ethnicity make any sense is to say that "at 5 generations back, approximately 30% of my ancestors were Scots". And even then we are ignoring the continuing genetic impact of Vikings and Romans and ....
Once you begin to consider post-migration generations, there is no value in describing ourselves as anything other than Heinz 57 (or bitzers, to use the Australian vernacular, as in "bits of this and bits of that").
Of course this common sense applies only on 363 days of the year. On Hogmanay and St Andrew's Day, we are 100% Scots.
Claiming a nationality not recognized by the nation in question, is worthless. Therefore if the jewish connection is through the paternal line, one has Jewish blood, but 0% Jewish nationality. If, though, it's through the maternal line, one might not have 100% Jewish blood, but would still have 100% Jewish nationality. In your case, you say your relatives have reasons to believe your mother was Jewish. Depending on those reasons, and your relatives' knowledge of the above, you have serious chances of being 100% Jewish yourself.
Actually it is all dependent on the immigrant of each of your nationality's. My father is an immigrant, straight from Hong Kong, China (making our family Cantonese, so we speak Cantonese and we make Cantonese food). He is 100% Chinese since my grandma and grandpa were also 100%. My mom's great great grandpa was straight from Germany and was 100% German. My mom's great great grandma and grandpa on the other side were from Czechoslovakia (100% Czechoslovakian).
So, from different nationality breeding (sorry to make it sound inhumane), the nationality loses some of itself. My mom's grandma (Czech) married a man of very mixed nationalities. The percentage loss itself by 50% for the other nationalities to who would be their daughter, my mom's mom (grandma) so my grandma was a bit grandma was 50% Czech and a bunch of other stuff. She married my grandpa on my mom's side whose great grandpa was a man from Germany, who married a woman who all we know was a bit Irish (like 25 percent, but from this calculation makes me hardly Irish). They got married and my mom's great grandpa was also 50% German and 50% other. All his nationality moved down and all we know is that my grandpa on my moms side is around 50% German, but married a woman who was also a bit German to, so we know that my mom is 50% German too. But wait, keep in mind that she is also Czech. That around 40-30% Czech degrades the German, because each generation loses automatically loses 10% of that nationality, she is now 40% German, but the that 40% Czech that carries on to her and makes her all in all, 30% Czech, 40% German and 30% other (from the unknown nationalities from her ancestors).
So, she married my father and since he's an immigrant, making me 1st generation Chinese along with my siblings. All in all, I am am 20% Czech, 10% German and 70% Chinese. But nationality is not just genetics, it is also culture and pride. I lean towards my fathers side because he passed his culture towards me more than my mom did (because she doesn't care about her culture) and the only culture that i have inside my mind and that I was taught was of my father and China and my family. That, giving my a mindset that I am 100% Chinese.
This may just be telling you what i am, but at least try to use this formula to help u calculate yours