We are indeed all Jock Tamson's bairns

With every generation you (nearly) double your number of ancestors because every individual has two parents – going back just 10 generations (200-300 years) you are likely to have around a thousand ancestors.

"When genetics researchers talk about common ancestry between people they usually mean that they are tracing the inheritance of particular sections of DNA or genes.And we know that different sections of our DNA have different patterns of genetic ancestry. This means that researchers can get very different estimates of how recently we share ancestors, depending on what they are looking at...
...look at mtDNA to follow ancestry passed along the female line. For mtDNA, everyone alive today shares a common ancestor who lived between 160,000 and 200,000 years ago.everyone alive today shares a common ancestor who lived between 160,000 and 200,000 years ago....
...look at Y chromosome DNA to follow ancestry through the male line, the most recent estimate is of a common ancestor who lived between 240,000 and 580,000 years ago...
...If, however, you look for the most recent person that everyone alive today is descended from, the best current estimate is that the individual lived only 3,500 years ago"

"Genetic ancestry testing presents a simplified view of the world where everyone belongs to a group with a label, such as ‘Viking’ or ‘Zulu’. But people’s genetics don’t reflect discrete groups. Even strong cultural boundaries, such as between the Germanic and Romance language groups in Europe, do not have very noticeable genetic differences. The more remote and less-populated parts of the UK, such as the Scottish Highlands, do have some genetic differences from the bulk of the population, but they are not big. There is no such thing as a ‘Scottish gene’. Instead groups show a story of gradual genetic change and mixing...
...Researchers use the genetic differences between Y chromosomes or mtDNA among a set of individuals to infer possible trees of relatedness. We can estimate the times of common ancestors on those trees, although these estimates lack precision. But it is not reasonable to make a leap from these DNA trees to mapping your ancestors onto geographical locations or past migrations. For example, a man in Scotland might have a type of Y chromosome that has been found more often in North Africa than elsewhere. However, this is based on populations in North Africa now, not in the past, and people have moved over the centuries. And, the same Y type may be found in other parts of the world – he could equally have inherited it from one of these. And even if some of his ancestors did come from North Africa, it does not show when they came to Scotland or how many of his millions of ancestors came from that region."

Prof Steve Jones, from University College London and author of some of the seminal books on genetics and evolution, said: "On a long trudge through history - two parents, four great-grandparents, and so on - very soon everyone runs out of ancestors and has to share them. "As a result, almost every Briton is a descendant of Viking hordes, Roman legions, African migrants, Indian Brahmins, or anyone else they fancy."