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The statistics given of ca. 350 ancestors just several hundreds years ago is NOT NECESSARILY CORRECT even by the criteria used.
It is ascribed to a DNA bottleneck.
What is a bottleneck?
The flow of liquid poured from a jar into a bottle gets narrower as it enters the bottle and then expands after entering it.
So too with ancestry.
About a hundred years ago there were more surnames than there are now even though we now have more people.
Surnames are transmitted by males. Some of the males bearing family names back then did not have children, others had females and no males. The tendency to have more males than females or vice versa may to some degree be hereditary.
This means that in every generation there are more and more people bearing certain surnames and less and less of those bearing others. Some surnames die out.
This may happen even when some of those whose names disappeared actually have quite a lot of descendants still walking around today. They just did not have enough boys generation after generation to keep the male-line going.
So too with females. Female DNA is measured by mtDNA. In every generation more and more girls have the same type of mtDNA while other types diminish. They keep bearing more boys than girls and their relative proportion of mtDNA in the general population diminishes.
Eventually it all evens out. The two sides converge with less variety in the mtDNA and less different types of the Y(male transmitted)DNA.
If we only take account of existing mtDNA and existing YDNA it looks like everybody was descended from only a few people because their particular mtDNA and particular YDNA will be the only types remaining.
Are you confused?
So am I.
Nevertheless, it is quite simple.
[It is even simpler than depicted above since in studies of this nature it is usually only the mtDNA that is measured.]
Take the following example.
We had six individuals that became 3 couples, i.e. Y1, Y2, Y3, coupled with mt1, mt2, m3. They gave us the following families and offspring:
1. Y1 & mt1 = 5Y1
2. Y2 & mt2 = 5mt2
3. Y3 & mt3= 3Y3, 2 mt3
We therefore had three couples and each had 5 children.
The results were:
No.1 all the mt1 died out, but there were 5 Y1.
No. 2 all the Y2 died out but there were 5 mt.
No.3 Both types (Y3, mt3) survived but they were both minorities when compared to the previous two.