I come from a town in Ontario, Caledonia, that was settled mainly by Scots. Still today, almost all the families have Scottish names. On Canada day, the Mackenzies and Thompsons alike can be seen in kilts, wearing their family tartans, and playing bagpipes.
But Thompson is a Lowland name. And family tartans are completely made up. (See the Sobieski-Stuarts.) To add to the confusion, kilts were invented by an Englishman.
Well, at least everyone can enjoy bagpipes.
Scotland's history goes back a long way. Of course a lot has changed, but if you're asking about the genetic makeup, you're going to find that the Scots are a real mix, not the single group that they're often thought of as today.
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1. Picts
Pictii to the Romans, in English we call them Picts. The Latin name is often translated as "painted ones" but, according to what I've read, it can designate different sorts of designs made on the body. Body paint (with woad or possibly red ochre) or tattooing have been suggested. (Caesar's comments about the Britons of seem to me to suggest tattooing. However, he was speaking of people in modern southeast England. But I digress.)
In any case, the Picts were the people the Romans encountered in Caledonia. Evidently not a very literary people, they have left nothing of their language and little of their culture apart from some interesting standing stones. (See, for example, the Aberlemno Stones.)
The Picts are generally taken to be a branch of the family of Celtic peoples. They were very ancient. They lasted well into the Dark Ages, but, rather curiously, they vanish into history. It is presumed that they blended with the Gaels.
2. Britons
Strathclyde was a British kingdom, which is to say that it was peopled by a folk closer to the Welsh than to the Gaels. Strathclyde was eventually absorbed into the Kingdom of Scotland in 1093.
3. Gaels
The Gaels came from Ireland, settling over much of the Highlands and Islands. The tribe of Gaels called Scotti are taken to have given their name to the eventual Kingdom of Scotland.
[EDIT: According to Wikipedia, Scotti is just a Latin name for the Gaels.]
4. Angles
It was mainly the Angles, not the Saxons, who settled in northeast England and southeast Scotland, founding the kingdoms of Bernicia and Deira. These united in 654 to form the Kingdom of Northumbria, which subsequently became a major power in Britain. The Northumbrians fought many wars with the British and Pictish kingdoms, who were still a major presence.
Edinburgh was an Anglian settlement. (Burgh means "fortified place".)
5. Norse
In Scotland the Norse, principally from Norway, had a huge impact as well. For example, DNA evidence indicates that the male populations of Orkney and the Shetlands were likely annihilated. Those Islands remained Norwegian possessions for centuries.
The Norse also settled extensively in the Hebrides as well as some of the mainland, leading to a kind of fusion Gaelic-Norse culture. (See the Lordship of the Isles, which was in effect a kingdom within a kingdom.)
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That's a summary of the ancient ethnic makeup of Scotland. There haven't been any great influxes of new peoples from the time of the Vikings through to modern times.
Things settled into two main parts: the Scots-speaking Lowlands, and the Gaelic-speaking Highlands and Isles. (However, this doesn't cover everybody, as has been pointed out.) Lowlanders viewed the Gaels as savages and raiders. The divide between Highlands/Isles and Lowlands is hard to imagine today, multiplied as it was by difference in religion, language, customs, clothing. It helps to explain the brutal crackdowns following Bonnie Prince Charlie's defeat at Culloden in 1746.
It may be worth mentioning that the leading figures of the Scottish Reformation were Lowland Scots.
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Your family was from Glasgow, which was, in ancient times, Celtic (Brythonic), but which by 1820 (a thousand years and more after what I've been going on about) was quite mixed.
If you provide some of the surnames from your family tree, it's probably easy enough to determine at least the names' origins (whether Highland or Lowland, for example). Genes are another matter entirely.