Hi!
It's good to see that you have an interest in your family history. My first suggestion is to pump any reletive still living for all the information on your family that you can as they are a resource that you will not always have. Seconsly there is no subsitution for documentation do the paperwork. Check family Bibles, birth and death certificate. etc.. .
Tha t being said I found some tidbits on the surname Moore at www.Ancestry.com in their surname origin section. The surname Moore is found in it's greatest numbers in Ireland (4236) then England (2873) and lastly Scotland(320) (these are the three top place of origin according to the site). As it has been said the Moore surname may be (is) a derivitive of Muir in Scotland and that may account for the low numbers in Scotland since the search only looks for Moore.
Good luck with your research!
Bill
May all your blessings be the ones you want and your friends many and true.
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