I have Petersens who moved from Chicago to Northern and Southern California, and at least one of them (my grandfather) identified himself as Scottish. None of them were truly Scottish, as genealogical research eventually proved. They were Danish.

Best thing to do is to use whatever tools are available to you, such as census or emigration records, to see how your grandparents or great-grandparents identified themselves when first in the USA. This may lead to a determination of their actual heritage.

Regards,
Debra
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http://sakionline.net/familypage