Dotty, I assume you had an mtDNA test - if so, this test describes your mtDNA. Your mtDNA is the same as your mother's and her mother's, etc. back in an unbroken all-female line. If Ruth DICKERSON b 1800 is in this direct all-female line, then she would have the same mtDNA. She would also be the only DICKERSON with that mtDNA. Her mother would have it, too, but presumably she would also have a different maiden name. That's the problem with using mtDNA in genealogy - the everyone in an ancestral line (and in the all-female lines of descent) has a different surname.
To connect to the male DICKERSON surname family, a male DICKERSON from that line needs to be Y-DNA tested. The markers, distance, etc are all confusing factors - just remember: if two men with the same SURNAME have matching Y-DNA, then they share a common ancestor. The same is true for mtDNA: if two women have matching mtDNA they probably share a common ancestor - it's just hard to find someone who matches because all the names are different. If you know of someone who should match, you can have them tested to see if they really do!.
You can also go to www.familytreedna.com and click on the FAQ Tab and scroll down to Family Finder. This is a new test for men or women and will report back to you the names and emails of anyone else who has taken that test and a large chunck of matching DNA to you - indicating a high probability that you are cousins from one of your ancestors. So far I've found 14 new cousins (3rd to 10th) with paper trails back to our 14 common ancestors that pass through 119 of my ancestors and 38 different surnames.I really like this new test!