Is this a perfect method?  No, but it’s a good way to get a general idea about where your ancestors were from.  Genealogical DNA tests can tell you a lot about your ancestry going back 300-500 years in time, for the most part.  They can also tell you a little bit about your ancestry going even further back.  This is why comparing your DNA to those whose families have stayed in a particular area for a long time is a fairly accurate way to perform the estimate.
AncestryDNA has the largest database to compare your results to when making matches, with 23andMe coming in second and FTDNA in fourth. MyHeritage DNA, although newer than the others, is catching up fast and numbers now surpass FTDNA. Current numbers can be seen in the chart above and are estimates based on available data. Each of these databases is growing, some of them quite rapidly.
Admixture percentages are one of the biggest reasons people choose to have their DNA tested. This report attempts to accurately match your DNA with population samples from around the world to tell you where your ancestors came from. Each of these companies has strengths and weaknesses when it comes to this calculation, and in the reports it provides to users.
The Y chromosome is a special chromosome, passed on from fathers to their sons, while mothers pass on mtDNA to both their sons and daughters. But mtDNA dies with men and so it survives only in the female line. This means that a man’s lineage can be followed along both paternal and maternal lines, while in a woman only her maternal or mtDNA line can be followed.
Good explanation, but I was a little distressed by the part of the analogy that says people know what work to do because "someone tells us." That statement makes people sound like robots and that we do not make decisions on our own. Maybe this is lost on me because I work for myself, but this paints the picture of a chain of people telling other people what to do and everyone following blindly. Something to think about when discussing this concept with children! Humans have free will... :)
Then comes the section about serious genetic variants. So far as “counselling” goes, previously, I’d waved away concern for my psychological welfare from the Observer’s science editor (“I’m a former goth,” I said. “My default setting is ‘doomed’”), but it turns out to be quite daunting. It doesn’t help that I initially mistake the full list of potential conditions for my own results, hence (thankfully briefly) thinking that I have higher risk factors for everything going. It makes me wonder – how many other people are going to do that?
Specific tests for your father’s family include ‘Y-DNA’ tests which focus on the ‘Y chromosomes’ in your cells’ nuclei, passed down from father to son. Specific tests for your mother’s family include ‘mtDNA’ tests which report on a subset of DNA found in the ‘mitochondria’ (your cells’ energy factories), passed down from mother to son or to daughter.
I hope this helps to clear things up. Ancestry DNA testing is not an exact science, and is limited by the fact that we don't inherit the exact same DNA our parents had, meaning that with each new generation, old DNA is lost. Ancestry tests can provide estimations of our genetic ancestry, and though they are improving all the time, they can't tell the whole story of our heritage.
The core feature of all ethnicity DNA tests is to show you a breakdown of the ethnic groups who have contributed to your autosomal DNA, normally as a list, pie chart, and/or map in an online account. It’s understood that these tests give you a picture of your ethnic heritage from the past five to six generations, and this is because the number of your ancestors increases exponentially the further back you go.
Living DNA and Findmypast are British companies joining forces to combine cutting-edge science with traditional family history research methods. We’ve made every effort to find a DNA company to partner with that provides the most benefit for those looking to explore their British and Irish roots. Living DNA's test results provide a regional breakdown that perfectly complements our unrivalled collection of British and Irish historical records. It’s this powerful combination that makes this partnership the perfect marriage of science and history.
After collecting spit and cheek cells, we mailed all of the tests at the same time and waited for results, noting all communications from the company in the meantime and how long it took each service to notify us that results were ready to view. We collected data based on testers’ impressions of their results, each service’s features and extras, how easy it was to use and navigate the service’s website, along with several other factors. We added this testing data to rigorous research and information gleaned from conversations with representatives from Ancestry, 23andMe, MyHeritageDNA, LivingDNA, Humancode (now owned by Helix) and 24genetics. 
I can’t help you with your question Robin, but you make a good point. I have had my DNA tested (only with MyHeritage so far) and the “North and West European” part is so broad (it could be anything from France, Belgium, the Netherlands, Luxembourg, Germany etc but it would have been important to have more detail as it is really what I would have loved to know more about) and then 0.8% Middle East…
We similarly advise caution for the multitude of non-diet health and wellness DNA tests, which offer insights into your sleep, food sensitivities, and vitamin and mineral levels. And double that for medical information found in consumer DNA kit test results. While medical insights learned from taking an at-home DNA test may be interesting, it’s best not to take them too seriously. If you have a concern about a genetic predisposition to a disease, it’s best to talk to your doctor instead of relying on a direct-to-consumer kit. 
If you're creeped out by how much information Facebook, Google and Amazon have on you based on your online browsing habits, just remember that these DNA testing services are getting what is effectively your medical history. Make sure of their policies before turning over that valuable data. Also, even if you don't share your DNA with a service, your familial DNA data may be available if a relative shared their genetic material. The privacy issues can get very complex.
When it comes to proving a biological relationship between a British citizen and a family member living abroad so that they may immigrate, DNA testing can greatly strengthen the case. However, DNA evidence alone does not guarantee a successful immigration application. If you’re considering taking a DNA test for immigration purposes, we recommend you take legal advice to ensure it’s used in the best possible way.
Alternatively, if you believe your "Italy/Greece" result indicated your known Italian heritage, it is possible that your son simply did not inherit the associated DNA from you. We all inherit roughly half our DNA from each parent, but the DNA we inherit is selected at random, and so even full siblings can have different ancestry results, depending on which genetic variants they inherited.
For our evaluations, we assembled a group of testers willing to spit into a tube on camera. We chose four individuals of varying backgrounds. Two had previously taken one or more DNA ancestry tests, and two had not. Two had fairly well-documented family histories to compare against, one was adopted, and one had information about one side of the family, but not the other. All of us took DNA tests from AncestryDNA, 23andMe, National Geographic and Family Tree DNA. One tester also took each of the five additional tests we reviewed. 
So what are you waiting for? If your family’s genetic signature hasn’t yet been tested, how about considering contributing to the genealogical record and resource for your family by finding one or two men to take a Y-DNA test. If you are a male S-NN-T descendant then please check out the Sinnott/Sennett (and variants) surname project at familytreeDNA.com – you even get a discounted rate for the Y-DNA37 test if ordered through the project. http://www.familytreedna.com/group-join.aspx?Group=Sennett
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