You may have missed the latest genetic discovery. As reported by The Daily Telegraph on Friday: "One million British men may be directly descended from the Roman legions". The story reappeared on Sunday, at the Who Do You Think You Are – Live event at London's Olympia, when it was repeated by Alistair Moffatt, the managing director of BritainsDNA, the company behind the claims.

Such stories are becoming increasingly common in newspapers, on television and radio. Last week on the BBC miniseries Meet the Izzards we were told that Eddie Izzard is a Viking descendant on his mother's side and an Anglo-Saxon descendant on his father's. Last year the Observer reported that Tom Conti has Saracen origins and is a relative of Napoleon Bonaparte.

And for upwards of £150 you too can have your DNA "tested" by any of a number of direct-to-consumer ancestry companies. But how reliable are these claims? The truth is that there is usually little scientific substance to most of them and they are better thought of as genetic astrology.

For some time it has been possible to compare DNA sections among individuals; and in a broad sense greater genetic similarity means greater relatedness. But you have inherited different sections of your DNA from different ancestors, and as we look back through time the number of ancestors you have almost doubles with each generation (it would double exactly were it not for the fact that we are all somewhat inbred).

This means that you don't have to look very far back before you have more ancestors than sections of DNA, and that means you have ancestors from whom you have inherited no DNA. Added to this, humans have an undeniable fondness for moving and mating – in spite of ethnic, religious or national boundaries – so looking back through time your many ancestors will be spread out over an increasingly wide area. This means we don't have to look back much more than around 3,500 years before somebody lived who is the common ancestor of everybody alive today.

And perhaps most surprisingly, it has been reasonably estimated that around 5,000 years ago everybody who was alive was either the common ancestor of everybody alive today, or of nobody alive today; at this point in history we all share exactly the same set of ancestors.

What does this say about the descendants of the Roman legions? It says almost everybody in Britain is one, as well as being the descendant of Vikings, Celts, Anglo-Saxons, Arabs, Jews, Saracens, Goths, Vandals, or whatever ethnic group you want to choose in Europe and its vicinity over the last few thousand years. Nobody is pure this, or pure that, and a substantial proportion of human ancestry is common to all of us. Ancestry is complicated and very messy.

Claims like those mentioned at the beginning of this article are usually based on only two sections of DNA: the mitochondrial DNA, which is inherited solely through the female line, and the male line equivalent, the Y chromosome. It is possible to compare these and make reasonable estimates of how long ago two individuals share a common ancestor through the male or female line, although such estimates are usually rather imprecise.

But saying where, and in what ethnic group, that common ancestor lived is considerably more speculative. In the hands of "genetic ancestry testing" companies this speculation almost invariably comes from the murky world of interpretative phylogeography – an approach to "reading" our genetic history that is easily steered by subjective biases, has never been scientifically shown to work and, in some forms, has been explicitly shown not to work.

So saying this or that Y chromosome came to Britain with the Roman legions, or that Oprah Winfrey has Zulu mitochondrial DNA, is just storytelling. Stories are fine if it is made clear that is all they are. But science isn't about telling stories, it's about testing them (in science we prefer the term hypothesis testing).

The simplicity of how Y chromosome and mitochondrial DNA are inherited is part of their appeal in ancestry testing: you don't have to worry about that inconvenient doubling of your ancestors with each generation back in time. You only have one father, one father's father, etc.

But the price of that simplicity is irrelevance: those two lineages represent a rapidly diminishing fraction of your ancestry the further back in time you go. It may be the case that your mitochondrial DNA lineage came to Britain with the Vikings – although that would be extremely difficult to demonstrate scientifically – but if true, this would still say very little about your origins.

There are some situations where Y chromosome or mitochondrial DNA information can be useful. It is, for example, reasonable to use large samples of these DNA types to say something about the histories of populations, if analyses are performed carefully and at the population level. Also, if genealogical research (parish records, surnames, etc.) suggests that two men share a common male line ancestor in the 16th century, the Y chromosome could be used to support or reject this claim. But individual Y chromosome or mitochondrial DNA types provide no more than the vaguest hint about where their ancestors lived hundreds, or thousands of years ago.

So why do newspapers report these claims and why do TV and radio programme makers base documentaries on them? After all, there are plenty of experts who are engaged in scientifically cautious research on our genetic history and will point out their absurdity. One reason is that, being simple "just so" stories, they have a popular appeal that cannot be matched by the more rigorous population level testing of migration histories. The bias is always towards the story rather than the science.

Another possible reason is that "ancestry testing" is aimed at individuals, although in reality the statements made are sufficiently general that they could be true for a large number of people. This is reminiscent of the "Forer effect" in psychology – the observation that individuals will tend to believe descriptions of their personality that supposedly are tailored specifically for them, but are in fact vague and general enough to apply to a wide range of people. The same effect has been used to explain the popularity of horoscopes.

Yet another possible reason for the popularity of genetic ancestry stories is that because of the inherently random nature of the way genes are passed on to offspring, and mutation – the processes that generate genetic differences – very different population histories can give rise to the same patterns of genetic differences between individuals. This means that one set of genetic similarities cannot equate to only one possible history and we have no choice but to use statistical models. But statistical models give the sort of probabilistic answers that only statisticians find sexy.

"Genetic Aastrology" stories are often promoted by people with financial interests in genetic ancestry testing companies, those concerning "Roman legions" and "Meet the Izzards" being no exception. On 9 July 2012 Alistair Moffat – co-founder of the ancestry testing company BritainsDNA and Rector of St Andrews University – appeared on BBC Radio 4's Today Programme to make several claims about genetic ancestry that were wildly inaccurate, so that even his business partners, under pressure, eventually accepted that errors had been made.

My colleague Prof David Balding and I wrote to the BBC and to the two main scientists at BritainsDNA – both of whom we knew – expressing our concerns about the claims being made. Our expressions of concern over accuracy were met with threats of legal action for defamation by Mr Moffat's solicitors.

Perhaps it is harmless fun to speculate beyond the facts, armed with exciting new DNA technologies? Not really. It costs unwitting customers of the genetic ancestry industry a substantial amount of hard-earned cash, and it disillusions them about science and scientists when they learn the truth, which is almost always disappointing relative to the story they were told.

Exaggerated claims from the consumer ancestry industry can also undermine the results of serious research about human genetic history, which is cautiously and slowly building up a clearer picture of the human past for all of us.

Many of the commercial companies plant stories in the media that sound exciting and seem scientific. But very often they are trivial or wrong, are not published in peer-reviewed scientific journals, and just serve as disguised PR for the company.

Mark Thomas is professor of evolutionary genetics at University College London