The most memorable moment of the 2010 general election leaders' debate was when David Cameron tried to justify a point about immigration by citing the agreement of "a 40-year-old black man" he had met in Plymouth who had served in the Royal Navy for 30 years (thus enlisting at the age of 10).

This impossibly young seaman was not the only person called as a witness in that debate: Nick Clegg had been talking to a ward nurse in a short-staffed hospital and a burglary victim from London; Gordon Brown had met a trainee chef and received a letter from a recovered cancer patient; Cameron had also recently met a crime victim from Crosby, a drug addict from Witney and a man suffering from cancer of the kidney. The latter reappeared in Cameron's 2012 speech to his party conference, and in a recent speech on social security Ed Miliband told stories of meeting a young unemployed man in Long Eaton and "somebody who had worked all his life, for 40 years" in the scaffolding business.

Are such stories a good form of argument? They seem to be popular with political speech writers and advertising copywriters who often use them to lend colour and "human interest" to a speech. But as the leaders' debate demonstrated, they can also sound such a false note that they distract from the claims you want to advance. To work well, stories must be in harmony with, and contributing to, your overall argument.

One way they can do this is by bringing to your argument "witnesses" who provide evidence that supports a particular claim. In school we learn to quote supposed "authorities" – writers who lend support to our case not simply because of the veracity of their findings or the eloquence they lend to our words but also because they have some kind of recognised standing which we hope to add to our own.

Know your witnesses

Outside school such citations are useful, but the range of potential sources is greater and the usefulness of any single one cannot be taken for granted. Different audiences value different sorts of "authorities" and a fundamental mistake is to refer to something your audience cannot evaluate or will not evaluate positively. Far from strengthen your case this will weaken it.

Use your anecdotes sparingly

Even good witnesses should be used sparingly and carefully. Excessive and obvious reliance on them will make it seem as if you can't think for yourself. And it can easily seem pretentious. Someone trying to persuade you simply by dropping names of powerful people they have met or of authors they have read is, to put it mildly, annoying.

Make anecdotes tell stories

Stories can also serve as examples – instances of reality which are presented as proofs of some kind of norm. They invite people to make an induction – to conclude that there is some kind of general underlying rule at work and of which we must take heed. When a child points out that their friend doesn't have to go to bed so early, or a teenager insists to a parent that "nobody else has to visit Grandma every weekend", they are trying to illustrate the presence of a rule or a norm from which their parents are unreasonably or bizarrely departing. In a similar way Cameron wants us to conclude from a single dramatic example that the NHS is bad for patients and Miliband that apprenticeships are working out well.

Anecdotal examples of this sort are a necessary and valuable part of everyday, public and political argument. That is because (climate change partially excepted) such arguments are rarely about the nature of physical reality but often about social reality. They concern partial and practical judgments about some aspect of our varied and complex cultures: whether or not people are on the whole trustworthy; the likelihood that people receiving social security are "striving" or "skiving"; whether exams are getting easier or harder.

To make strong claims about the social reality of these things you will need to present examples and these can be made vivid if expressed in the form of stories. The most effective – combined with other evidence and information – help bring clearly to mind something you want the audience to think about more, to sympathise with or to see in a new way. They help to establish a picture of a situation and a definition of reality on the basis of which conclusions may be drawn.

Stories come in many genres. They may be little comedies or tragedies, homely confirmations of what "everybody knows" or unexpected revelations. It is important to be sure that your story is emotionally in tune with the rest of your argument (rather than a substitute for it). And it certainly must not dominate.

Don't make yourself the story

One of the more annoying techniques of politicians is to use purely personal experience as an anecdotal exemplar – as if, just because the politician "got on their bike", we must accept that everyone could or should do likewise. Stories about the speaker may be fine for entertaining dinner speeches (on the way to which a really funny thing happened), but they have a limited place in argumentative speeches where the good character of a speaker should be evident and not need explicit mention.

The problem with Cameron, Clegg, Brown and Miliband's stories is that often they aren't a contribution to the main argument but an attempt to convince the audience that these are great guys, men of the people, on our level. They have forgotten that a good argument is always about the audience to whom it is addressed and not the person making it. If you make yourself the story your argument will fail.