Opinion polls for the presidential elections… with margins of error!
In the last few days, a lot of opinion polls have been released about the next presidential elections in France, to be held in April and May 2012. They feature three oponents, Nicolas Sarkozy, National Front’s Marine Le Pen, and one of the Socialist Party leaders, either Dominique Strauss-Kahn, Martine Aubry, Ségolène Royal or François Hollande. Here are some of the results cited in a recent Le Monde article
- DSK 29 %, Nicolas Sarkozy 23 %, Marine Le Pen 21 %
- Martine Aubry 24 %, Nicolas Sarkozy 24 %, Marine Le Pen 22 %
- Nicolas Sarkozy 24 %, François Hollande 23 %, Marine Le Pen 22 %
- Nicolas Sarkozy 24 %, Marine Le Pen 22 %, Ségolène Royal 19 %.
A crazy thing is that most attention of the debates is focused on the rank of the candidates, but a look on margins of error shows that the relative positions are not significant in most of the cases.
A standard opinion poll has a sample of size n=1000. Let the true score of a candidate be , the choice of voter i be (=1 if intends to vote for the candidate, 0 otherwise), their mean . Then the 95%-confidence interval is approximately (which means that it contains the computed score with a probability of (approximately) 95%).
The score p is around 20%, so the margin of error is . Most of the polls cannot really tell who is gonna reach the second round…