Cheat sheet on polling margin of error

The “margin of error” in a poll is the number you add and subtract to get a 95% confidence interval for the underlying proportion (under the simplest possible mathematical model for polling).  Pollers typically quote the “maximum margin of error”, which is the margin of error when the reported value is 50%. When the reported value is 0.7%, reporting the maximum margin of error (3.1%) is not helpful.  The Conservative Party is unpopular, but it’s not possible for them to have negative support, and not likely that they have nearly 4%.

Here is a cheat sheet, an expanded version of one I posted last year. The first column is the reported proportion and the remaining columns are the lower and upper ends of the 95% confidence interval for a sample of size 1000 (Here’s the code).   The Conservative Party interval is  (0.3%,1.4%), not (-2.4%, 3.8%).

       l    u
0.1  0.0  0.6
0.2  0.0  0.7
0.3  0.1  0.9
0.4  0.1  1.0
0.5  0.2  1.2
0.6  0.2  1.3
0.7  0.3  1.4
0.8  0.3  1.6
0.9  0.4  1.7
1.0  0.5  1.8
1.5  0.8  2.5
2.0  1.2  3.1
2.5  1.6  3.7
3.0  2.0  4.3
3.5  2.4  4.8
4.0  2.9  5.4
4.5  3.3  6.0
5.0  3.7  6.5
10   8.2 12.0
15  12.8 17.4
20  17.6 22.6
25  22.3 27.8
30  27.2 32.9
35  32.0 38.0
50  46.9 53.1

As you can see, the margin downwards is smaller than the margin upwards for small numbers (because you can’t have fewer than no supporters). By the time you get to 30% or so, the interval is pretty close to what you’d get with the maximum margin of error, but below 10% the maximum margin of error is seriously misleading.

You can get a reasonable approximation to these numbers by taking the number (not percent) of supporters (eg, 0.7% is 7 out of 1000), taking the square root, adding and subtracting 1, then squaring again: (then converting back into percent: ie, dividing by 10 for a poll of 1000).

    approx l approx u
0.1     0.00     0.40
0.2     0.02     0.58
0.3     0.05     0.75
0.4     0.10     0.90
0.5     0.15     1.05
0.6     0.21     1.19
0.7     0.27     1.33
0.8     0.33     1.47
0.9     0.40     1.60
1       0.47     1.73
1.5     0.83     2.37
2       1.21     2.99
2.5     1.60     3.60
3       2.00     4.20
3.5     2.42     4.78
4       2.84     5.36
4.5     3.26     5.94
5       3.69     6.51
10      8.10    12.10
15     12.65    17.55
20     17.27    22.93
25     21.94    28.26
30     26.64    33.56
35     31.36    38.84
50     45.63    54.57

which is pretty easy on a calculator, or with an Excel macro. For example, for 1000-person polls, if you put the reported percentage in the A1 cell, use =(sqrt(A1*10)-1)^2/10 and =(sqrt(A1*10)+1)^2/10