Editing Margin of error (section)

== Comparing percentages ==
Imagine multiple-choice poll <math>P</math> reports <math>p_{a},p_{b},p_{c}</math> as <math>46%, 42%, 12%, n=1013</math>. As described above, the margin of error reported for the poll would typically be <math>MOE_{95}(P_{a})</math>, as <math>p_{a}</math> is closest to 50%. The popular notion of ''statistical tie'' or ''statistical dead heat,'' however, concerns itself not with the accuracy of the individual results, but with that of the ''ranking'' of the results. Which is in first?

If, hypothetically, we were to conduct a poll <math>P</math> over subsequent samples of <math>n</math> respondents (newly drawn from <math>N</math>), and report the result <math>p_{w} = p_{a} - p_{b}</math>, we could use the ''standard error of difference'' to understand how <math> p_{w_{1}},p_{w_{2}},p_{w_{3}},\ldots</math> is expected to fall about <math> \overline{p_w}</math>.  For this, we need to apply the ''sum of variances'' to obtain a new variance, <math> \sigma_{P_{w}}^2
</math>,

:<math> \sigma_{P_{w}}^2=\sigma_{P_{a}- P_{b}}^2 = \sigma_{P_{a}}^2 + \sigma_{P_{b}}^2-2\sigma_{P_{a},P_{b}} = p_{a}(1-p_{a}) + p_{b}(1-p_{b}) + 2p_{a}p_{b}
</math>

where <math>\sigma_{P_{a},P_{b}} = -P_{a}P_{b}</math> is the [[covariance]] of <math>P_{a}</math> and <math>P_{b}</math>.

Thus (after simplifying),

:<math> \text{Standard error of difference} = \sigma_{\overline{w}} \approx \sqrt{\frac{\sigma_{P_{w}}^2}{n}} = \sqrt{\frac{p_{a}+p_{b}-(p_{a}-p_{b})^2}{n}} = 0.029, P_{w}=P_{a}-P_{b}</math>

:<math> MOE_{95}(P_{a}) = z_{0.95}\sigma_{\overline{p_{a}}} \approx \plusmn{3.1%}</math>

:<math> MOE_{95}(P_{w}) = z_{0.95}\sigma_{\overline{w}} \approx \plusmn{5.8%}</math>

Note that this assumes that <math>P_{c}</math> is close to constant, that is, respondents choosing either A or B would almost never choose C (making <math>P_{a}</math> and <math>P_{b}</math> close to ''perfectly negatively correlated''). With three or more choices in closer contention, choosing a correct formula for <math> \sigma_{P_{w}}^2</math> becomes more complicated.