User:Smaines/percent-difference-study

Comparing percentages
Imagine multiple-choice poll $$P$$ reports $$p_{a},p_{b},p_{c}$$ as $$46%, 42%, 12%, n=1013$$. As described above, the margin of error reported for the poll would typically be $$MOE_{95}(p_{a})$$, as $$p_{a}$$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 poll $$P$$ over subsequent samples of $$n$$ respondents (newly drawn from $$N$$), and report result $$w = p_{a} - p_{b}$$, we could use the standard error of difference to understand how $$ w_{1},w_{2},w_{3},\ldots$$ is expected to fall about $$ \overline{w}$$. For this, we need to apply the sum of variances to obtain a new variance, $$ \sigma_{w}^2 $$,


 * $$ \sigma_{w}^2=\sigma_{p_{a}\plusmn 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}

$$

where $$\sigma_{p_{a},p_{b}} = p_{a}p_{b}$$is the covariance of $$p_{a}$$and $$p_{b}$$. Note also that $$\sigma_{x-y}^2 = \sigma_{x+y}^2$$.

Thus (after simplifying),


 * $$ \text{Standard error of difference} = \sigma_{\overline{w}} \approx \sqrt{\frac{\sigma_{w}^2}{n}} = \sqrt{\frac{p_{a}+p_{b}-(p_{a}-p_{b})^2}{n}} = 0.02, w=p_{a}-p_{b}

$$
 * $$ MOE_{95}(p_{a}) \approx 1.96\sigma_{\overline{p_{a}}} = \plusmn{3.1%}$$
 * $$ MOE_{95}(w) \approx 1.96\sigma_{\overline{w}} = \plusmn{5.8%}$$

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