## Is Trump’s Lead Significant?

You have to interpret more carefully than that. Roughly 25% of polled Republicans currently prefer Trump *over the others*. That last bit is important: that many other Clowns are vying for the prize matters in the interpretation of Trump’s 23.5 percent.

Since there are fifteen Clowns, an even distribution of favorability would be 6.7% per Clown, so 23.5% is in fact a pretty big outlier. How big? The mean of this favorability distribution is $\mu = 6.0$%, pretty close to the 6.7% expectation. The standard deviation of this distribution of Clown favorability ratings is $\sigma = 5.6$%. Trump’s $p = 23.5$% therefore is a $\Delta = \dfrac{\left|p\, – \mu\right|}{\sigma} = 3.1$-sigma outlier, which is statistically significant. What this means is that the chance of that being just a statistical fluke (i.e., the likelihood that a random choice from among a Gaussian distribution with $\mu = 6.0$% and $\sigma = 5.6$% would land you at 23.5% or higher) is $1 – \mathrm{erf} \left(\dfrac{\Delta}{\sqrt{2}}\right) = 0.0018 = 0.18$ percent.

$$\mathrm{erf}(z) = \dfrac{1}{\sqrt{\pi}} \int_{-z}^z e^{-t^2} dt$$

and is the probability of a random variate lying between $-z$ and $+z$ in a distribution with zero mean and standard deviation $\frac 12$. Now, the 0.2% result above would hold if the favorability distribution were a normal (i.e., Gaussian) distribution, which it almost certainly is not. But the conclusions should correspond closely enough to reality to use as a rough guide.

The next candidate down is Carson at 10.3% (and Bush at 9.8% — essentially the same). Carson is only 0.8 sigma out from the mean, which corresponds to the likelihood of his favorability rating being where it is or higher due to random chance is 45 percent (for Bush it’s 50%).

**Conclusion:** Carson and Bush’s $\sim10$% scores represent, for each, a coin toss in terms of preference. In other words, for Republicans, Bush and Carson are both *meh* for now. But Trump’s 23.5% favorability, given this particular Field of Clowns, signifies a clear and large preference.