Amber Watkins gave me a suggestion on the animation for the ratio estimation, and I think this is a good topic for my **animation** package. I’ve finished writing the initial version of the function `sample.ratio()`

for this package, which will appear in the version 1.1-2 a couple of days later.

As we know, the benefit of ratio estimation is that sampling skewness may be adjusted for, because the estimation of $latex \bar{Y}$ will make use of the information in the relationship of *X* and *Y*: $latex \bar{X} \cdot (\bar{y}/\bar{x})$. Here is a demo (we can see the ratio estimate, denoted by the red line, generally performs better than $latex \bar{y}$):