Grow Average© was a Macintosh (Classic) program from the 1990s for demonstrating the Law of Large Numbers and the Central Limits Theorem. The program has two running modes: "grow sample" and "grow average.
The mode "grow average" allows the user to construct sampling distributions (distributions of sample averages) of various sizes. These distributions are saved so that sampling distributions based on different-sized samples can be visually compared. The important insight, of course, is that statistics based on larger samples are less variable than statistics based on smaller samples. This property is referred to as the "Law of Large Numbers."
The mode "grow sample" demonstrates what happens to the mean (or median) of a sample as it grows in size. When a sample is still relatively small, its average can be seen to fluctuate, sometimes wildly, as new observations are added. As the sample gets larger, its average stabilizes near the population average. We designed this mode to help suggest to students a "mechanism" behind the Law of Large Numbers: namely, that whenever the sample average differs from the population average, there are generally a greater number of population elements to sample from that will bring the sample average closer to the population average than there are population elements to sample from that will take the sample average farther from the population average. Furthermore, the farther the sample average is from the population average, the higher in general the probability that the next observation will bring the sample average closer to the population average.
In developing this program, we have attempted to make it useful for a variety of instructional settings and levels. The program can display not only means (and SDs) but medians (and IQRs), sample with or without replacement, has several different levels of sample size and various sampling speeds, and can use any data stored as a text file (with certain limitations).
See also the Chance Plus Project.
Grow Average can still be used on Macintosh computers from the 1992-2007 era. They must use the PowerPC processor and be able to run MacOS Classic programs.