PageRank

In the first hand, we must separate two things: the real PageRank and the PageRank green bar shown in Google Toolbar and other online and offline PageRank tools. The Google bar PageRank (I'll be calling it the green PageRank, or gPR) is merely an indicator. The real PageRank of a website (I'll be calling it the PageRank, or PR from now) is a mathematical value reflecting the probability of a visitor randomly following the links on websites to open this particular website. The value of 1 means 100% probability, that is a visitor randomly surfing the web will always open the website. Sooner or later. On the opposite, the value of 0 means that a random visitor never comes to that particular website through a link on some other site.

I won't get deep into mathematics of the PageRank since this info can be easily found on the web. I'll only underscore the key moments of the PageRank statistical nature.

1. First of all, you must understand the following: the number of websites grows each day, while the overall PageRank value always stays the same: 1 (one). In other words, there is a 100% probability of the fact that a visitor opens SOME site on the web. But the odds of each particular website are going lower and lower every minute. If you have 3 apples and two of them are maggoty, what are your odds to take a good apple? They are 1/3 or 33%. If you have to choose one apple of 100 you only have 1% probability. That's the case with the PageRank - it lowers naturally every day.
   
2. Due to the pt.1 and the overall enormous number of indexed websites it is not possible to show the exact PageRank value every minute. That is why we need the green PageRank which is updated every 3 or 4 months and shows the PageRank value in a more comprehensible form: as a number from 0 to 10. This number correlates to the actual PageRank very little, it only shows the basic trends.
   
   Also, the gPR scale is non-linear. One may think that a website with PR2 is two times more popular (or at least has two times more chances to get that random visitor we were talking about earlier) than its unlucky brother with PR1, but that is not true. In fact it is more likely to be that a PR2 website is 10 times more popular than PR1, but 10 times less popular than PR3. Something like this, but the number 10 is only an example here, since we don't know the exact formula.
   
3. The PageRank models a random user who is surfing the web and following random links on websites. From the practical point of view this means: the more links all over the web points to your website, the higher its PageRank is.

So, now you know that the key off-page ranking factor is the number of inbound links to a website and the green PageRank is the indirect indicator of that number. However, the PageRank mathematical mechanism considers only a quantity of links, while in fact there is also a quality factor. This is implemented via different filters and value dumping factors that Google applies to each link before including it into the PageRank calculation.