are no statistics that can prove that horses helped the Spanish to defeat the Incas.  There are no statistics that can prove that writing or centralized government did so.  These are claims that seem self-evident, but which cannot really be proven statistically.

Finally, and perhaps most importantly, Diamond needs to use anecdotes for the sake of keeping his readers’ interest.  Diamond is writing for a popular audience.  He has to use anecdotes to make the book more interesting and accessible.  He has to catch his audience’s attention before he starts to make more use of statistics in later chapters.

These are three of the more important reasons why Diamond would not use statistics or other “hard” types of evidence at this point in the book.

When Diamond proposes that history become more scientific, he is talking about the history of broad processes in human history.  He is not talking about the details of what we might call "micro" history.  As he says, he is trying to look at why Europeans conquered the Native Americans, not why Kennedy won the 1960 election.

So, Diamond hopes to learn more about the larger scope of human history.  He hopes to be able to discern "general principles" (as he says on p. 421 of the paperback edition) in history.  He wants to be able to answer questions about major trends and factors that affect the whole of human history.  He is trying to learn more about the "big picture," not about the smaller details of history.

In the last pages of the book, Diamond addresses the issues in a very systematic way in which he compares history to various sorts of sciences.  Diamond thinks that history should be studied more in the way that the sciences are studied.  He realizes, however, that this sounds odd to many people.  Therefore, he looks at how history compares to various sciences.  He says that history cannot be like a science like chemistry where experiments are possible.  However, he says it can be like sciences such as climatology or paleontology where experiments are not possible.  Diamond systematically examines ways in which "historical sciences" are different from other sciences.  Thus, we can say that Diamond addresses these issues in a systematic and comparative way.

To understand Diamond's proposal, you should read in the Epilogue.  Diamond's discussion of the idea of human history as a science begins on p. 420 in the paperback edition of the book.

Basically, Diamond says that historians should try to be more scientific.  Historians, he says, should not give up on science simply because they cannot run experiments the way that chemists do.  Instead, they should look to sciences like climatology or ecology where experiments are not possible.  In doing so, they should try to find natural experiments such as the one discussed in Chapter 2.  They should understand that this sort of technique will work only in the long term and cannot be used to predict or explain specific events.  

Since Diamond is a scientist by training, it is not surprising that he uses science to construct history at many points in this book.  Some examples of how he does this include:

• His discussion of natural selection and the natural history of various plants in Chapter 7.  There, he uses science to determine what kinds of plants would have been feasible and/or attractive for people to domesticate.
• His discussion of how diseases evolve in Chapter 11.  There, he uses science to explain how and why some diseases evolve to become epidemic diseases that persist in a population.  The presence of these diseases in the "Old World" and their absence in the "New World" are very important to Diamond's history.
• His discussion in Chapter 5 of radiocarbon dating.  Here, he uses science to help determine when the domestication of plants began in various places.

In these ways and others, Diamond uses science to help construct a history of how some areas came to have "guns, germs and steel" while others did not.

In Chapter 3 of Guns, Germs, and Steel, Diamond is trying to explain to us why Pizarro and his Spanish soldiers were able to defeat the Incas even though they were badly outnumbered.  He wants to establish the idea that “guns, germs, and steel” were the factors that allowed the Spanish to win.  The best way to explain this is through anecdotes.  There is no other good way to do it.

In particular, let us think about the use of statistics.  Let us try to imagine what statistics Diamond could possibly deploy that would prove his point.  Let us also think about what statistics are even available.  One point that Diamond is trying to make is that the smallpox epidemic helped defeat the Incas.  Imagine that Diamond wanted to say how many Incas were killed.  There were no censuses at the time and so there is no way to know how many Incas died in the epidemic.  Even if Diamond could tell us, what good would that do?  There is no way to present a statistic to prove that smallpox helped the Spanish win.

I would argue, then, that Diamond uses anecdotes in Chapter 3 largely because A) few statistics are available and B) statistics would not really prove his point any more effectively than anecdotes do.