# In search of ... the exact doubling time

So far, we've been guesstimating doubling time from a graph of population over time, or we've calculated it directly from a table. But there are a few problems...

First, guesstimating a parameter from a graph is not really approved scientific procedure. It would be hard to imagine publishing a report in the Journal of the American Medical Association that started out "we looked at the graph, squinted a little, and decided that the doubing time was 23 minutes...".

Secondly, when we calculated the doubling time from the table, that only worked because the table happened to include an exactly doubled population. If the table does NOT contain a doubled population entry, then you're in trouble. In other words, we can figure out the exact doubling time for this E. nice below, but not for E. naughty:

time E. nice E. naughty
10:00 10 million 10 million
10:20 20 million 15 million

At first glance, it appears that the population got "halfway" to doubling, so doubling time should be 40 minutes. Let's see if that's true -- if the population keeps growing at the same rate, will it double in 40 minutes?

time E. messy
10:00 10 million
... multiply by 1.5 to get
10:20 15 million
... multiply by 1.5 to get
10:40 22.5 million

Oops, we got to 22.5, not 20 million. Granted, this is a small difference, but it's still not the right answer. Like the old joke goes, a million here, a million there, pretty soon you're talking about a real epidemic. What we need is a foolproof way to determine exactly what the doubling time is.