Instead of a moving average window, what about an exponential decay window?
x today + (1-x) yesterday
x + (1-x) yesterday's weight
where "before the start" has a weight of 0.
x determines the "half life" of the contribution of older lengths...
And, of course, a simple correlation measure between adjacent days might be interesting.
I would have expected a more bell-like distribution.
Bell curves occur when you take the distribution of the averages
of large numbers of uncorrelated samples, each drawn from the same distribution (aka, the central limit theorem).
It seems unlikely that the probability of each character being added is very uncorrelated.
Now, presuming the length of each "book" of diaries is uncorrelated with the others (or lightly correlated), and each book is large enough, then the average of the lengths of entries in each book should form a bell curve (once again, CLT).