one takes the mean or average of the two. But why?
For the result may certainly be worse than the first measurement.
Suppose that the events I am measuring are in fact fairly described by
Fig. II, although (at the outset) I know nothing about them; and that my
first measurement gives _p_, and my second _s_; the average of them is
worse than _p_. Still, being yet ignorant of the distribution of these
events, I do rightly in taking the average. For, as it happens, 3/4 of
the events lie to the left of _p_; so that if the first trial gives _p_,
then the average of _p_ and any subsequent trial that fell nearer than
(say) _s'_ on the opposite side, would be better than _p_; and since
deviations greater than _s'_ are rare, the chances are nearly 3 to 1
that the taking of an average will improve the observation. Only if the
first trial give _o_, or fall within a little more than 1/2 _p_ on
either side of _o_, will the chances be against any improvement by
trying again and taking an average. Since, therefore, we cannot know the
position of our first trial in relation to _o_, it is always prudent to
try again and take the average; and the more trials we can make and
average, the better is the result. The average of a number of
observations is called a "Reduced Observation."

We may have reason to believe that some of our measurements are better
than others because they have been taken by a better trained observer,
or by the same observer in a more deliberate way, or with better
instruments, and so forth. If so, such observations should be
'weighted,' or given more importance in our calculations; and a simple
way of doing this is to count them twice or oftener in taking the
average.

Sec. 6. These considerations have an important bearing upon the
interpretation of probabilities. The average probability for any
_general class_ or series of events cannot be confidently applied to any
_one instance_ or to any _special class_ of instances, since this one,
or this special class, may exhibit a striking error or deviation; it
may, in fact, be subject to special causes. Within the class whose
average is first taken, and which is described by general characters as
'a man,' or 'a die,' or 'a rifle shot,' there may be classes marked by
special characters and determined by special influences. Statistics
giving the average for 'mankind' may not be true of 'civilised men,' or
of any still smaller class such as 'Frenchmen.' Hence life-insuranc