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-insurance offices rely not merely on statistics of life and death in general, but collect special evidence in respect of different ages and sexes, and make further allowance for teetotalism, inherited disease, etc.  Similarly with individual cases:  the average expectation for a class, whether general or special, is only applicable to any particular case if that case is adequately described by the class characters.  In England, for example, the average expectation of life for males at 20 years of age is 39.40; but at 60 it is still 13.14, and at 73 it is 7.07; at 100 it’s 1.61.  Of men 20 years old those who live more or less than 39.40 years are deviations or errors; but there are a great many of them.  To insure the life of a single man at 20, in the expectation of his dying at 60, would be a mere bet, if we had no special knowledge of him; the safety of an insurance office lies in having so many clients that opposite deviations cancel one another:  the more clients the safer the business.  It is quite possible that a hundred men aged 20 should be insured in one week and all of them die before 25; this would be ruinous, if others did not live to be 80 or 90.