To make a single measurement of a phenomenon does not give one much confidence. Another measurement is made; and then, if there is no opportunity for more, 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.