1873O * n Experimental Research. 3 



named, then the theory enables the observer to determine 

 the degree of likelihood that his average is within such 

 limit. Thus it is found that the probability of the average 

 of 1051 being within 50 of the truth (whatever the truth 

 may really be) is o*66 ; and therefore the observer should 

 contemplate the possibility of his average being within 50 of 

 the truth, or, which is the same thing, that the truth lies 

 somewhere between the limits 1051 ± 50. This he would 

 assume with the same degree of confidence, neither more 

 nor less, that he would yield to a witness who is known to 

 speak truth 66 times and falsehood 34 times out of every 

 hundred. 



If the calculation had been made for the limits 1051 ± 10, 

 the resulting probability would have been much less than 

 0'66 ; and if for 105 1 ± 100, the resulting probability would 

 have been much greater. This is in accordance with com- 

 mon sense. 



It must also be very important to the observer, when he 

 has made different sets of observations, to know how best to 

 combine their respective averages. For both purposes the 

 average of the set, or of each, may be "weighted" by 

 means of the formula — 



..2. 

 W 



where n — the number of observations and He 2 = the sum 

 of the squares of the successive differences obtained by sub- 

 tracting each observation from the arithmetic mean (average) 

 of the whole. 



The term weight is almost self-explanatory. Of a series 

 of observations there may be one which the observer consi- 

 ders to have been obtained under more favourable conditions 

 than the others, and to which, in the balance of judgment, 

 he should accord greater " weight." For if we suppose a 

 satisfactory experiment to give 5, and one of unequal weight 

 6, it would be obviously unfair to take the average as 5J ; 

 but it would be more reasonable to give the result 5 the ad- 

 vantage of supposing it to have occurred, say, three times 

 to the occurrence of the result of 6 once. This would be 

 giving the observations 5 and 6 the weights of 3 and 1, and 

 the average would be 55-. Such a mode of reasoning gave 

 rise, before mathematicians had constructed the theory of 

 probabilities, to a rule for finding the average, which may be 

 quoted as follows : — Weigh every observation, multiply it by 

 its weight, take the sum of the products, and divide this 

 sum by the sum of the weights. But it was found that the 



