64 



NA TURE 



[May 19, 1904 



depends upon a portion on each side of the middle ordinate 

 which is at a measurable distance from that ordinate. 

 Where the practice is accurate, the portion of the whole 

 figure that may be safely excluded in calculating the area 

 is much larger than in the cases where the shooting is wild. 

 A measure of the accuracy of the marksmen is obtained 

 by drawing an ordinate to divide into equal parts the half 

 area to the right or left of the middle ordinate, and estim- 

 ating the distance between these two ordinates. 



The whole area under consideration represents the total 

 number of shots, and is therefore the same in the case of 

 each curve. For the sake of simplicity we may suppose 

 that 100 shots are fired. It is not true that that number of 

 shots will in any case give the exact curve. We should 

 only obtain its precise form by firing an infinite number 

 of shots and then reducing the whole to a percentage. But 

 for the sake of simplicity in our argument we will talk of 

 100 shots as the number that has been fired, and say that 

 the area is proportional to that number. We see, then, 

 that all the areas enclosed by each of these curves re- 

 spectively and the base line are equal ; and this gives us 

 a simple way of plotting any one series if a single curve 

 has been drawn. It is only necessary to suppose the curve 

 to be stretched to a certain 

 extent in either the hori- 

 zontal or vertical direction, 

 and to be contracted to a 

 proportionate extent in the 

 other direction, in order to 

 pass to another curve of 

 the series. In fact, if one 

 of the curves w'ere painted 

 on a stretched india-rubber 

 sheet, all the other curves 

 could be got from it by 

 pulling the sheet in one 

 direction and slacking it 

 off in the other. 



Another plan would be 

 to bend a loop of wire into 

 the form of one of the 

 curves, and to place a 

 lamp behind it so as to 

 throw the shadow upon a 

 screen. The loop and lamp 

 might then be easily made 

 to move in such a manner 

 that the shadows in the 

 successive positions gave 

 the whole series of curves. 



You will notice in the 



figure the points which 



show the intersection of 



neighbouring curves with 



one another. This is called, 



in mathematical language, the envelope of the family of 



curves. In this case it is a portion of two rectangular 



hyperbolae. 



Now, instead of our performers with the pistol, let us 

 take the case of a series of examiners. As soon as I had 

 observed that the curves of good examiners tended to 

 approximate to the curve of errors, I cast about for the 

 reason of this similarity. It is not far to seek. If we 

 consider one particular candidate as the mean candidate, 

 that is, a candidate such that there are as many above him 

 as below him, we shall see how natural it is that the candi- 

 dates should group themselves about this central figure as 

 the pistol shots about the mean shot. It is clear that the 

 curve of the good examiner should resemble the curve of 

 the bad shot. The object of examination is to separate 

 the candidates from one another as widely as is permissible 

 under the given conditions, while the object of the target- 

 practice is to get as many shots near the central line as 

 possible. 



Here we come to a most important limitation. You 

 have already noticed that the curves we have been consider- 

 ing never touch the base line, that is to say, given a 

 sufficient number of candidates, there will always be one 

 or two removed to an extraordinary degree from the bulk 

 of their fellows. But the examiner is obliged to give marks 

 within certain limits, which he fixes arbitrarily as o and 



Deviation in inches. 

 Fig. 3. — Se^es of Curves of "Error." 



NO. 1803, VOL. 70] 



too. If he were to place his zero point at a very great 

 distance from the middle point, representing 50 marks, he 

 would be able, no doubt, to make allowance for extra- 

 ordinary candidates; on the other hand, the bulk of the 

 candidates would be placed so close together that he would 

 not be able to distinguish between them in any satisfactory 

 manner. He is therefore bound to choose points such that 

 the areas enclosed between the base line and the curve 

 which lies beyond those points are very small compared 

 with the areas up to the middle line. All the candidates 

 beyond those points must be considered as having either 

 nought or full marks. 



Now you will see, I think, how an examiner in English 

 composition, especially if he is a university man who has 

 become acquainted with the finest examples of literature, 

 tends to get a very steep form of curve (Fig. 4). He looks at 

 some one paper, which differs to a considerable extent as re- 

 gards both style and matter from the mean paper, and says, 

 "This paper should have 80 marks at least." But then 

 he thinks, perhaps unconsciously, " How do I know that,, 

 before finishing the pile of papers before me, I shall not 

 find a budding Milton or Addison or Charles Lamb? If 

 I give this candidate 80 marks, shall I be able to assign 

 its true value to a composition of such extraordinary 

 merit? " So he only 

 awards 60 marks to the 

 composition, and finds 

 almost certainly, when he 

 comes to the end of his 

 pile, that no candidate has 

 received any mark near 

 100. It is too late now' to 

 begin marking the papers 

 all over again, and accord- 

 ingly he sends in returns 

 which do not serve to dis- 

 tinguish between the candi- 

 dates in English composi- 

 tion to the same extent as 

 they are distinguished in 

 geometrical drawing, for 

 example. The result is 

 that a good candidate in 

 the former subject is 

 treated unfairly as com- 

 pared with a good candi- 

 date in the latter subject. 



Again, we see why a 

 curve (Fig. 5) based on 

 marking dictation papers 

 by the system of deductions 

 is so abnormal. In this 

 case, the examiner, with- 

 out considering minor defects, makes a certain deductior» 

 for each mistake in spelling . If 10 marks are taken off for 

 each mistake, all candidates having more than ten errors- 

 receive no marks, whereas if we were to assign negative 

 marks, the curve of errors would almost certainly be repro- 

 duced, the mean ordinate being below the zero point. The 

 divergence which you perceive near the point representing; 

 full marks is due to there being a good many candidates who 

 make no important mistakes. If minor defects, such as re- 

 finements of punctuation, were considered, and the scale- 

 stretched beyond 100, this divergence would also disappear. 



The problem which presented itself was how to bring 

 these very different results into some accord. In order to 

 give equal weight to various subjects having the same- 

 maximum, it seemed to me necessary that the examiners- 

 should have a common standard to work up to. Accord- 

 ingly, during the latter period of my service with the Civit 

 Service Commission, I caused such a diagram as has been 

 placed before you to be printed on the sheet containing the 

 examiner's report of his work. Qn that diagram, also, was- 

 printed a curve resembling a moderate sized gendarme's 

 hat. If, as often happened, the examiner had 1000 papers 

 to mark, he was requested to go through a batch of 100 

 taken at hazard, and to plot his curve upon the diagram. 

 After a few examinations an old hand would probably find 

 that his curve for the first 100 resembled closely the standard 

 curve before him, but a fresh examiner might find himself 

 altogether beside the mark. In such a case he was askedi 



Id f& ftrt> 



Number of marks gained. 

 'iG. 4. — English Composition. 



