On The Measurement Of A Physical Quantity Whose Magnitude Is Influenced By Primary Causes Beyond The Control Of The Observer And On The Method Of Determining The Relation Between Two Such Quantities

248 STATISTICS: W. A. SHEWHART P'ROC. N. A. S. ON THE MEASUREMENT OF A PHYSICAL QUANTITY 'WHOSE MAGNITUDE IS INFLUENCED BY PRIMARY CAUSES BEYOND THE CONTROL OF THE OBSERVER AND ON THE METHOD OF DETERMINING THE RELATION BETWEEN TWO SUCH QUANTITIES By W. A. SHEWHART RESEARCH LABORATORIES Or THE AMERICAN TZLOPHONS, AND TELEGRAPH COMPANY AND THE WOSTURN ELCTRIC COMPANY, INC. Read before the Academy, April 25, 1922 In order to relate the physical and chemical properties of carbon to the microphonic properties of this material, it has been found necessary to make measurements on physical quantities which are influenced by primary causes beyond the control of the observer. In many problems of the physical and engineering sciences it is possible for the observer to control within narrow limits, the causes of variation of a quantity while it is being subjected to measurement. Certain problems arise, however, in these sciences as in the field of economics and biology wherein it is impossible to do this. In general let Xi and X2 represent two quantities related to others U1, U2, ... Ua; Vi, V2, ... Vb; W3, W2 ... Wb in the following way: (1) Xi F1 (Ui, U2, ... Ua, Vl, V2, ... Vb) (2) X2= F2 (U1, U2, ... Ua, W1, W2, ... Wb) where F1 and F2 represent unknown functional relations. In most physical experiments it is possible to hold the U's, V's, and W's, practically constant while a measurement is being made on either X1 or X2. In the last analysis, however, the U's, V's, can never be held constant and in many cases, particularly where these symbols represent molecular phenomena, the variations about their mean values may become quite large. Thus, if X1 and X2 represent two microphonic characteristics of granular carbon which are functions of the physical and chemical properties represented by the U's, V's, and W's, it has been found necessary to apply certain statistical criteria to determine the nature of the cause complex controlling a single quantity such as either Xi or X2 and also to determine quantitatively the degree of relation existing -between the two microphonic properties. As a result of such a study involving an analysis of thousands of measurements of the above type, 'certain conclusions which .are of general interest have been reached in respect to the practical application of statistical methods in connection with physical measurements of this character. VOL. 8, 1922 STATISTICS: W. A. SHEWHART 249 In order to study the nature of the complex of causes controlling a single quantity such as X1, one of the first problems is to determine whether or not the causes represented by the U's, and V's satisfy the following conditions: (1) That all of the causes, n in number, are effective at the time of each observation, (2) that the probability, p, that a cause will produce a positive effect is the same for all of the causes. (3) That the probability, p, remains the same for all of the observations, (4) that the effect, Ax, of a single cause is the same for all of the causes. If these conditions are fulfilled, the distribution in Xi can be represented by the successive terms of the expansion (p + q)n where the ordinates are separated at intervals of 2 Ax. For most of the problems it has been found convenient to compare the observed distribution with the theoretical distribution consistent with the above random conditions. In general, the following procedure has been followed: For each observed distribution two factors k = -3--/2 and 12 - 2 have been calculated and compared 22 -23 with similar factors consistent with the above mentioned binomial expansion, where the first four corrected moments of the observed distribution about the mean are represented by the symbols jul, jf, p3, and &4. It will be noticed that k and 32 are independent of the units used in making the measurements. F