SECTION 3.0 



GENERAL DISCUSSION OF MATHEMATICAL ERRORS 



In order to understand more completely the various classifications into which the Bathy- 

 thermograph errors have been placed, it is advantageous to start with a general discussion of 

 mathematical errors. 



3.1. True Values and Probable Values 



In determining an instantaneous temperature somewhere in the oceans one must 

 assume that there is a true value which can be called an absolute value without error. 

 However, the true value of most quantities can never be known because of t?ie un- 

 avoidable errors in measurement and calculation, no matter how precise the instru- 

 ments employed are. 



In lieu of obtaining a true value one usually obtains the most probable value and 

 along with the probable value a probable error. One can say that the probable value 

 will not differ from the true value by more than the probable error. 



3.2. Errors 



The word "error" is a most common term used to describe a multitude of dif- 

 ferent conditions. One type of error has already been discussed, the probable error. 

 In addition errors can be classified as determinate errors and indeterminate errors. 



The term "true error" is often used, which mathematically defines the difference 

 between a true absolute value and any single measured value. However, the true 

 value is never known in the ocean, and therefore one needs to concern himself only 

 with the probable error. 



3.2.1. Determinate Error 



Any error that is discovered and allowed for in magnitude and sign in the form 

 of a correction for its effect is a determinate error. For example, comparison of an 

 ordinary thermometer with a standard may reveal errors in the graduation of the 

 former and result in a standard calibration. Every temperature measured with this 

 thermometer would then be subject to an error of definite magnitude and sign which 

 could be determined by reference to the standard calibration. 



3.2.2. Indeterminate Errors 



All errors that either cannot or are not properly allowed for in magnitude and 

 sign are known as indeterminate errors. It is obvious that the correction for deter- 

 minate errors will themselves be subject to errors and thus constitute one class of in- 

 determinate errors. 



3.2.2.1. Accidentia! Errors 



A particularly important class of the indeterminate errors is that of accidental 

 errors. Such errors are due to the combined effect of a large number of undeter- 

 mined causes. Experience has shown that these deviations are inevitable in all meas- 

 urements and that they result from small unavoidable errors of observations due to 

 more or less fortuitous variations in the sensitivity of our measuring instruments 

 and of our senses of perception. 



