The Standard, Probable, and Other Errors

Consider a measurement or series of measurements that gave the result R and its estimated error E. The experimenter reports the result as 

Whether either Eq. 2.68 or 2.69 is used, the important thing to understand is that R E does not mean that the correct result has been bracketed between R — E and 7 -I- . It means only that there is a probability that the correct result has a value between R — E and R + E. What is the value of this probability There is no unanimous agreement on this matter, and different people use different values. However, over the years, two probability values have been used more frequently than others and have led to the definition to two corresponding errors, the standard and the probable error. 

The standard error. If the result of a measurement is reported as R and is the standard error, then there is a 68.3 percent chance for the true result to have a value between R - and R + E.  

Both standard and probable errors are based on a Gaussian distribution. That is, it is assumed that the result R is the average of individual outcomes that belong to a normal distribution. This does not introduce any limitation in practice because, as stated in Sec. 2.10.2, the individual outcomes of a long series of any type of measurement are members of a Gaussian distribution. With the Gaussian distribution in mind, it is obvious that the definition of the standard error is based on Eq. 2.62. If a result is R and the standard error is E, then E=cr. 

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