Propagation of Uncertainty from Random Error

We can usually estimate or measure the random error associated with a measurement, such as the length of an object or the temperature of a solution. The uncertainty might be based on how well we can read an instrument or on our experience with a particular method. If possible, uncertainty is expressed as the standard deviation or as a confidence interval, which are discussed in Chapter 4. This section applies only to random error. We assume that systematic error has been detected and corrected. 

For most experiments, we need to perform arithmetic operations on several numbers, each of which has a random error. The most likely uncertainty in the result is not simply the sum of the individual errors, because some of them are likely to be positive and some negative. We expect some cancellation of errors. 

Suppose you wish to perform the following arithmetic, in which the experimental uncertainties, designated eh e2, and e3, are given in parentheses. 

The arithmetic answer is 3.06. But what is the uncertainty associated with this result  

To find the percent relative uncertainty in the sum of Equation 3-4, we write 

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