When discussing the use of analytical procedures. What is meant by the \"precision of the expectation\"? in applying this notion to an analytical procedure, how might an auditor calculate a tolerable difference?

The quality of an expectation is referred to as the precision of the expectation. Precision is a measure of the potential effectiveness of an analytical procedure; it represents the degree of reliance that can be placed on the procedure. Precision is a measure of how closely the expectation approximates the unknown "correct" amount. The degree of desired precision will differ with the specific purpose of the analytical procedure. The precision of the expectation is a function of the materiality and required detection risk for the assertion being tested. If the assertion being tested requires a low level of detection risk, the expectation needs to be very precise. However, the more precise the expectation, the more extensive and expensive are the audit procedures used to develop the expectation, resulting in a cost/benefit trade-off.
The second step in the substantive analytical procedures decision process is to define or calculate a tolerable difference. Since the expectation developed by the auditor will rarely be identical to the client's recorded amount, the auditor must decide the amount of difference that would require further investigation. The size of the tolerable difference depends on the significance of the account, the desired degree of reliance on the analytical procedure, the level of disaggregation in the amount being tested, and the precision of the expectation. Auditors often use rules of thumb such as, "tolerable difference is 10% of the predicted amount."