Chapter 11 Language of Descriptive Statistics Section 11.1 Terminology and Language
11.1.3 Remarks on the Rounding Processes
As the following considerations and examples will show, we have to be very careful when calculating with rounded results. Let us consider the set of all non-negative real numbers. On this set, let us define the multiplication
by the calculation rule
i.e. the product is calculated by calculating the usual product first and subsequently rounding the result mathematically to two fractional digits.
The law of associativity no longer applies to the rounded multiplication. For the numbers , , and , we have, for example,
Changing the brackets, however, results in
Since calculators (and computers) always calculate using rounded results, this means that the law of associativity does not apply unrestrictedly to multiplication on calculators.
Likewise, false results can be caused by careless rounding. Let us consider the numbers and . Then, we have
Furthermore, we have
Rounding after each multiplication to fractional digits results (due to ) in the wrong result: