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Benchmarking is the process of evaluating and comparing devices or systems. Processors are frequently benchmarked, in the hopes of coming up with a single number that can capture the value of the CPU and let it be easily compared to others. The value of benchmarking is that it allows this comparison. The problem with it is that there are so many ways to do it, and the answers different benchmarks give about the same processors aren't always consistent.
There is a difference between benchmarking processors and benchmarking whole systems. System benchmarking attempts to evaluate "real world" performance of the whole system, which involves much more than just the processor. Processor benchmarks show isolated performance of processors relative to one another.
Even in just looking at processors, there are many types of benchmarks, which can result in many different scores. Since the processor doesn't operate in a vacuum, the scores will always depend to some extent on the system that is used when performing the benchmark. Some benchmarks are more able to isolate the processor's performance from impact by other system components than others. In addition, benchmarks that are based on Windows or other multitasking operating systems, are subject to accuracy problems induced by how the software is set up, and what other programs might be in operation when they are run.
Some benchmarks can only be used by doing relative comparisons; since the absolute number that is produced depends on the configuration, it can realistically only be used to compare two processors when used in the same exact configuration (much the way the "P rating" calculation is done.) An absolute number from this sort of benchmark can only be given as an average of scores from many system configurations used to run it, or only given with also providing information about the configuration used. Other benchmarks provide roughly the same score for the processor across a wide array of system configurations.
In addition, the benchmark's score depends on what types of code it is executing to do the test. Is it 16-bit or 32-bit code? Many loops or just a few? There are many different types of code and some architectures are better than others at certain types. The power of some of the newer processors is understated when they are tested using older benchmarks (such as the original Norton SI index.) Also remember that benchmarks typically contain some floating-point code, which will mean older processors with no floating point unit will get lower scores.
Here are some relatively common processor tests, which I use on this site to compare processor performance. In doing so I am not claiming that these are necessarily the best measures, but I believe that combined they provide a reasonable picture of processor speed without getting overly bogged down in system-level dependencies: