One in a series of articles to discuss the merits of various proposals on the agenda for the 2008 ISU Congress.
The following proposal was submitted by Skate Canada:
Rule 353, paragraph 1, k) & m), i) Amend as follows:
i) The panelís points for each Program Component are the multiplied by factors which are as follows:
The factored results are rounded to two decimal places and added. The sum is the Program Component Score
Reason: Currently, the ISU calculation software takes this step of rounding the trimmed mean of program component scores before factoring. This step is necessary, however is not articulated in the current rules. This amendment will allow the rule to properly articulate the practice.
In a commentary on the proposal for rule 353, Sonia Bianchetti discusses the implications of the wording "is not articulated in the rules" in the reason for this proposal. In this article we would like to focus on the phrase "This step is necessary."
Mathematically, no, this step is not necessary at all. This step, in fact any intermediate rounding steps used in the ISU calculation method, is incorrect and indicates a basic lack of understanding of significant digits and how they affect the precision of the calculation for final results.
The following is take from a lesson on significant digits posted on the web site of the Physics Dept. of the University of Guelph in Guelph, Ontario, Canada
Extra Digit in Intermediate Answers
When doing multi-step calculations, keep at least one more significant digit in intermediate results than needed in your final answer.
For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations. If you round-off all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. (This phenomenon is known as "round-off error.")
Had the writers of this proposal investigated, they would have found that intermediate rounding steps are not only not necessary, they contribute to an error in the final answer that can adversely alter the results of a competition.
So, if intermediate rounding compromises the mathematical precision of the final result, why are any intermediate rounding steps included in the calculation method?
The only reason we can think of is that it is cosmetic -- that is, intermediate rounding is used to make the Protocols look "pretty."
Competition protocols include a number of intermediate calculations that are presented with two decimal places. These include the panel GoEs, the panel element scores, the panel Program Component scores, the Total Element Score (TES), and the Program Component Score (PCS). In order that the intermediate numbers printed on the protocol sum up to the final score printed on the protocol, the ISU software carries all the intermediate rounding errors forward and includes them in the final score.
On the current ISU protocols, there is rounding in each intermediate results and there is a cumulative rounding error in the total result so that all the numbers on the protocol look consistent. Apparently the ISU did not want to field questions like "My element scores and Program Scores add up to 174.63, but my total score is 174.62, why are they different," and so decided to cook the books instead, thinking that a few hundredths of a point here or there would not matter. Unfortunately for the skaters, they do matter.
Since a few hundredths of a points in calculation error has the demonstarted potential to skew the results, another approach is needed other than what is currently in ISU 353 and what Skate Canada is currently proposing. We offer two possibilities.
1. Do an exact calculation for the total score rounding to two decimal places only at the end of the calculation, show intermediate results on the protocols rounded to two decimal places, and add the following disclaimer to the protocols: "Due to rounding of intermediate results, the sum of intermediate results may differ from the total score. The total score is an exact calculation and is the most precise value for the total score," or words to that effect.
2. Do an exact calculation for the total score rounding to two decimal places only at the end of the calculation, show intermediate results on the protocols rounded to three decimal places. No disclaimer on the protocols is then needed. A quick and dirty example of what the numbers would look like on such a protocol is shown here.
In any event, this proposal only makes a bad rule worse, and is without mathematical justification. This proposal should be rejected, and ISU rule 353 should instead be revised to require an exact calculation to at least three decimal places, with rounding of the total score only as the last step of the method. Other steps can then be take to make the protocols and other event reports clear to the skaters and the public, without compromising the precision of the results.
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Copyright 2008 by George S. Rossano