# Phelps vs Spitz: z-scores tell all

So, yesterday I suggested that, given improvements in training and equipment, Olympic athletes of today should be compared to those of the past using z-scores, rather than raw performance data. This was specifically with reference to comparing swimmer Michael Phelps and the historical performance of Mark Spitz, but I couldn’t find enough data from Spitz’s events in the 1972 Olympics to calculate the standardized z-scores.

(For those just joining us, z-scores use information about a distribution of data points to calculate a “universal” measure of how much one point stands out from the rest – in this case, how much Spitz or Phelps stands out from those among contemporary swimmers.)

Anyway: after another round of digging on Google, I’ve found detailed results (i.e., the final times for the top eight competitors) for the men’s 200-meter butterfly in 2008 and 1972. To convert Phelps’s and Spitz’s times to z-scores, I estimated the parameters of a distribution from the other seven men in the top eight by by taking the average (arithmetic mean) and standard deviation of those times in good ol’ Microsoft Excel [.xls file]. The z-score is just the difference between a single score and the average, divided by the standard deviation.

And …

Spitz wins! His z-score is -3.67, compared to -2.27 for Phelps. (The numbers are negative because the times are, of course, lower than the average from the other seven.) So, even though Phelps is considerably faster than Spitz, Spitz outperformed his competition by a greater margin than Phelps did.