*Baseball Research Journal*that used z-scores to compare home-run sluggers of different eras.

I shared the activity with two listserve discussion groups, those of the APA Division of Evaluation, Measurement, and Statistics and the Society for Personality and Social Psychology, offering to provide the raw data and documentation on how to conduct the exercise. I'm pleased to report that over 100 people have requested these materials to use in their own statistics classes. The materials can still be requested, via my faculty webpage (see link in the right-hand column). I framed the exercise as follows, in the documentation:

*Michael Phelps, with eight gold medals in the 2008 Beijing Olympics (on top of six golds from the 2004 Athens games), and Mark Spitz, with seven gold medals in the 1972 Munich Olympics, are swimming’s two greatest champions.*

The two swam many of the same events. Though the respective times by Phelps are several seconds faster than Spitz’s, the 36 years between 1972 and 2008 are a long time for improvements in training, technique, nutrition, and facilities. A statistic known as the z-score allows us to see which swimmer was more dominant

The two swam many of the same events. Though the respective times by Phelps are several seconds faster than Spitz’s, the 36 years between 1972 and 2008 are a long time for improvements in training, technique, nutrition, and facilities. A statistic known as the z-score allows us to see which swimmer was more dominant

**relative to his contemporary peers.**Phelps and Spitz had three individual (non-relay) events in common, the 200-meter freestyle, 200-meter butterfly, and the 100-meter butterfly. Because I have a relatively small class and wanted to have groups of three or four students each work on a different segment of the data, we looked at only the first two of the aforementioned events. Two considerations to note are that (a) times were converted into total seconds to facilitate computations; and (b) where an athlete swam multiple races of the same event (i.e., heats, semifinals, and finals), his fastest time was used. Here are the results:

**200 FREESTYLE**

2008

Mean = 109.42 seconds

SD = 3.20

Phelps time = 102.96 seconds (1:42.96)

Phelps z =

**-2.02**

*[For non-statisticians who may be reading this, z = an individual's value minus the mean, with the difference then divided by the standard deviation. The latter represents how spread out the data are.]*

1972

Mean = 120.33 seconds

SD = 4.57 seconds

Spitz time = 112.78 seconds (1:52.78)

Spitz z =

**-1.65**

**200 BUTTERFLY**

2008

Mean = 117.85 seconds

SD = 3.01

Phelps time = 112.03 seconds (1:52.03)

Phelps z =

**-1.93**

1972

Mean = 129.51 seconds

SD = 5.32

Spitz time = 120.70 seconds (2:00.70)

Spitz z =

**-1.66**

Note that negatively signed z scores are a "good" thing, indicating by how much Phelps or Spitz was faster (i.e., consuming less time) than his respective competitors. As can be seen, Phelps was more dominating against the 2008 fields of his events, than Spitz was against the 1972 fields. It would also be interesting to look at the 100-meter freestyle, which of course, Phelps won by the narrowest of margins.

I thank Nancy Genero of Wellesley College, a fellow University of Michigan Ph.D., for sharing the results from her class; by comparing our respective data files for possible typographical errors, we were able to reconcile some minor differences. Also, as a technical note, an "outlier" swimmer who had a time of 2:33.75 in the 1972 200 freestyle (when the next slowest time was around 2:13) was excluded. An extreme value would have affected both the mean and SD, of course.

Unlike the above analyses, which used all competitors in an event (regardless of whether they reached the finals or even the semifinals), one could also look exclusively at the finals. To the extent that qualifying rules for the Olympics may have changed between 1972 and 2008, or that other factors were operative, the proportion of weak swimmers (in a world-class context) in the fields might have been different in the two Games, again possibly affecting the z-score results. University of Nevada Reno graduate student Irem Uz indeed analyzed only the finals, and these were his results:

Phelps 200 free z = -1.92

Spitz 200 free z = -1.34

Phelps 200 fly z = -1.62

Spitz 200 fly z = -2.01

Under this method, there's a little redemption for Spitz. Examining the results of Spitz's 200 fly win in 1972, his dominance is clear:

1. Mark Spitz 2:00.70 WR

2. Gary Hall 2:02.86

3. Robin Backhaus 2:03.23

4. Jorge Delgado, Jr. 2:04.60

5. Hans Faßnacht 2:04.69

6. András Hargitay 2:04.69

7. Hartmut Flöckner 2:05.34

8. Folkert Meeuw 2:05.57

The mean was roughly 2:04, putting Spitz 3.30 seconds faster than it. Meanwhile, the extremely tight clustering of the fourth- through eighth-place swimmers served to keep the overall SD small (1.63). The upshot is a very big z for Spitz.

**ADDENDA**

The

*Wall Street Journal's*"Numbers Guy," Carl Bialik, provided some other types of Phelps-Spitz comparisons as this year's Olympics were going on.

The

*New York Times*created an amazing slide show of graphics, showing how the swimming times of Phelps and Spitz stacked up against each other, and also how each fared against his respective competition.

Another blogger, Jeremy Yoder, independently came up with the idea to analyze z-scores for Phelps and Spitz. Yoder's results are different from the comparable analyses reported above, for some reason.

Here's a 2014 application of

*z*-scores to golf.