Is a Basketball Free-throw Sequence Nonrandom? A Group Exercise for Undergraduate Statistics Students

I describe a group exercise that I give to my undergraduate biostatistics class. The exercise involves analyzing a series of 200 consecutive basketball free-throw attempts to determine whether there is any evidence for sequential dependence in the probability of making a free-throw. The students are given the exercise before they have learned the appropriate statistical tests, so that they can come up with ideas on their own. Students spend a full class period working on the problem, with my guidance and hints. In the next class period, we discuss how each student group approached the problem. I then present several alternative ways to analyze the data, including a runs test and a contingency table analysis of transition frequencies

Estimating Maximum Performance: Effects of Intraindividual Variation

Researchers often estimate the performance capabilities of animals using a small number of trials per individual. This procedure inevitably underestimates maximum performance, but few studies have examined the magnitude of this effect. In this study we explored the effects of intraindividual variation and individual sample size on the estimation of locomotor performance parameters. We measured sprint speed of the lizard Sceloporus occidentalis at two temperatures (20 degrees C and 35 degrees C), obtaining 20 measurements per individual. Speed did not vary temporally, indicating no training or fatigue effects. About 50% of the overall variation in speed at each temperature was due to intraindividual variation. While speed was repeatable, repeatability decreased slightly with increasing separation between trials. Speeds at 20 degrees C and 35 degrees C were positively correlated, indicating repeatability across temperatures as well. We performed statistical sampling experiments in which we randomly drew a subset of each individual\u27s full set of 20 trials. As expected, the sample\u27s maximum speed increased with the number of trials per individual; for example, five trials yielded an estimate averaging 89% of the true maximum. The number of trials also influenced the sample correlation between mean speeds at 20 degrees C and 35 degrees C; for example, five trials yielded a correlation coefficient averaging 90% of the true correlation. Therefore, intraindividual variation caused underestimation of maximal speed and the correlation between speeds across temperatures. These biases declined as the number of trials per individual increased, and depended on the magnitude of intraindividual variation, as illustrated by running sampling experiments that used modified data sets