Page:Sm all cc.pdf/42

 === Example 2: race between the hare and the tortoise. === In an update of the ancient race between the hare and the tortoise, the tortoise won the race and yet the hare got a speeding ticket. Since the tortoise won, its ‘average’ speed must have been faster than the hare’s more erratic pace. Use of a mean and standard deviation would be quite inappropriate for the hare. The hare had a bimodal speed of either zero (resting) or -- rarely -- extremely fast; probably the mean would be closer to the dominant zero peak and the standard deviation would imply some negative speeds. Sampling the hare’s speed at uniform time intervals would give a completely different picture than if its speed were sampled at uniform distance intervals: according to the former it was usually resting, but according to the latter it was usually breaking the speed limit.

Example 3: percentage of high school students that graduate, by state.
We cannot expect values of any variable for different states of the U.S.A. to be truly independent: adjacent states or states with similar industries could be expected to give more similar values than distant states with different economic bases. We will proceed anyway, because such examples are illustrative and because it is fruitless to respond to a question like “What is the average percentage of students that graduate from U.S. high schools?” with the answer “It is impossible to say, because it is invalid to average such data.” Figure 5 shows that the distribution of high-school graduation rates appears to be approximately normal. Indeed, it looks more like a bellshaped or Gaussian distribution than do Figures 1 and 2 (which are known to come from a normally distributed parent population). Furthermore, skewness is low, and Chauvenet’s criterion does not reject any data. Thus it is relatively safe to conclude that the calculated average graduation percentage is 75.1% and that the ‘true’ average is 75.1±2.1%. Nonparametric statistics are neither needed nor as appropriate as parametric statistics for this dataset. The mean value of 75.1 is close to the median of 76.2, at least in comparison to the high standard deviation of 7.4, again suggesting normality.

Example 4: population of U.S. states (1990 census).
The populations, in millions, of the U.S. states obviously diverge from a normal distribution (Figure 6a). Our ‘quick-and-dirty’ technique of comparing mean to median indicates a non-normal distribution: the mean is almost 50% larger than the median, and examination of Figure 6a suggests that one anomalously high value is at least partially responsible for pulling the mean so far to the right of the median. The distribution has a strong positive skewness of 2.4, with no left tail and a long right tail.