Example 3: Monthly averages of temperature for Anchorage, Alaska.
The histogram of monthly temperatures in Anchorage (Figure 10a) is strongly bimodal, with equal-sized peaks at 10-25° and at 45-60°. Skewness is zero because the two peaks are equal in size, so the mean is close to the median and both are a good estimate of the true average. Many bimodal distributions have one dominant peak, however, causing a distribution that is skewed and biasing both the mean and median.
Nonparametric statistics are much more appropriate here than parametric statistics. Neither is an acceptable substitute for investigation of the causes of a bimodal distribution. For this example, the answer lies in the temporal trends. Again we have a time series, so a plot of temperature versus time may lend insight into data variability. Months of a year can define an ‘ordinal’ scale: order along a continuum is known but there is neither a time zero nor implicitly fixed values. Here I simply assigned the numbers 1-13 to the months January-December-January for plotting, keeping in mind that the sequence wraps around so that January is both 1 and 13, then I replaced the number labels with month names (Figure 10b). A circular plot type known as polar coordinates is more appropriate because it incorporates wraparound (Figure 10c).
Consider the absurdities of simply applying parametric statistics to datasets like this one. We calculate that the average temperature is 35.2° (i.e., cold), but in fact the temperature almost never is cold. It switches rapidly from cool summer temperatures to bitterly cold winter temperatures. Considering just the standard deviation, we would say that temperature variation in Anchorage is like that in Grand Junction, Colorado (16.8° versus 18.7°). Considering just the mean temperature, we would say that the average temperature of Grand Junction (52.8°) is similar to that of San Francisco (56.8°). Thus temperatures in Grand Junction, Colorado are statistically similar to those of San Francisco and Anchorage!
Crossplots
Crossplots are the best way to look for a relationship between two variables. They involve minimal assumptions: just that one’s measurements are reliable and paired (xi, yi). They permit use of an extremely efficient and robust tool for pattern recognition: the eye. Such pattern recognition and its associated brainstorming are a joy.
