Some statistical methods have been mentioned in the preceding discussion, and consideration of statistical concepts is unavoidable when considering how best to visualize numerical data. As noted earlier, because we can lie with statistics, we can also lie with statistical maps. Indeed, maps have been used throughout history as propaganda tools (Campbell, 1993, pp. 229-235), so potentially we can have honest error as well as pure cartographic deceit. Perhaps the greatest danger in the mapmaking process is that people tend to believe the information in maps (what MacEachren, 1995, p. 337) called the connotation of veracity), and they also believe that maps are unbiased (the connotation of integrity).

But mapmakers, like other elements of society, are culturally conditioned, selectively including and excluding data according to the values of the responsible parties. Given that maps can harbor many possible errors and biases, both intentional and accidental, it is incumbent on the crime analyst to be aware of possible sources of error and to work to avoid them. Nowhere is there more scope for distortion and misinterpretation than in the preparation of maps based on numerical data. This is due to the potential complexity of the information and the infinite set of display permutations, whether in raw form or as some derivative measure such as a rate or percentage.

Mapmakers can gain a preliminary understanding of what the numbers mean through the process of exploratory spatial data analysis. It is quite helpful to understand what the distribution of a set of numbers looks like when expressed graphically. Is this a normal (bell-shaped) distribution with most observations clustering around the mean, or average, and a few very low and a few very high values? Is it a skewed distribution with extreme values to the right (high values) or the left (low values)?

In the unlikely event of a normal, symmetrical, bell-shaped distribution, maps created by all of the classing methods look similar. Almost always, however, real-world data are somewhat skewed, and different classing methods produce maps that look different and convey different visual impressions to readers.

Consider in some detail what will happen when different methods are applied to a data set that has a strong positive skew (figures 2.16 and 2.17).

A Note on Skewness A normal distribution is the familiar bell-shaped curve that is seldom seen in crime data. Most crime data are positively skewed, meaning there is a long right "tail" representing a few high values. Hot spots (high crime areas are geographic expressions of skewness, which presents difficulties in mapping numerical data. See the box, How Much Exploration? and figures 2.15 and 2.16.

Let's review each histogram (or frequency curve) and map in figures 2.15 and 2.16, method by method.

• Equal count. On the histogram, the right tail (highest values of the distribution) is prominent because there are few extremely high values. Thus, the program has to seek the lower rank-ordered data values (farther to the left on the histogram) to come up with the 13 observations for the class. (Note that the number of observations per class is uneven, ranging from 13 to 17.) The resulting map tends to visually exaggerate the seriousness of the problem because color saturates more map areas.

Best Choice? Generally, natural breaks or equal intervals will be the best methods for creating area-type maps in crime analysis.

• Equal range. Because the distribution is right-skewed, equal range will tend to favor lower data values. The two lowest classes have 23 and 26 members, respectively, while the higher classes have 7 and 3. The map contrasts with the equal count version, now visually minimizing the problem.

• Natural break. This method appears to have struggled to come up with natural breaks, which is a problem, along with breaks in awkward places. The result here is quite similar to the equal range breakdown, with cuts between classes shifted to the left (lower values) as compared with the equal range. It comes as no surprise that the equal range and natural break maps are quite similar.

• Standard deviation. Here, the breakdown of class intervals is set with reference to the average, or mean, so that an interval of 1 SD is established to the left of (below) the mean (blue line, 0.47), and above the mean at the same distance. The effect of this on the right-skewed distribution is a symmetrical breakdown, with about as many observations in the lowest (10) and highest (9) classes and in the two middle classes (22 and 18). The visual impression conveyed by the associated map is close to the severity of the equal count method. This is due to the similar number of observations in the top category.

Only the immediate objective and the available tools limit the amount of exploration and preprocessing crime mappers do. Perhaps the single most important exploratory step is the creation of a histogram, box plot, or comparable graphic with which to visualize the shape of the data and answer the fundamental question: Is it severely skewed or in some other way not normal (e.g., bimodal or double peaked)? How will this affect maps, and what type of map will permit a presentation that minimizes distortion and accurately portrays the data? Again, this examination of the data is the ideal. Not all analysts will have the tools or the time to go through this step. Nevertheless, these possibilities are outlined here to raise awareness of what constitutes the best practice.