Range has several definitions found in statistics books. It may be the highest and lowest numbers observed for a set of observations or a single number which is the lower number subtracted from the higher number. Some texts describe it as this difference plus one. It is too simple, it seem, for high-powered computers and software that can do so much. It is too unstable for people who want the average and some expression of variability. In range however, we may be able to find the answers to some of the animal and plant questions that are of great interest to all of us.

I live with the premise that the mean, the arithmetic mean, may never occur. I count animals and I invariably get statistics with fractional animals. Knowing that none exist is not the point. I take the average of 4 animals observed and 6 observed and get 5...and a group of 5 has never been observed! The same occurs for almost every set of observations made by ecologists or economists. Perhaps the mean is not the number I want or need for certain decisions. That the mean may not exist is a premise that begets caution and a new (at least alternative) view of data.

Most ecologists, at least naturalists, like to know the biggest, the tallest, the fastest...anything. Zero is commonplace in ecology so the range becomes the same as the value of the maximum. When means and standard deviations are used, these statistics may hide such extremes. They may hide them because the selection of the number of standard deviations only brackets a limited number beyond the mean. "The mean plus or minus 2 standard deviations" means that even if the population is well described by the bell shaped curve, only about 68 percent of the values observed and like to be observed are included within the stated limits. The extremes are there, part of the computations, but ignored or dropped in the statement about the population. In my observations, most ecological populations are not normally- or bell-shaped and thus normal statistical thought (that about "bells") can be very misleading. The range, usually the maximum value, but both lower and upper limits, are needed to conceive of the shape of the skewed distribution. A return to thinking about and using the range may be useful.

When I try to reduce sample sizes to reduce costs (since budgets have become mystical), then I reduce my requirements for confidence (from 5 to 15 %), the acceptable inaccuracy from 5 to 10 %, and then I look at the variance (the third major factor in most sample-size determination equations). I recall that the range, not the sample size, is the controlling factor in making that estimate. The variance is often difficult (and expensive) to get. (To do so justifies all of those pot-boiler studies, preliminary works, and seed-money requests.) The range for normal populations is about 6 standard deviations and the variance is merely the square of the standard deviation. We can make progress (not as well as with a lot of money) at low cost by using what range tells us in the context of some reasonable assumptions about sampling to inform managerial work in a highly variable field (with few penalties for being wrong.)

I usually imagine that there are three dominant independent variables affecting almost any wildlife variable of interest -- habitat, population, or people. If I plot those three variables as axes of a box, and mark off the range of each, then I can imagine the topic of interest, the dependent variable floating around, driven by changes, but always within the box. The dominant factor box gives me a feeling of control, of understanding. It limits my chances for being wrong; it suggests new questions and ways to perceive the complex world of work.

In analyses of streams and rivers, the maximum flood height is reported and often expressions of the 10-, 50-, or 100-year flood crest are used. This value results from cumulative observations over time. People measure the height of the flood and then, year after year, measure the height...but only plot the height of the water if it exceeds the highest water on record. They are clearly interested in the range, the highest that the stream will ever flood...for safety, etc. Most rivers and streams now have such records. A logarithm, a log-log graph, or a simple exponential function readily approximates the graph with time as the independent variable (such as in species-area curves now often used by ecologists). I suspect we do not have such records for many animals (either weight or population sizes), for the biomass of plants, or for the biomass of communities. By judicious use of records (and with the same software that is used for determining this curve for stream crest estimates) we could estimate the maximum weight of warbler X, or elk Y. Ecosystems, by some theory, are energy maximizers. Why then, should most of our observations be of the mean and its deviations? Why shouldn't we evaluate the difference from the estimated maximum (or minimum, which for most animals and plants is zero)? Cannot the same curves with ranges be used to estimate the maximum and minimum temperatures for GIS map pixels? For fish weights? For hours of lake fishing? For miles of recreational hiking? For estimating extreme ranges that are used in GIS systems to screen and clean data. There are few things in the environments in which I live that have values less than zero. There are few that have infinitely great values. In the field, I wish to deny the facile assumptions of sophomore math and return to estimating the real limits. When I make decisions in the wildlands, I want to see what would happen if likely, extremely great, and extremely small events occurred. By working with ranges, using them in maps and simulators (often with the uniform or flat distribution), or in combinations of these, I can reduce my risks and generally improve my decisions.

In GIS work, I think each of three unusual artificial layers will be found to be useful as independent variables for making an estimate or prediction of an occurrence of ecological variables. These three are the maximum value (or proportion) in a 9-cell window, the minimum value in the window, and the difference (the range). When doing landscape-scale work such as comparing watersheds or viewscapes, I want a map layer with the range in each polygon or map cell.

The range sounds too gross to most field analysts straight out of school. They have never worked with the GIS! Map the range with slope, aspect, elevation, boundaries, landform, roads, lakes, and lake zones. The complexity, the noise, and the unfathomable relations in a map of so many factors when combined will suggest that any discrimination that the range makes possible will be very welcome. Perhaps the GIS experts can help field people see that range itself is the factor of importance to be estimated as related to locators of elevation, latitude, and longitude and that such discrimination opens new vistas for understanding the worlds of wild plants and animals and the people who use them.

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