When I entered valuation work, my first appraisal assignment involved the cash-out refinance of three quick lube garages in various counties of East Tennessee.
As a newcomer to the field, I made the rookie mistake of wearing dress slacks and three-inch heels to our inspections. Navigating through the garage pit, the oil and dirty water nearly ruined my shoes.
At the time, I was still very new to appraisal work (and quick lubes, for that matter). When I learned that we were evaluating the properties by price per bay rather than price per square foot, something clicked in my novice appraisal mind. Curious, I asked my mentors why this approach was used. The response was something along the lines of "it just typically works better that way," which lacked the quantitative rigor I was looking for. While I noticed that price per garage bay sale prices seemed more consistent than price per square foot sale prices, I couldn't quite grasp the reason behind it.
A few months later, during my first Appraisal Institute course, I learned about the coefficient of variation (COV) and its role in selecting the unit of comparison. The objective is to minimize the COV between different units of comparison. Suddenly, it became clear that this was what my mentors had been doing: minimizing the internal dispersion between various units of comparison without explicitly naming or calculating it. Many appraisers instinctively apply this principle when valuing apartment buildings by price per unit, hotels by price per room, or commercial land by price per front foot.
But what exactly is the coefficient of variation, and why is it so effective? The COV is the standard deviation of a data set normalized by the mean. This normalization is crucial because it expresses the standard deviation as a percentage of the mean, allowing for a consistent scale across sets with different units. This approach provides a clearer, more comparable measure of variability, making it a powerful tool in appraisal work.
Consider the following data sets:
Set 1: {1, 1, 2, 3, 5, 8, 13}
Set 2: {100, 100, 200, 300, 500, 800, 1300}
It should be evident that the only difference between Set 1 and Set 2 is a linear factor of 100. At first glance, neither set appears to be more inherently dispersed than the other. However, if we compare the (sample) standard deviations of the two sets:
Set 1: Std = 4.42396
Set 2: Std = 442.396
Set 2 has a higher standard deviation than Set 1, but this is due to the different scales of the sets, not because Set 2 has more internal dispersion. If we only consider the standard deviation, it might seem that Set 2 is significantly more dispersed than Set 1.
To get a clearer comparison, we should normalize the standard deviations of both sets by their respective means, representing them as percentages of their means:
Set 1: COV = std/mean = 4.42/4.71 = 93.9%
Set 2 COV = std/mean = 442/471 = 93.9%
Now, the mathematics accurately reflect the intuitive sense of dispersion between these two sets—they have the same relative dispersion. The COV clearly provides a better measure of internal dispersion than the standard deviation alone.
You might be wondering, "Why does this matter? What’s the point?"
Consider adjustments in appraisal. We typically aim to minimize adjustments as represented by a gross percentage of unit prices. By choosing the unit of comparison with the least internal dispersion, the adjustments required are smaller. Essentially, you are selecting the unit of comparison that minimizes adjustments on a gross percentage basis, which significantly enhances the credibility of your final value conclusion.
Additionally, as demonstrated in our example, changing a set of data by a linear factor doesn’t alter the internal dispersion as reflected by the COV. Although multiplying one set by 100 results in a larger sample standard deviation, the COV remains unchanged.
In other words, the COV is invariant under scale transformations.
This concept applies to valuing land by price per acre compared to price per square foot: the two differ only by a linear factor of 43,560. Therefore, one is not inherently better than the other in terms of normalized internal dispersion. There’s no need to compare the two to determine which has less internal dispersion, as they only differ by a scaler factor.
Thus, the COV provides a consistent and reliable measure of internal dispersion, which is crucial for making accurate comparisons across different units of measurement. This understanding is particularly valuable when selecting the unit of comparison that minimizes adjustments and enhances the credibility of the final value conclusion.
The application of the COV is not limited to a specific property type but is a fundamental principle that can be used across various appraisal contexts, whether it be valuing quick lube garages by price per bay, apartment buildings by price per unit, or land by price per acre versus price per square foot. By recognizing that these units of comparison only differ by a linear factor, appraisers can confidently choose the most appropriate unit without the need for additional comparisons, ensuring a more rigorous and credible valuation process.
