Section 5.10 of the Property Tax Code requires the Comptroller to conduct and publish an annual study of appraisal districts to determine the median level of appraisal and the uniformity of appraisal in each major property category in each appraisal district in the state. In addition, the Tax Code requires the agency to publish other standard statistical measures it considers appropriate. These measures are described later in this section. This section also requires the Comptroller to use the data collected from the annual school district study for the annual appraisal district study.Calculating the Median Appraisal Ratio

Section 1.12(c) of the Property Tax Code defines the median appraisal ratio as follows:Measuring Appraisal Uniformity"The median appraisal ratio for a sample of properties is, in a numerically ordered list of the appraisal ratios for the properties: (1) if the sample contains an odd number of properties, the appraisal ratio above and below which there is an equal number of appraisal ratios in the list; or (2) if the sample contains an even number of properties, the average of the two consecutive appraisal ratios above and below which there is an equal number of appraisal ratios in the list."

The value of individual properties does not influence the median ratio; only the ranking of individual ratios within the sample matters. The median ratio falls at the middle of a group of ratios ranked from highest to lowest or lowest to highest.

Table Five uses 19 sales (marked "S1" to "S19") and six appraisals (marked "A1" to "A6") to show how to identify the median ratio. In this table, the appraisal ratios are ranked from the highest ratio to the lowest. Twenty-five properties make up the sample. The median ratio, 0.81, is thirteenth on the list. Twelve properties are ranked above it, and 12 are ranked below it.

An easy way to find the median for a sample containing an odd number of properties is to divide the total count by two, then round the result upward to the nearest whole number. The sample shown in Table Five contains 25 parcels. Dividing 25 by 2 yields 12.5. Rounding upward to the nearest whole number produces 13. The thirteenth ratio is the median.

For an even-numbered sample, the median is the average of the two middle ratios. If there were 24 properties in the sample, the median would be the average of ratios 12 and 13. Eleven ratios would be above 12 and below 13. Regardless of the number of parcels in a sample, the median will be the same whether the ratios are ranked from highest to lowest or lowest to highest.

Staff follows these steps to calculate a median level for each major category of property in each appraisal district, provided there were at least five properties in the sample. Staff then combines the properties making up the sample for each category into a larger sample of all properties in the appraisal district. The median ratio from the larger sample is listed as the "overall" ratio for the appraisal district.

The median level of appraisal measures the accuracy of an appraisal district's appraisals in relation to the standard of 100 percent of market value. The Property Tax Code also requires the agency to calculate a "coefficient of dispersion" around the median for each major property category. The coefficient of dispersion (COD) measures appraisal uniformity.Technically, the COD expresses as a percentage of the median the average absolute deviation of the appraisal ratios in a sample from the sample's median. The COD indicates how tightly the individual ratios are clustered around the median ratio for the sample. A high COD indicates high variation-few ratios close to the median and low appraisal uniformity. A low COD indicates low variation-ratios clustered tightly around the median and high appraisal uniformity.

The International Association of Assessing Officers (IAAO)

1990 Standard on Ratio Studiescontains standards for CODs. These are as follows: (1) single-family residential in newer homogeneous areas: 10 or less (in older, less homogeneous areas: 15 or less); (2) vacant land: 20 or less; (3) income properties in large, urban jurisdictions: 15 or less; and (4) income properties in small, rural jurisdictions: 20 or less. The IAAO does not publish standards for other real and personal property, but notes that they vary with local conditions.The COD measures appraisal uniformity independently of the median level of appraisal. As a result, CODs allow comparison of appraisal uniformity among districts or property categories where median levels of appraisal differ significantly.

Calculating a COD requires six steps:

- Subtract the median ratio for the sample from each individual ratio making up the sample. The result is the deviation for each ratio.
- Convert each deviation to its absolute value.
- Total the absolute values of each deviation.
- Divide the total deviation by the number of properties in the sample to get the average absolute deviation.
- Divide the average absolute deviation by the median ratio.
- Multiply the result by 100.
Table Six shows a sample calculation for a COD.

Comptroller staff calculates a COD for each major category of property in an appraisal district if the sample has at least five properties. The staff combines the samples for each category into a larger sample to calculate the overall COD.

In addition to the COD, the staff calculates three other measures of appraisal uniformity-the percentage of properties within 10 percent of the median, the percentage of properties within 25 percent of the median and the "Price-Related Differential" (PRD) (discussed below). The percentages are computed if the sample contains at least six properties. The PRD requires at least five properties.

To calculate the first of these, the staff multiplies the median appraisal ratio times 10 percent. Adding this result to the median yields the ratio that exceeds the median by 10 percent. Subtracting the result from the median yields the ratio 10 percent below the median. The staff counts the number of properties in the sample that have ratios equal to or between these two numbers. Dividing that count by the total number of properties shows the percentage within 10 percent of the median.

To calculate the percentage within 25 percent of the median, the staff multiplies the median times 25 percent and then adds and subtracts the results to find the upper and lower end of the range. In Table Six, the properties in the sample that have ratios between 89.1 percent and 72.9 percent are within 10 percent of the median, and properties that have ratios between 101.2 percent and 60.7 percent are within 25 percent of the median. In Table Six, all properties fall within 25 percent of the median.

The COD and the percentage of properties within 10 and 25 percent of the median are measures of "horizontal" ratio dispersion. They measure how consistently appraisal districts are appraising properties at the same level (percentage of market value) without regard to the value of the properties. A low COD and high percentages indicate equitable appraisals, while a high COD and low percentages indicate inequitable appraisal.

Another form of inequity may arise from systematic differences in the appraisal of low-value and high-value properties. According to the IAAO

1990 Standard on Ratio Studies, "When low-value properties are appraised at greater percentages of market value than high-value properties, assessment regressivity is indicated. When low-value properties are appraised at smaller percentages of market value than high-value properties, assessment progressivity results. Appraisals made for tax purposes, of course, should be neither progressive nor regressive." Progressive and regressive appraisal are forms of inequity called "vertical" inequity.Staff reports a measure of vertical dispersion called the "Price-Related Differential" (PRD) for each property category on the CAD Summary Worksheet. The PRD is calculated by dividing a sample's mean ratio by its weighted mean ratio. The IAAO standard for this measure is .98 to 1.03, with PRDs below this range indicating progressivity, and measures above this range indicating regressivity. A PRD inside this range indicates that low-value and high-value properties are being treated uniformly in regard to level of appraisal. Table Seven shows a sample PRD calculation. In this example the PRD is 1.01, which indicates uniformity.

The IAAO warns that the PRD is not a reliable statistic when the sample is small or when the sample is heavily influenced by extreme sales prices. For this reason, staff publishes the sample size on the CAD Summary Worksheet. Irrespective of sample size, however, the PRD is only an indicator; it alone cannot prove vertical equity or inequity. Additional tests are required to prove vertical inequity. Together, the median level of appraisal, the COD, the percentage of properties within 10 or 25 percent of the median and the PRD enable the Property Value Study to address the legal requirements that appraisals be equal, uniform and at 100 percent of market value.