The Property Tax Division (PTD) determines total taxable value in a school district by estimating market value or by accepting the local appraised value in each property category in the district and then adding these category values for an overall school district value. To estimate category values, division staff obtains a representative sample of properties in each category, computes a weighted mean ratio from this sample and divides this ratio into the school district's self-reported appraisal roll value for the category.Stratified Weighted Mean Appraisal RatiosComptroller staff selects property samples randomly whenever possible to ensure that the samples reasonably represent the larger universe of properties in each category. A census of all sales is used when the number of sales is smaller than (or does not greatly exceed) the target sample size. Comptroller staff also stratifies samples to improve sample representativeness if stratification data are available and if stratification is feasible. (Stratification is discussed in detail in the next section.)

An appraisal ratio for an individual property is the ratio of the property's appraised value as shown on the appraisal roll to its market value. The market value is indicated by the sales price of the property or staff-appraised value. Table One shows appraisal ratios for a sample consisting of both sales and appraisals as indicators of market value. For example, Sale Number 1 in this table has an appraisal roll value of $65,834 and an adjusted sale price of $83,113. Dividing $65,834 by $83,113 yields an appraisal ratio of 0.79 for this parcel.

Table One also shows the computation of a non-stratified weighted mean appraisal ratio. A weighted mean appraisal ratio, as opposed to an unweighted mean appraisal ratio, takes into account the different values of the individual properties making up the sample. It is calculated by summing the appraisal roll values, summing the sales prices and staff appraisals and dividing the first sum by the second. As shown in the table, the total appraisal roll value for this sample is $2,007,285, and the total value of sales and appraisals is $2,443,170. Dividing the former by the latter produces the weighted mean appraisal ratio of 0.8216. Finally, dividing the district's total self-reported appraisal roll category value of $27,621,400 by the weighted mean appraisal ratio of 0.8216 produces an estimated category market value of $33,619,036.

As mentioned earlier, PTD uses value-stratified weighted mean appraisal ratios whenever feasible to estimate market values for residential properties (Categories A and B), vacant lots (Category C) commercial properties (Categories F1 and L1) and minerals (Category G). These ratios are stratified by value class within each category if reasonably accurate stratification data are available.Using Confidence Intervals in School District Value EstimatesThe distribution of appraisal roll values by value class is obtained from stratification surveys, the appraisal rolls, or the prior year stratification surveys, depending upon availability. If stratification data are not available for a school district, stratified weighted mean appraisal ratios cannot be calculated. If the data to calculate a value-stratified ratio becomes available at any time during the process, including the protest process, a value-stratified ratio may be calculated.

A value-stratified weighted mean appraisal ratio is a mechanism to automatically adjust the sample to be representative of the population from which it is taken. For example, low-valued properties tend to be clustered in certain geographic areas, while mid-range and high-valued properties tend to be clustered in others. Similarly, construction types tend to vary with value class. A value-stratified weighted mean appraisal ratio adjusts for location effect and for the effects of varying construction types. In addition, it is a particularly useful tool for enhancing sample representativeness when appraisal levels in a category vary significantly between lower-valued and higher-valued properties.

PTD has established a value-stratification procedure, which results in six strata. For the most part, the value ranges within the strata vary from school district to school district, and from year to year depending entirely on the distribution of property value within each school district.

The six strata are:

Stratum #1- The low value stratum. After sorting all the properties in the category from lowest value to highest value, and beginning with the lowest valued property, this stratum contains the low-valued properties that collectively equal 5 percent of the category's total appraised value. PTD does not study this stratum. Instead, PTD accepts the locally determined value by defaulting to a ratio of 1.00.

Stratum #2- This stratum contains all properties that individually exceed 20 percent of the value in the property category. PTD may or may not study these high-valued properties.

Stratum #3- After the remaining properties (properties not included in stratum one or stratum two) are sorted from lowest value to highest value, properties representing about the first 25 percent of the remaining appraisal roll value in the category comprise stratum 3.

Stratum #4- Properties representing about the second 25 percent of the remaining appraisal roll value in the category comprise stratum 4.

Stratum #5- Properties representing about the third 25 percent of the remaining appraisal roll value in the category comprise stratum 5.

Stratum #6- Properties representing about the fourth 25 percent of the remaining appraisal roll value in the category comprise stratum 6.PTD generally studies strata 3-6 by random sampling procedures discussed elsewhere in this publication.

Refer to PTD's

Stratification Surveyfor detailed instructions on stratifying property.Table Two, Three and Four show how a stratified weighted mean appraisal ratio is calculated and how it differs from a weighted mean and a simple mean appraisal ratio. The stratified weighted mean appraisal ratio for a category is calculated by (1) grouping sample properties by appraisal roll value stratum, (2) calculating a weighted mean appraisal ratio for each value stratum, (3) dividing the weighted mean appraisal ratio into the CAD total appraisal roll value for each value stratum to estimate a market value, (4) summing these individual market value stratum estimates and (5) dividing the sum of the CAD values in each stratum by the sum of PTD's individual market value stratum estimates.

Table Two lists the properties in a hypothetical random sample. The sample properties are grouped in six strata (see preceding text for definitions of the various strata). A ratio is calculated for each property, by dividing the CAD value by the PTD appraisal value or sale price. A weighted mean ratio is calculated for each stratum by dividing the sum of the CAD values by the sum of the PTD appraisal or sale amounts. A weighted mean ratio is calculated for the entire property category by dividing the sum of the CAD values in every strata by the sum of the PTD values in every strata. A simple mean ratio is calculated by summing all the individual property ratios in the entire category and dividing by the number of ratios. The weighted mean and simple mean are calculated for comparison to the stratified weighted mean to be calculated in Table Four and for use in calculating the price-related differential (PRD). The PRD is calculated by dividing the simple mean by the weighted mean.

Table Three lists the strata shown in Table Two and shows for each stratum: the number of sample parcels, the CAD value of the sample properties, the PTD value of the sample properties and the weighted mean ratio. Table Three also shows how the weighted mean stratum ratios are calculated by dividing the CAD value in each stratum by the PTD value in each stratum.

Table Four lists the strata shown in Table Two and Table Three and shows for each stratum: the number of parcels in the stratum, the CAD value in the stratum, the stratum ratio (from Table Two or Table Three), and the PTD market value estimate. Table Four also shows the calculation of the stratified weighted mean ratio by dividing the sum of the CAD values for each stratum by the sum of the PTD market value estimated for each stratum. This stratified weighted mean ratio is divided into the appropriate self-reported category total to develop the PTD's market value estimate for the category (refer to your ISD Summary Worksheet to see this final calculation).

As shown in the hypothetical example in Table Two, there are substantial differences in the level of appraisal among value strata. Lower-valued properties are appraised at higher levels than higher-valued properties, as indicated by a price-related differential well above 1.03. Using a stratified weighted mean appraisal ratio will adjust for these differences so that they will not bias the sample ratio and the resulting market value estimate for the category.

The six value ranges used to compute stratified weighted mean appraisal ratios in each of Categories A, B, C, F1, and L1 are the same as those computed in the previous year's stratification survey or from the previous year's electronic appraisal roll submission.

Because the current study year's value ranges are unknown when staff selects sample properties, staff selects the sample based on value ranges from the previous study year. In calculating the stratified weighted mean, staff uses the current year's appraised values in the previous year's value ranges. To obtain the value ranges and amounts, staff uses either the stratification survey data or stratification from the electronic appraisal roll unless notable differences exist between the category totals shown on the reports of property value and the category totals from another source.

In some school districts, staff finds certain properties in a category sample sufficiently different from the remaining sample properties to warrant treatment as "exception" properties. Properties in samples smaller than the minimum sample size are also treated as exceptions. An exception property is a property placed in its own separate value class. The staff's rationale is to offset the potential bias that an exception property might have on the estimated ratio.

Staff set the minimum sample size in each stratum at the lower of five, or 25 percent of the number of properties in the stratum population. The 25 percent rule has the effect of lowering the minimum sample size when the number of properties in the stratum population is very small (fewer than 18).

PTD's samples of properties may sometimes include outliers. The IAAO's

Property Appraisal and Assessment Administrationstates "Outliers are properties with very high or low sales ratios. ...Particularly when the sample is small, outliers can distort ratio studies and should be reviewed carefully."If PTD staff determines the outlier is the result of an appraisal district error or unusual market variability, the outlier remains in the study. If the outlier was caused by a clerical error, a property mismatch or an error in appraisal judgment, PTD staff attempts to correct the error so that the property can remain in the study. If staff finds, however, that the outlier is a non-market transaction, staff excludes the outlier from the sample. PTD may exclude extreme outliers that remain after the process described above is concluded.

Comptroller staff uses confidence intervals to determine whether local value is assigned to a school district. To compute a confidence interval, staff adds the PTD value estimates for tested categories. Tested categories include randomly sampled categories and the value of censused properties from Categories D2 (productivity value of acreage), J (Utilities ), and the value of exception properties. This sum is PTD's value estimate for tested categories. Staff then computes a confidence interval around this estimate of value. (One-half of the confidence interval, expressed in percentage terms, is theAggregating School District Study Data for the Appraisal District Studymargin of error.) If the district's summed values for the tested categories fall within this confidence interval, then staff assignslocal valueto the district. Conversely, if the district's summed values for these tested categories fall outside this confidence interval, then staff assigns state value to the property categories it studied in the district. The division uses a uniform margin of error as a planning tool to allocate its resources. The margin of error as used in the study is a range of values within a school district that are equally likely to be correct. This range is expressed as a percentage of PTD's estimate of school district taxable value. The margin or error discussion in this section assumes a planned margin of error of 5 percent. The planned margin of error is set by management and may vary from year to year. A 5-percent planned margin of error can be illustrated by the following example.If PTD attains a margin of error of 5 percent in a school district and estimates a value of $100 million for that district, the margin of error is $95 million to $105 million. If the local value falls anywhere within this range, PTD certifies the district's local value to the Texas Education Agency for school funding purposes.

The size of the actual margin of error is determined by the sample that is used in the study. PTD estimates how many sample parcels will be required to achieve the planned margin of error. Sometimes the estimated number of parcels is too high or too low to reach the planned margin of error and produces an actual margin of error lower or higher than planned.

In past studies, PTD has collected enough samples to reach a lower margin of error than planned in some school districts. In other districts, PTD was not able to collect enough samples to meet the planned margin of error so that PTD's actual margins of error were larger than planned.

In response to appraisal district and school district requests, the division will hold districts harmless when actual margins of error are less than planned. At a 5-percent, planned margin of error, this policy means that even if PTD's actual margin of error in a district is 4 percent, 3 percent or less, the district will be treated as if the margin of error is 5 percent for the purpose of determining value. For example, if PTD collects enough samples to reach a highly accurate actual 2-percent margin of error and the district's local value falls between 2 and 5 percent of PTD's value, the district's local value will be certified.

On the other hand, if PTD's actual margin of error is more than 5 percent, PTD will use the actual margin of error to decide whether to assign local value. For example, if a district's local value is 5.5 percent from PTD's value and the actual margin of error is 6 percent, the district's local value will be certified.

PTD attempts to obtain the proper number of samples in each school district property category to attain the planned margin of error in each school district. The two variables that affect staff's ability to attain the planned margin of error in each school district are appraisal variability and sample size. Variability is a measure of the district's appraisal uniformity or ability to appraise properties at the same percentage of market value. The coefficient of dispersion (COD) is one measure of variability. Sample size refers to the number of sales and appraisals included in a school district's property value study.

School districts with high variability require large sample sizes to attain the planned margin of error. The greater the variability, the larger the required sample size. PTD adjusts its sample sizes upward in an attempt to compensate for high variability and attain the planned margin of error. In some school districts, however, the variability is so large that hundreds or even thousands of appraisals are necessary to reach the planned margin of error. So, in some school districts, PTD may not allocate the prohibitive level of resources necessary to obtain the planned margin of error.

In most school districts, variability in the sample turns out to be slightly higher or lower than the variability used to calculate the sample sizes assigned to field staff. In these districts, even though PTD staff obtained the planned number of parcels, the margin of error will be higher or lower than planned but will be close, as compared to districts that have very high variability and prohibitive sample sizes. In any school district property stratum with excessive variability (stratum COD in excess of 20), PTD automatically limits the sample size to 30, and excludes the stratum from the margin of error calculation. This procedure prevents non-uniformly appraised value strata from contributing to larger school district margins of error.

More detailed explanations of the confidence interval and margin of error computations can be found in the Questions and Answers portion of this booklet on page 31.

Samples collected for the school district study and aggregated to the appraisal district level provide the basis for computing appraisal district performance measures, by category. The methods used to compute these performance measures are discussed in the next section.Samples from each category are aggregated to the appraisal district level, with one exception. The ratio derived for agricultural acreage receiving productivity appraisal is not a median derived from a property sample. Consequently, staff does not calculate measures of appraisal uniformity for acreage receiving productivity appraisal. The appraisal district performance measures listed under "D. Rural Real-Market Value" are derived from the property samples used to compute the weighted mean appraisal ratios for estimating the market values of non-qualified acreage and farm and ranch improvements.