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APPENDIX C
Margins of Error and Confidence Intervals
in the Property Value Study
Definitions

95% confidence interval: The 95% confidence interval or range of values means that, on average, 95 out of 100 samples would result in a value that lies within the computed range of values. The correct value is assumed to be within the computed range of values.

standard error: A “standard error” is a commonly used statistical term. It is a measure of the differences between an average and all the numbers that go into determining that average. Conceptually, it is somewhat similar to a coefficient of dispersion.

“t-value”: The “t-value” is an adjustment factor that increases the margin of error as the sample size decreases.

1. What is the margin of error? How is it calculated?
A margin of error (as computed in the Property Value Study) is approximately twice the “standard error” of a school district’s estimated value (in the property categories “tested”), expressed as a percentage of such value. Consequently, the margin of error indicates statistical reliability.

The following procedures are used to calculate the PTD margin of error:
 (a) Calculate the “standard error” (SE \$) of the school district’s estimated value. (b) Multiply the “standard error” (SE \$) by the appropriate t-value at the 95% “confidence interval.” (See definition above.) (c) Divide the product of the standard error (SE \$) and the t-value (See definition.) by the school district’s estimated value. formula = (SE \$ * t-value) / ISD \$ estimate

2. How is a margin of error related to a confidence interval?
The margin of error is equal to one half of the confidence interval expressed as a percent of total value “tested” in a school district. For example, assume that PTD staff estimates market value in sampled and censused property categories in school district ABC to be \$100 million (before exemptions). The margin of error is computed to be plus or minus 5 percent of \$100 million. Market value plus 5 percent is \$105 million; market value minus 5 percent is \$95 million (the \$100 million estimate is known as a “point estimate”; the confidence interval of \$95 million to \$105 million is often called an “interval estimate.”)

3. What is the purpose of a confidence interval?
A confidence interval provides one measure of whether the state’s estimate of value in a school district is statistically significantly different from the self-reported appraisal roll value (i.e., “local value”) in that district. In other words, a confidence interval is a measure of the reliability (or precision) of the Comptroller’s estimate of school district value.

Assume that Comptroller staff estimates market value in ABC school district to be \$100 million with a margin of error of 5 percent at the 95 percent confidence level. This means that the actual market value in ABC school district is probably somewhere between \$95 million and \$105 million. This range constitutes the 95 percent confidence interval. The 95 percent confidence interval means that, in repeated sampling of this school district, approximately 95 of every 100 computed confidence intervals would be expected to contain the true market value, which staff has estimated to be \$100 million, while only five of these would not.

If the local value in the ABC school district lies within the calculated confidence interval, then the difference between the local value and the “point estimate” of value is statistically “insignificant.” This means that the Comptroller has not “disproved” local value. In this case, the Comptroller certifies ABC’s local value to the commissioner of education. If the local value lies outside the confidence interval, the Comptroller’s estimate of value is certified to the Commissioner of Education. If local value lies outside the confidence interval, the Comptroller has “disproved” local value because the difference between the local value and the Comptroller’s estimate is statistically significant.

The study contains a “hold harmless” feature. This feature means that if the school district’s tested value is calculated to be within 5 percent of the PTD estimate of value, the PTD will automatically certify the school district’s value. Also, if the school district’s margin of error is calculated to be less than 5 percent, then the PTD will calculate (i.e., widen) the confidence interval as if it were 5 percent for purposes of certifying value. The actual percentage used in the calculation is set by management and could vary in future studies.

4. Is the target margin of error the same in every school district?
Yes. The target margin of error is also referred to as a “planned” margin or error.

5. If the target margin of error is the same in every district, is the target confidence interval the same in every district?
No, because they are expressed in different units. For example, the margin of error is expressed in percentage terms while the confidence interval is expressed in dollar terms. Assume there are two districts, ABC and XYZ. The Comptroller estimates the total value (in tested property categories) to be \$100 million (in ABC) and \$500 million (in XYZ). If the margin of error is 5 percent in both districts, the confidence interval of ABC would be \$95 million to \$105 million, while the confidence interval for XYZ would be \$475 million to \$525 million. Although the margin of error is the same for both districts, the “widths” of the confidence intervals are different because the districts’ values are different. However, even if two school districts have identical margins of error and/or confidence intervals, this does not determine whether local or state value will be certified. The critical test is whether local value lies within the PTD computed confidence interval for the district.

6. Are the confidence interval and margin of error for a school district computed on the basis of all value in the district?
No. In computing a confidence interval for a school district, staff only includes property categories whose values were estimated from representative (i.e., random) samples taken in that school district. If a property category is not tested, that category value is excluded from the confidence interval and margin of error calculations for that school district.

For example, assume a school district with a Comptroller estimate of market value of \$106 million before exemptions. Total local value in the district as shown on the self report is \$98 million. The estimated margin of error is 5 percent. Assume further that staff does not sample any properties in Multi-family (Category B) and Vacant Lots (Category C) in the school district because they constitute less than 5 percent of value. The combined value of these “non-sampled” (i.e., non-tested) categories is 6 million. “Non-sampled” property categories are assigned local value.

The confidence interval for this district is computed as follows:

\$106 million less \$6 million = \$100 million (the point estimate)

\$100 million - 5% and \$100 million + 5% = \$95 million and 105 million (the confidence interval).

Since the local value for the “sampled” property categories (excluding Categories B and C) lies within the confidence interval, the Comptroller would certify local value for the district.

Remember that the Comptroller computes confidence intervals before deducting exemptions. If a school district’s local value, before exemptions, lies within the Comptroller’s computed confidence interval, then the Comptroller certifies local taxable value, after exemptions, to the commissioner of education.

7. Are “technical” properties treated differently than “local” properties?
Yes. In many cases, technical properties are treated as censused (i.e., “non-random”) categories rather than sampled categories. (In a census, one studies every unit in a group to determine some characteristic of the group. In a sample, one studies a portion of the units in a group to estimate some characteristic of the group. Sampling requires far fewer resources than conducting a census.)

Censused properties are not used to calculate the confidence interval, but they are used to calculate the margin of error. All properties in a census are studied so there is no sampling error since the variance and standard error for censused properties is zero.

On the confidence interval detail sheet, censused properties are shown as “non-random” properties. To compute the margin of error, staff adds the value of censused properties to the combined value of the sampled property categories. One half of the confidence interval (as computed from sampled and censused properties) is divided by this total to produce the margin of error for the school district.

In effect, the censused (non-random) properties collectively comprise a separate subcategory.

All properties in the “J” Category (Utilities) as well as the “D2” Category (qualified agricultural acreage) sample are treated as censused properties.

8. How does the Comptroller’s use of confidence intervals affect the methodology used to select and appraise properties for the Property Value Study?
It has no effect. Confidence intervals for each school district’s market value are calculated after all sales and appraisals are entered into the system and all market values are calculated.