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MEMORANDUM 

SUBJECT: Use of the Upper Prediction Limit for Calculating MACT Floors 

DATE: October 10, 2014 

FROM: David Putney 

 Office of Air Quality Planning and Standards 

TO: Docket ID No. EPA-HQ-OAR-0797 

This document explains EPA’s use of the UPL, which is the statistical methodology EPA uses as 

the primary tool to account for emissions variability when setting emissions standards under 

these sections, and explains how the UPL is used to calculate the average emissions limitation 

achieved over time by the best performing source or sources. 

I. Emissions Standards under Section 112 and 129 

For major stationary sources of hazardous air pollutants (HAP) under section 112 and for sources 

regulated under section 129,1 the CAA requires emission standards that represent the maximum 

achievable control technology, or “MACT” level of control. The MACT standard setting process 

is a two-step process that starts with the agency determining the level of control that is currently 

achieved by the best-controlled similar source (for new source standards) or the average of the 

best-performing sources (for existing source standards). That level of control is the minimum 

level of control required by the CAA regardless of cost or feasibility, and it is commonly called 

the “MACT floor.” This document focuses on that first step in the MACT standard setting 

process, as it is in that step that the agency uses the UPL to both evaluate average emissions 

performance and estimate variability to determine the level of control sources currently achieve.2

For new sources located at facilities that are major sources of HAP, section 112 of the CAA 

requires EPA to establish standards that are no less stringent than the emissions control that is 

achieved in practice by the best controlled similar source. The statute does not define “achieved 

in practice” nor does it dictate the manner in which the agency determines which source is the 

 

1

 A “major source” is a stationary source or group of stationary sources at a single location and under common 

control that emits or has the potential to emit ten tons per year or more of any hazardous air pollutant or 25 tons per 

year or more of any combination of hazardous air pollutants. Any stationary source that is not a major source is an 

“area source.” CAA § 112(a). For area sources of HAP, EPA has the discretion to set emissions standards under 

section 112(d)(5) based on generally available control technology in lieu of MACT standards. 

2

 The second step in the MACT standard-setting process is the evaluation of whether beyond-the-floor controls are 

appropriate, as provided in section 112(d)(2) of the Act. This second step is not addressed in this document. 
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best controlled similar source, instead leaving it to the agency’s discretion to make those 

determinations. For existing sources located at facilities that are major sources of HAP, standards 

must be no less stringent than the average emissions limitation achieved by the best performing 

sources for which EPA has emissions information. Again, the CAA leaves to EPA’s discretion 

the manner in which to calculate “the average emissions limitation achieved” by the best 

performing sources. Under section 112, the CAA recognizes that source categories and 

subcategories have differing numbers of sources and provides category size-specific instructions 

on determining standard stringency for existing sources. Specifically, standards for categories or 

subcategories with fewer than 30 sources must be based on the average emissions limitation 

achieved by the best performing 5 sources, and standards for categories or subcategories with 30 

or more sources must be based on the average emission limitation achieved by the best 

performing 12 percent of sources for which EPA has emissions information. Section 129 requires 

EPA to establish standards for new sources that are no less stringent than the emissions control 

that is achieved in practice by the best controlled similar source for nine specified criteria and 

hazardous air pollutants, and standards for existing sources that are no less stringent than the 

average emission limitation achieved by the best performing 12 percent of sources in the 

category or subcategory for the same nine pollutants. 

II. Background 

EPA promulgated rules covering major source industrial, commercial, and institutional boilers 

and commercial and industrial solid waste incinerators (CISWI) rule on March 21, 2011, and 

subsequently made certain revisions to the rules on January 31, 2012. On August 20, 2013, the 

D.C. Circuit Court of Appeals issued its decision in National Assn. of Clean Water Agencies v. 

EPA, which involved challenges to EPA’s MACT standards for sewage sludge incinerators, 

issued under section 129 of the Act. See 734 F.3d 1115. In the NACWA decision, the D.C. 

Circuit remanded certain aspects of the rule for further explanation, including the question of 

how the UPL represents the MACT floor for new and existing units, as required by the CAA. 

The sewage sludge incinerator rule was issued on the same day as the boilers and CISWI rules, 

and used the same general methodology for calculating the MACT floors. For this reason, EPA 

requested an opportunity to supplement the record in pending challenges to the boilers and 

CISWI rules, to provide the explanation of the agency’s analysis of variability in setting the 

MACT floor standards that the court believed was needed in the record for the sewage sludge 

incinerator rule. The court granted EPA’s motion for a remand of the record on May 15, 2014. 

This document provides that additional explanation, which also applies to MACT standards that 

are part of the rulemaking for the Primary Aluminum Reduction Plant source category. 

There are several key points, addressed in more detail below, that underlie EPA’s methodology 

for calculating MACT floor standards through the use of the UPL. First, the floor standards 

reasonably account for variability in the emissions of the sources used to calculate the standards. 

This variability occurs due to a number of factors, including measurement variability (both 

sampling and analysis) and short term fluctuations in the emission levels that result from shortterm changes in fuels, processes, combustion conditions, and controls. Second, because the 

emissions data available to EPA is in the form of short-term stack tests and the standards must be 

complied with at all times, the agency uses the UPL to estimate the average emissions 

performance of the units used to establish the MACT floor standards at times other than when 

the stack tests were conducted. Thus, the UPL results in a limit that represents the average 
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emissions limitation achieved by the best performing sources over time, accounting for 

variability in emissions performance. 

III. Overview of MACT Floor Methodology 

In establishing MACT floors, we use the available information to determine the average 

performance of the best performing sources (for existing source floors) and the average 

performance of the best-controlled similar source (for new source floors). Each MACT standard 

is based on limited data from sources whose emissions are expected to vary over their long term 

performance. For this reason, and because sources must comply with the MACT standards at all 

times, consideration of variability is a key factor in establishing these standards. This variability 

in emissions is due to numerous factors, including operation of control technologies, variation in 

combustion materials and combustion conditions, variation in operation of the unit itself, and 

variation associated with the emission measurement techniques. In order to account for 

variability that is reflected in the available data that we use to calculate MACT floors, we use the 

UPL. 

The NACWA court recognized that EPA has significant discretion in establishing the MACT 

floors, including how to account for variability. See NACWA v. EPA, 734 F.3d at 1142-43 

(“EPA can decide what value the MACT floors are supposed to represent, as long as that 

decision is a reasonable interpretation of the statute.”) However, the court expressed concern 

about the interpretation it believed EPA was taking of the “average” emissions performance of 

the best performing sources. Specifically, the court asked for further clarification of how 

“’average’ means the average of a future 3-run compliance test.” Id. at 1143. As explained 

below, EPA does not interpret the term “average” as used in section 112(d)(3) in this manner. 

Rather, EPA interprets the average to mean the average emissions over time, based on both the 

calculated average of all emissions test data from the best performing source or sources and the 

available information regarding the variability of emissions. The UPL, which EPA uses to 

account for variability among the best performing sources, reflects an upper limit for the 

emissions of those sources at times other than when the emissions tests occurred. This not only is 

a prediction of the emissions performance of those sources in tests conducted in the future, but is 

also an indication of the range of current average emissions performance of those units. 

In calculating MACT floors, EPA analyzes all available emissions information and then uses the 

emissions information for the best performing source or sources to calculate a value that 

represents the level achieved in practice by the best controlled similar source for the new source 

MACT floor and a value that represents the average emission limitation achieved by the best 

performing twelve percent of sources for the existing source MACT floor. EPA must establish 

the MACT floor standards based on the emissions information before the agency, and the 

standards must be met at all times by sources subject to them. The available emissions data are 

generally in the form of short term, three-run stack tests, with each test run lasting for between 1 

and 4 hours. Those short term tests cannot encompass the emissions performance of a source 

over time. EPA interprets “emissions performance,” as used in section 112(d)(3) to mean the 

emissions of a source over the long term, rather than just during a short-term stack test. This is a 

reasonable interpretation of the statutory directive that the MACT standard be no less stringent 

than the average emissions limitation “achieved by” the best performing sources, since the 

emissions level of a source is not expected to always be the same as it is during stack testing. 
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Therefore, EPA applies a methodology that predicts the actual emissions levels the source is 

achieving at times other than when stack testing was conducted. This enables EPA to calculate 

the average emissions performance of the best performing sources over time. Therefore, the 

MACT floor calculation takes into account the inherent variability in emissions performance to 

more accurately reflect the range of the best performing sources’ emissions over time. The 

remainder of this discussion focuses on why the statistical methodology that we used - the UPL - 

provides results that reasonably represent the level achieved in practice by the best-controlled 

similar source for new sources, and the average emissions limitation achieved by the best 

performing emissions units for existing sources. 

IV. Use of the Upper Prediction Limit 

The UPL is a value derived from widely accepted and commonly used statistical principles, and 

represents the upper end of a prediction interval.3 The UPL is used in a wide variety of industries 

and professions, such as manufacturing, finance, healthcare and others.4 The UPL is calculated 

using one of several equations that must be selected based on certain characteristics of each 

dataset. The most important parameters in the UPL calculation are the mean (average) 

performance of the unit or units, the distribution of the dataset, the variance in the measured 

emission values (that is, the individual data points), and the sample size. The equations that are 

used to calculate the UPL are explained in more detail below. 

In the context of development of MACT floors, the UPL is a value, calculated from a dataset, 

that identifies the average emissions level that a source or group of sources is meeting and would 

be expected to meet a specified percent of the time that the source is operating. That percent of 

time is based on the confidence level used in the UPL equation, as explained in more detail 

below. For a single best controlled emission source that serves as the basis for a new source 

MACT floor, the 99 percent UPL is the emissions level that the source would be predicted to be 

below during 99 out of 100 performance tests, including emissions tests conducted in the past, 

present, and future, based on the short term stack test data available for that source. It is 

important to note that the UPL does not represent the worst emissions performance of the best 

performing units at any time, but rather the average level expected to have been achieved over 

time, as the first element of the UPL equation is the average of the short-term emissions test data 

from the best performing units. In other words, the 99 percent UPL is the level of emissions that 

we are 99 percent confident is achieved by the average source represented in a dataset over a 

long-term period based on its previous, measured performance history as reflected in short term 

stack test data.

In sum, the UPL predicts the level of emissions that the sources upon which the floor is based are 

expected to meet over time, considering both the average emissions level achieved as well as 

emissions variability and the uncertainty that exists in the determination of emissions variability 

given the available, short-term data. EPA did not have emissions information from sources at all 

times each source was operating, and therefore determined it was necessary to apply a 

 

3

 National Institute of Standards and Technology, NIST/SEMATECH e-Handbook of Statistical Methods, 

<http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/predlimi.htm>. 

4

 See, e.g., Gibbons, Robert D. and Coleman, David E. (2001). Statistical Methods for Detection and Quantification 

of Environmental Contamination, New York: Wiley, pp 21-31, and Luko, S.N. and D.V. Neubauer, Statistical 

Intervals Part 2. Standardization News, October/November 2011. <http:www.astm.org/standardization-news/datapoints/statistical-intervals-part-2-so11.html>. 
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methodology that addressed the fact that the data were not complete. Statistical tools are 

commonly used in science and industry because they permit predictions to be made based on the 

available data. As explained in more detail below, the UPL equations include a term that adjusts 

the average emission level of the units to account for both the emissions variability observed in 

the data and the uncertainty that exists in that estimate of variability for a sample of the given 

size, to estimate the performance of the units over time. As shown by the t- and z-statistics 

incorporated into the UPL equations, the uncertainty regarding emissions variability is higher in 

small samples and is lower in large samples, but it is always accounted for in UPL. 

The 99 percent UPL value reasonably estimates the level currently achieved in practice by the 

best controlled similar source and the average emission limitation currently achieved by the best 

performing 12 percent of sources by estimating the level achieved by the best source (for new 

sources) or by the average of the best sources (for existing sources) ninety-nine percent of the 

time the sources are operating. It is reasonable to interpret the statutory requirement that the 

MACT floor level reflect the “emission control that is achieved in practice” by the best 

controlled similar source and the “average emission limitation achieved by” the best performing 

12 percent of sources as a level that the average level the best performers are meeting on a 

consistent basis over time, not just at the single point in time during which emissions test data 

were collected. If EPA were to simply take the average of the stack test data from the best 

performing source or sources, the MACT floor would not represent the levels of emissions 

achieved in practice over time because the MACT floors would not reflect variations in material 

inputs (e.g., fuels), control device performance, operating unit performance (e.g., changes in 

combustion conditions that impact emissions), and other factors that affect unit performance over 

time. This is demonstrated by the fact that even single three run tests, which are performed over 

a short period of time, typically show different emissions levels during each individual test run. 

It would not be reasonable to establish a MACT floor standard that the best performing sources 

would exceed routinely. But neither would it generally be reasonable to establish a standard that 

all the best performing 12 percent of existing sources can meet without any modification because 

the statute requires the agency to establish the standard at the average level of performance of 

the best 12 percent of sources. If EPA simply calculated the average (mean) of the emissions 

data from the best performing 12 percent of sources, that would only lead to a limit that can be 

met by all the existing sources in the floor at all times in the unlikely hypothetical situation 

where all the sources have identical emissions test data and controls and operate in exactly the 

same manner– i.e. the sources would have to be essentially identical and use identical fuels or 

other material inputs. Since that situation does not represent real-world source categories, the 

agency takes the available data and applies a statistical formula to account for variability in 

emissions of the best performing source or sources.

 
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V. Establishment of MACT Floors 

At first glance, developing a HAP emissions limit in accordance with the statutory requirements 

seems a straightforward task, requiring simply that EPA collect all available valid data from all 

the sources in a specific category or subcategory, rank the data to select the best performing 

twelve percent of the category or subcategory, and calculate the average emission level achieved 

by that group of sources for a particular pollutant. However, multiple factors complicate these 

calculations. First, emission test results taken during a three-run stack test, as is the case with 

virtually all the HAP emission data available to EPA, represent three “snapshots” of a source’s 

operations during the test period and those results generally will not represent a source’s full 

range of normal operations or emissions levels. However, because there are three separate runs in 

a test, a single test will in most cases show some of a particular source’s variability over the short

period of time during which testing was conducted. Further, EPA generally has limited data with 

which to establish MACT standards, primarily because the data needed are not generally 

collected by sources prior to regulation (particularly for HAP, but also for other pollutants such 

as nitrogen oxides, sulfur dioxide, and carbon monoxide regulated under section 129). The 

collection of such data is a time- and resource-intensive effort, as emissions tests can take more 

than a week to perform, analyses of samples from the tests can take more than a month, 

preparation of test reports typically takes several months, and costs range up to $40,000 per test 

per pollutant. For example, during our data collection efforts for the boiler and CISWI rules we 

required hundreds of facilities to conduct emissions testing, and we estimated an average cost of 

about $90,000 for each facility that was required to conduct testing. 

As explained above, EPA establishes MACT floor standards under sections 112 and 129 as 

limits that reflect the level achieved in practice by the best controlled similar source, or the 

average emissions limitation achieved by the best performing 12 percent of sources over time, 

but we necessarily must do that based on the emissions information that is available. In this case, 

that information consisted of emissions test data that represents the specific instances in the 

sources’ operations when testing was conducted. The UPL is a common statistical technique that 

allows EPA to use emissions test data and the data characteristics, such as the distribution and 

sample size, along with the intrinsic variability associated with those data,5

 to calculate an 

emissions limit based on a specified level of confidence such that an average best performing 

existing source would not be expected to exceed the limit a specified number of times. For 

instance, if an emission limit were set at the 50 percent UPL, we would expect the average unit 

to have emissions above the level of the standard 50 percent of the time. 

This section explains how EPA applied the UPL to the emissions data from the best performing 

source or sources to calculate MACT floor limits that represent the average emissions 

performance of those sources, while accounting for variation in those sources’ emissions over 

time. 

A. Ranking Sources and Establishing the Distribution of Data 

After collecting emissions data, EPA analyzes the data both to identify and rank the best 

performing sources and to evaluate the data distribution. This evaluation is necessary because the 

 

5

 Gibbons, p 30. 
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UPL equation that is used to account for variability and calculate the MACT floor standard 

depends on the distribution of the data. Typical data distributions are normal and lognormal, as 

shown in Figures 1 and 2. Therefore, the first step in applying the UPL is to rank sources (based 

on lowest emissions) to identify the best performing sources and then analyze the distribution of 

the emissions data from those best performing sources. The ranking is used to establish which set 

of units make up the top 12 percent of performers for each pollutant and subcategory and which 

unit EPA considers the single best-controlled for each pollutant and subcategory. Once these 

units are identified, EPA determines the data distribution as discussed below. 

In order to select statistical equations (to calculate the UPL) that are appropriate for each 

individual dataset, EPA evaluates each individual dataset to determine the data distribution, as 

noted above. The individual datasets that we evaluate include the datasets for the best performing 

unit for each pollutant and subcategory for the new source MACT floor standards and the 

datasets for the best performing 12 percent of units for each pollutant and subcategory for the 

existing source MACT floor standards. To evaluate the distribution of the data, EPA checks each 

dataset for distribution characteristics by applying skewness and kurtosis tests. Skewness is a 

measure of symmetry (or the lack of symmetry) of a distribution, and is useful in determining the 

distribution because normal distributions are symmetric while lognormal distributions are not. 

Kurtosis is a measure of whether a dataset has a similar, more distinct, or flatter peak than a 

standard normal distribution.6 Figures 1 and 2 show simple examples of normal and lognormal 

distributions. In each figure for the purpose of EPA’s analyses, the x-axis is the level of 

emissions and the y-axis represents the probability of a particular value occurring (established 

based on the number of occurrences in the dataset at each level of emissions). If both the 

skewness and kurtosis tests suggest the data exhibit a normal distribution, EPA evaluates the 

emissions data for the source category using statistical equations for normally distributed data. If 

the test results are inconsistent regarding normality, or if both test results suggest non-normality, 

EPA evaluates the emissions data for the source category using statistical equations for 

lognormally distributed data. 

EPA uses this approach because, absent evidence of a normal distribution, air emissions data 

have historically been shown to exhibit lognormal distribution characteristics such as 1) run 

values always above zero; 2) a majority of run values will be low; and 3) a few run values will be 

high but will have a low probability of occurring.7,8,9 These characteristics yield a left-skewed 

distribution curve, meaning the distribution has a long tail to the right. 

 

6

 National Institute of Standards and Technology, NIST/SEMATECH e-Handbook of Statistical Methods, 

<http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm>. 

7

 Ott, Wayne R. (1990) A Physical Explanation of the Lognormality of Pollutant Concentrations, Journal of the Air 

& Waste Management Association, 40:10, 1378-383, DOI: 10.1080/10473289.1990.10466789.

8

 Gilbert, Richard O. (1987), Statistical Methods for Environmental Pollution Monitoring, New York: Van Nostrand 

Reinhold, p 152. 

9

 Gibbons, Robert D. and Coleman, David E. (2001), Statistical Methods for Detection and Quantification of 

Environmental Contamination, New York: Wiley, p 369.
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Figure 1. Normal Distribution10

Figure 2. Lognormal Distribution11

 

10 http://www.psychstat.missouristate.edu/introbook/sbk10m.htm.

11 http://math.tutorvista.com/statistics/normal-distribution.html.
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The following example demonstrates the characteristics of a lognormal distribution curve. 

Consider a dataset obtained from three filterable particulate matter (PM) emissions tests, each 

consisting of three test runs, from three independent sources. The nine runs produced the 

following values measured in pounds PM per million British Thermal Units (lb PM / MMBTU): 

0.00134, 0.00139, 0.00153, 0.00319, 0.00387, 0.00417, 0.00970, 0.0128, and 0.0243. This 

dataset, referred to as Case A, yields both skewness and kurtosis consistent with a lognormal 

distribution. Figure 3 shows Case A data as solid green squares plotted on the lognormal 

distribution determined from the characteristics of the run data. Based on those characteristics, 

one would expect 50 percent of all filterable PM values to be less than 0.0039 lb PM / MMBTU. 

In addition, one would expect about 95% of all filterable PM values to be less than 0.0188 lb PM 

/ MMBTU. The long tail to the right, as shown in Figure 1, is consistent with the expectation that 

high values can exist, but they have a low probability of occurring – below one percent for 

values greater than 0.0255 lb PM / MMBTU. 

FIGURE 3. Lognormal Data Distribution Example. The smooth curve represents the density 

function of the lognormal distribution determined from the characteristics of the filterable PM 

emissions test data (shown as squares) from Case A.

B. Calculating the MACT Floor 

Once EPA identifies the best performing single source or best performing twelve percent of 

sources and the distribution of each dataset, we calculate the UPL, as described below, to 

identify an emissions limit that represents the level the average unit among the best performing 
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units has met, is meeting, and would be expected to meet in practice with a specific confidence 

level.

12 The confidence level, in this case 99 percent, is the percentage of measurements (past, 

present, and future) that are predicted to fall at or below the UPL value. Stated differently, the 99 

percent UPL is the level of emissions that we are 99 percent confident is achieved by the average 

source represented in a dataset over a long-term period based on its previous, measured 

performance history as reflected in short term stack test data. The UPL value is based on the 

available emissions data from the best performing single source (for new source standards) or the 

best performing 12 percent of sources (for existing source standards). The UPL incorporates the 

intrinsic variability of the individual run measurements used to calculate an emission test 

average.13 While the UPL can be calculated at different confidence levels, we selected the 99 

percent level in order to provide reasonable assurance that the limit can be met at all times by a 

source with emissions at the average level achieved by the best performing source or sources.

C. Applying the UPL Equations 

Once the distribution is established and the confidence level selected, a UPL value is calculated 

using six components. Four of these components are derived from the emissions dataset: the 

average (or mean), the variance of the dataset, the t-statistic or z-statistic (depending on the 

distribution characteristics), and the number of samples, which is associated with the total 

number of emissions test runs. For example, if there are results from six emissions tests, there 

typically are at least 18 emissions test run values (or samples), since an emissions test generally 

consists of 3 test runs.14 The other two components are the number of run values used to 

calculate the average and the confidence level. 

UPL equations are data-distribution specific, so that the normal distribution UPL equation is 

different from the lognormal UPL equation. Even though they differ due to separate 

mathematical properties associated with each distribution, the UPL equations share a common 

format that is composed of two parts which are added together. Equation 1 shows the UPL 

equation for a dataset with a normal distribution. 

Equation 1. ×
 = ̅ + 
,


 





+







Where: α = level of significance expressed as a decimal (e.g., 1% significance = 0.01), note 

that confidence level = 100 −  × 100
; 

 = average or mean of test run data; 

 = t score, the one-tailed t value of the Student’s t distribution for a specific degree 

of freedom and level of significance; 

 

12 Gibbons, p 28. 

13 Gibbons, pp 30-31. 

14 EPA test methods typically require 3 test runs. In cases where a test included fewer than 3 valid test runs, we 

typically do not include the test as a valid test for the purpose of MACT floor calculations. However, if limited or no 

other data are available for a particular source and pollutant, we consider the use of such data on a case-by-case 

basis. 
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 = number of test runs; 





 = variance of test run data; and 

 = number of test run values used to calculate the test average (generally 3). 

The first part of the UPL equation, ̅ is the average of the best performing sources’ emission 

measurements. Thus, the average of the emission measurements from the best performing source 

or sources is a fundamental element, and is in fact the starting point, of the UPL calculation. The 

second part of the equation, i.e., everything to the right of ̅, is associated with the variance of 

the run data, 



;	the total number of runs in the dataset, ; the confidence level, calculated from 

α; and the number of run values used to calculate the emissions average for each test result,	. 

As shown in Equation 1, the UPL is directly related to the sources’ emissions average. In other 

words, as the emissions average goes up or down, so does the UPL value. The UPL also is 

directly related to the confidence level and to the variance, meaning that as either of these values 

go up or down, so does the UPL value. However, the UPL is inversely related to the number of 

runs in the dataset and to the number of runs averaged for each emissions test. Therefore, as 

either of these values increases, the UPL value decreases and as either of these values decreases, 

the UPL value increases. A primary reason for this result is that the amount of uncertainty 

associated with the UPL is higher when sample sizes are smaller. Larger datasets provide better 

estimates of the average, the standard deviation and the distribution of the data. A lower 

boundary for the UPL exists at every significance level. This lower boundary corresponds to 

what the UPL would be if an infinite number of samples was available because the uncertainty 

component of the UPL becomes negligible as the sample size increases. In practical terms, this 

lower boundary represents a value that includes almost no uncertainty when the dataset is large. 

In other words, even if we had an infinite number of samples, the 99 UPL would not be below 

this lower-bound value. Therefore, the first half of the UPL equation takes the average emissions 

value of the sources in the MACT pool and the second half of the equation addresses the 

variability of that average emissions level based on the factors identified in the variables in that 

portion of the equation. 

Equation 2 shows the UPL equation for a dataset with a lognormal distribution. 

Equation 2. ×
 = 

!"#

$"

%

% +

&'()




 !"#*"%

+ *"% − 1, +  

!"#*"%



*"%



+

*"-






Where: α = level of significance expressed as a decimal (e.g., 1% significance = 0.01); note 

that confidence level = 100 −  × 100
; 

 = base of the natural logarithm (≈ 2.718282); 

.̂ = mean of the log transformed run data (= 



∑ ln	
3



34 ); 

56 = variance of the log transformed run data (= 



∑ ln	
3 − .̂


 

34 ); 
Page 12 of 15

7 = z score, the one-tailed z value of the z distribution for a specific level of 

significance; 

 = number of run values used to calculate the test average (generally 3); and 

 = number of runs. 

The widely-accepted statistical procedure for calculating a UPL for log-normally distributed data 

involves performing a log-transformation of the data. As with Equation 1, the first part of the 

UPL equation, 

!"#

$"

%

% ,	is associated with the average of the best performing sources’ emissions. 

The second part of the equation, i.e., everything to the right of 

!"#

$"

%

% , is associated with the 

variance of the run data, 56;	the number of runs in the dataset,	; the confidence level, calculated 

from α; and the number of run values used to calculate the test average,	. 

UPL Equation 2 operates in a manner similar to Equation 1. The first part of UPL Equation 2 is 

directly related to the emissions average and corresponds to an estimate of the average of the 

lognormal data. In other words, as the emissions average goes up or down, so does the UPL 

value. The second part of the UPL equation is directly related to confidence level and to the 

variance of the average of the lognormal data, meaning that as either of these values go up or 

down, so does the UPL value. As in the normal case, the second part of the UPL equation is 

inversely related to the number of runs in the dataset and to the number of runs averaged for an 

emissions test. Therefore, as either of these values increase, the UPL value decreases and as 

either of these values decrease, the UPL value increases. 

Consistent with Equation 1, a lower boundary for the UPL exists for Equation 2 at every 

significance level when an infinite number of samples is provided. In practical terms, this lower 

boundary represents the minimum amount of variability associated with a future, past, or present 

emissions test average.15 Furthermore, the selection of a higher confidence level for a UPL 

calculation widens the interval of results, meaning that a higher confidence level increases the 

UPL value. As with Equation 1, the first half of the UPL equation takes the average emissions 

value of the sources in the MACT pool and the second half of the equation addresses the 

variability of that average emissions level based on the factors identified in the variables in that 

portion of the equation. 

To provide an example of how the UPL calculation is performed, the application of the UPL to a 

specific set of data from a set of best performing sources in EPA’s sewage sludge incinerator rule 

is explained below (Case B). 

Case B shows how one takes data from the best performing sources and translates that into an 

emissions limit. There are 150 existing units in the multiple hearth subcategory for sewage 

sludge incinerators, so there are 18 units that are best performing sources for Case B since 12 

percent of 150 sources is 18 sources. Charts 1 and 2 show PM emissions data for each run of 

testing for the 18 best performing sources (note that we do not always have the same number of 

test runs for each source). 

 

15 Gibbons, p 31. 
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Run

Unit PM Emissions, mg/dscm 

1 2 3 4 5 6 7 8 9 

1 2.08 1.99 0.82 2.87 16.02 15.25 25.95 41.72 38.90

2 0.50 0.55 5.57 15.42 12.02 18.31 26.90 39.64 30.40

3 1.05 2.73 5.41 8.31 15.84 19.08 20.05 20.40 32.24

4 18.55 28.59 40.18

5 18.82 23.66 40.46

6 2.70 29.56 39.24

7 19.10

8 20.60

9 21.40

Chart 1. PM emissions data, mg/dscm, for Units 1 – 9 by individual run. 

Run

Unit PM Emissions, mg/dscm 

10 11 12 13 14 15 16 17 18 

1 47.11 33.76 49.65 42.53 48.90 45.99 72.26 80.51 70.31

2 43.86 41.11 34.07 40.52 46.64 124.20 60.89 80.46 88.68

3 27.71 44.49 28.54 37.87 70.58 45.33 78.61 69.49 83.01

4 27.91 56.75 89.06 63.62 73.99 74.90

5 52.90 48.47 28.92 94.64 79.78 69.80

6 37.16 24.03 55.09 63.06 75.36 85.83

7 45.03 81.62

8 32.13 77.11

9 55.30 67.15

Chart 2. PM emissions data, mg/dscm, for Units 10 – 18 by individual run. 

Using Excel software, the characteristics of the best performers from this source category are as 

follows: 

 = mean of test run data = 41.33; 

 = number of test runs = 90; 





 = variance of test run data = 736.24; 

Kurtosis = -0.31; and 

Skewness = 0.47. 

As the kurtosis and skewness results are both near zero, we conclude that the data exhibit a 

normal distribution. Given a normal distribution, we use Equation 1 to calculate the UPL. 

Because we are using a confidence level of 99, the significance level, , calculated using the 

confidence level is 0.01 (1-0.99). With 90 run values, the  value as determined from Excel 
Page 14 of 15

software is 2.37 (8
,.
 = 2.37). Since emission test averages, such as those used to 

demonstrate compliance with the UPL-based limits, include 3 individual test runs, the value of m 

is set at 3. The 99 percent UPL is calculated from the best performers’ dataset for a normal 

distribution as follows: 

×
 = ̅ + 
,
=

 >

1



+

1



?

.×
 = 41.33 + 8
,.
=736.24 >1

3

+

1

90?

88 = 41.33 + 2.37√253.59

88 = 41.33 + 37.74 = 79.07.

As mentioned earlier, this 99 percent UPL is composed of two parts: the first part which is 

directly related to the emissions average of the dataset (41.33) and the second part which is 

related to the variance, level of confidence, number of runs, and number of runs that constitute 

an emissions test (37.74). Thus, the equation includes both the average emissions level achieved 

by the best performing sources, and also a consideration of the variability of those emissions, 

including the variability associated with conducting an emissions test at other times. The 

calculation shows that the best performing sources would be expected to have an emissions test 

value that would not exceed 79.07 mg PM/dscm, based on the average of the best performing 12 

percent of sources’ emissions data and the consideration of variability within the data. 

Most, but not all, of the emissions test data from the best performing twelve percent of sources is 

below this value. As shown in Figure 4, values of individual test run data from these sources 

vary, with 10 values above the 99 percent UPL. The range of individual unit values is related to 

an individual unit’s variability. Individual run values above the 99 percent UPL demonstrate that 

even units among the best performers may need to apply some combination of additional control 

devices, modified process operations, or modified raw materials in order to be able to meet an 

emissions limit set at the 99 percent UPL. This is to be expected, and further demonstrates that 

EPA’s use of the UPL to calculate the MACT floor is reasonable, since the MACT floor 

represents the average emission level achieved by the best performing sources, not the worst

emission level achieved by those sources. For instance, units 15, 16, 17, and 18 (the “bottom 4” 

out of 18 units in the top 12 percent in this example) all have at least one test run value above the 

99 percent UPL for the entire dataset, but no source among the best performing 12 percent of 

sources had an overall average above the 99 percent UPL, although several sources had an 

average that is close to the UPL and one source had a 3-run test average above the UPL. 
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FIGURE 3. Filterable PM data from 18 multiple hearth sewage sludge incineration units. 

Individual run data shown as open red diamonds, unit average emissions test data shown as solid 

pink squares, the average of all runs shown as the dashed green line, and the 99 percent UPL is 

shown as the solid blue line. 

VI. Conclusion 

In conclusion, EPA’s methodology for establishing the MACT floors is reasonable and 

represents the average emissions achieved by the best performing sources with consideration of 

the variability in the emissions of those sources. The methodology first identifies the best 

performing sources by ranking sources from lowest to highest emissions. Next, the distribution 

of the data is evaluated in order to identify which UPL equation to apply. After that, the average 

of all the short-term stack test data from all the best performing sources (or the single bestcontrolled source, for new sources) is calculated. Finally, the UPL is applied to that calculated 

average value to account for variability such that the MACT floor standard can be met by a unit 

with emissions at the average level of the best performing source or sources.