Measuring Potential for Quality Education

Running Header: POTENTIAL FOR QUALITY EDUCATION METRIC 1
Potential for Quality Education Metric
Jonathan Fowler
Adams State University

POTENTIAL FOR QUALITY EDUCATION METRIC 2
INTRODUCTION
Education has long been seen by many societies to be the single most important social
program. Philosophers throughout history have debated the role of education and its access. The
concept of improving the quality of life for all through educational institutions has become a well
accepted philosophy in modern civilization. Therefore, it has also become important to analyze
the quality in which education is being provided.
To understand some of the issues plaguing developing educational system for large
societies, one only has to look as far the American education system. For decades this country
has attempted to provide quality education to all citizens, and spends the most per capita on
educational resources, yet it falls short at delivering the most well educated students. American
students often fall in the middle of the pack in a comparison of test scores across all subjects
among developed nations. Part of this continuing and growing issue is the prominence of the
Achievement Gap in various socioeconomic and ethnic echelons.
To even begin understanding what causes American education to fail at closing the
Achievement Gap, we must first define what quality education is and how we can measure it.
There are many ways that we can define quality in education, of which Don Adams (1993)
narrows down to six different perspectives: quality as reputation, resources and inputs, process,
content, outputs and outcomes, and “value added”. The focus of this research mainly focuses on
the quality of process, defined by Adams as,
“Quality as process suggests that not only inputs or results
but also the nature of the intra-institutional interaction of
students, teachers and other educators, or ‘quality of life’ of
the program, school or system, is valued.”

This approach of understanding the process is essential, because this project proposes that is
equally essential to look at the results when determining quality as it is to look at initial potential
for such quality to have a propensity to prosper.
If we assume that the goal of education spending and reforms are to improve and
maintain the quality of education in schools, then it is equally important to understand the
propensity for quality education to be cultivated in a school or region. Though quantitative
metrics in educational research can be difficult to develop, I believe it will be possible to
measure the potential for education by measuring a limited number of key statistics in a
community, region, or district.
By developing this potential for quality education metric (PQE metric or PQEM), several
key uses can be concluded. The greatest advantage proposed is that the PQEM will allow
researches and policymakers to understand how to most effectively utilize educational resources.
For example, if the PQEM is calculated to be low for a school district, then it would be a “red
flag” for policymakers to investigate the causes for the lower metric, and possibly solve those
issues, instead of blindly giving resources to the district with little or no directive to solve the
issues inhibiting students to effectively be educated.
Once this PQE metric is refined, examined, and proven to be effective, further research
can be done to continue improving the quality of educational systems. Researching correlations
between incomes in a region and PQE can be investigated. Violence rates, quality of life,
property value, and a myriad of other statistics could also be opened up to correlated studies
surrounding the PQEM. The wealth of possible data is great and is worthy of investigating the
development of this new metric.

LITERATURE REVIEW
Before a measure on potential educational quality can be made, it will be necessary to begin
defining education quality. This is a difficult task, but using the works of Adams (1993) and
Cheng (1997), an indication of how this definition can be boiled down is possible. Using these
documents as a backing, a clear indication of important quality indicators come from relative
teacher pay, pupil-teacher ratios, free and reduced lunch ratios, and density of gang and violent
criminal activity. Although these indicators are not alone, the literature provided shows their
strong correlation to educational quality and academic performance.
Teacher salary has been a contentious issue when it comes to academic performance
outcomes. Many economists have argued about whether raises base pay of works improves
productivity, effectiveness, and quality among candidates for positions. Eric Hanushek has
argued in many of his works about how direct increases of teacher salary does not correlate to
improve academic performance of students (1994; 1996; 2007). Yet, there are many researches
that show how important teacher pay is towards increasing the potential of high educational
quality. Jennifer Imazeki (2005) and Dolton & Marcenaro-Gutierrez (2011) show empirical
evidence of the importance of teacher salaries to the retention of both veteran and less
experienced teachers that show high potential as quality educators, while showing performance
gains in academic achievement. Lee & Barro (2001) also show how relative pay of teachers is an
important factor in developing higher academic performance.
Beyond teacher salary, the pupil-teacher ratio is also a controversial issue surrounding the
measurement of educational quality. Finn & Achilles (1999), Lee & Barro (2001), and Sequeira
& Robalo (2008) all show a high level of empirical evidence on the importance of the pupil-
teacher ratio in determining the quality of education a student receives. Despite many who

contradict this evidence, I believe the literature will provide a strong backing of the theory that
pupil-teacher ratios are important in this realm of educational quality. Furthermore, investigating
the educational philosophy of culturally relevant pedagogy (Ladson-Billings, 1995) is a strong
corollary indicator of need lower pupil-teacher ratios due to the necessity of developing strong
relationships with all students.
Along with understanding the economics of the teachers’ side of the equation, it may be
imperative to consider the economics of the community in which the students are immersed. A
significant amount of research has been done to show the importance of socioeconomics in the
achievement gap of students (Brooks-Gunn & Duncan, 1997; Lee, 2002; Reardon, 2011).
Research has also shown a strong correlation between socioeconomic statuses and the ratio of
free and reduced lunches in schools and districts (Caldas & Bankston, 1997; Harwell & LaBeau,
2010). Therefore, this ratio has the potential to be an important factor in measuring the potential
for quality of education in a region.
Finally, I came across the idea in my literature search of including criminal and gang
activity density as an indicator of educational quality. This is an important factor, as many
sociopsychological theorists have long claimed the importance of understanding “Broken
Window” theory and how our surrounds environments effect how we all behave (Wilson &
Kelling, 1982). Therefore, is a student is engulfed in a subculture of violence and crime then they
are more likely to be apart of that culture and thus have a lower chance of performing well
academically (Felson, Lisk, & South, 1994). This is also supported by research done by Case &
Katz (1991) on how family and communities interactions that are negative in nature will also
effect minors in a negative fashion as well. Through understanding these negative effects and

analyzing them, there should be some inclusion of them in any metric developed to measure
potential for quality education.
METHODOLOGY
The mathematical model I propose to measure the potential for quality education revolves
around weighted normalized statistical measures. Researches and theorists in the educational
field have cited numerous indicators that effect the quality of education. To develop this model,
it seems prudent to start small and easily measureable. Complex indicators can always be
researched and added into the model if increased increased efficacy is deemed necessary.
Of all the statistics available, the pupil-teacher ratio (PTR) has come under scrutiny for
whether or not it is of importance to quality education. A larger body of evidence shows its
importance throughout the history of education research and most detractors are arguing against
the economics of reducing class-size. The educational philosophy of Culturally Relevant
Pedagogy (CRP) introduced formally by Ladson-Billings (1995), entrenches itself in the
importance of pupil-teacher relationships, which require reasonable class sizes. Therefore, the
PTR is included as an effective indicator in the PQE model. Collection of this data comes form
district statistics reporting, the Colorado Department of Education, or through the mathematics:
!"#$%&'( =
+,-./01-2
+30'&4052
To normalize this metric, it will be compared to the national PTR average:
!"#$%&'( ≡ Normalized Teacher-Pupil Ratio
!"# =
!"#$%&'(
!"#7'-8%1'(

The average national pupil-teacher ratio reported by the National Center for Education Statistics
was 16.0 for the year 2013 (U.S. Department of Education, 2015), making the normalized
metric:
!"# =
!"#$%&'(
16.0
This normalized metric can then be weighted into the overall model. A step that will be covered
at the end of this chapter.
The second indicator to be included in this model is the ratio of students eligible for free
and reduced lunch. Most schools public report their statistics on this ratio, as well as the
Colorado Department of Education and National Center for Educational Statistics.
Mathematically, the free and reduced lunch ratio can also be calculated:
=#>$%&'( =
+?@$
+,-./01-2
The metric can then be normalized using the national average of the free and reduced lunch ratio:
=#> ≡ Normalized Free and Reduced Lunch Ratio
=#> =
=#>$%&'(
=#>7'-8%1'(
The national average ratio of students eligible for free and reduced lunch was reported by the
National Center for Educational Statistics as being 0.513 for the year 2013 (U.S. Department of
Education, 2014), making the normalized metric:
=#> =
=#>$%&'(
0.513

With these redundant sources of data, it improves the potential for the free and reduced lunch
ratio to be a reliable statistic to be included in this model.
The final indicator that will be measured for this research is the violent crime density for
the region. Using the Federal Uniform Crime Reporting Statistics, the number of violent crimes
in a city with a population of 10,000 or a county with over 25,000 can easily be obtained on a
yearly basis. If we define this metric of per capita violent crimes as U$%&'( and compare it to the
national average, described as U7'-8%1'(, a normalized metric can be represented as:
U ≡
U7'-8%1'(
U$%&'(
To account for natural fluctuations in these statistics in these statistics, an average of the violent
crimes per capita can be averaged for both local and national statistics over three years:
UV ≡
U7'-8%1'(,V
U$%&'(,V
This normalized can then be weighted and added to the PQEM.
With the normalized statistics, !"#, X, and UV at hand, they can be weighted and summed
into the final PQEM. Since all the statistics are normalized, a value for each, greater than 1,
would indicate a higher than average statics in a positive manner. Lower than 1 would indicate a
lower than average statistic. Using a weight for each metric we can define three simplified
variables of the weighted metrics as:
YZ ≡ !"# ∗ Z
Y] ≡ X ∗ ]

YV ≡ U ∗ V
where Z, ], and V are the weighing variables, such that Z + ] + V = 1. With the
weighted metrics calculated, the PQEM can finally be defined as:
!_`a ≡ Y1
1
8bZ

!_`a = YZ + Y] + YV
The final hang-up is determining how to weigh these metrics accurately, and calibrate it
so that it is properly measuring potential for quality education. Using a Python script and the
linear regression analysis Application Programming Interfaces included from the Python libraries
NumPy and SciPy, calibrated weights will be calculated. Once the test sample of statistics are
collected, the matrix of results will be measured against average ACT scores for the district and
high school graduation rates, which are reported by the Colorado Department of Education. The
weight configuration that shows the strongest correlation will the be used for the final analysis.
The test sample with involve the school districts with student populations of 10,000 or
greater. Based on current statistics, this would include twenty school districts, providing a
statistically significant sample with a variety of demographics. This sample will make collected
data more accurate. To analyze the efficacy beyond this sample will require further research
beyond the initial scope of this project.
It should be noted that all the metrics provided in this chapter can also be zero normalized
to develop metrics that show positively correlating indicators as positive numbers and negatively
correlating indicators as negative numbers. The mathematics for this version of the model is
provided in the Appendix A.

Finally, during the literature review it was argued that teacher salary should be included
in this model. Unfortunately, due to the limitations of the Cost of Living Index accuracy that was
available at the time of this research, it was not included in this iteration of the model. An
argument and the mathematics behind using teacher salary in such a model is included in
Appendix B for reference.
RESULTS
Using the methodology stated in the previous chapter, the PQEM statistics were collected
for the twenty Colorado school districts included in the study. Looking at the most basic results,
using equally weighted metrics, a moderate correlation can be observed using a coefficient of
determination in a linear fit to the data (Figure 1). Although this model with equally weighted
metrics does not show the perfect fit that one would hope for in a predictive model, it does show
that there is a correlation between the PQEM and the composite value of education quality for
each district.
Figure 1: A moderate (R² = 0.52) correlation can be found between the equally weighted PQEM and the composite quality of
education values.
R² = 0.52
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
PQEM
Composite Quality of Education
PQEM: Equal Weighted
Z = 0.33
] = 0.33
V = 0.33

When the data is run through the Python weighting optimization script, a slightly stronger
correlation can be observed (R2
= 0.57) using the weighted metrics w1 = 0.03, w2 = 0.79, and
w3=0.79. This iteration can be seen in Figure 2. Also two strong outliers from Douglas County
Public Schools and Academy District 20 appear in this iteration.
Figure 2: A slightly stronger moderate correlation (R² = 0.59) can be observed when optimized weights are used for the metrics
in the PQEM. The strong outliers from Douglas County Public Schools and Academy District 20 are circled.
If the two strong outliers are removed from the dataset and it is run through the script
once more, a pronounced increase in the correlation of the PQEM to composite quality of
education occurs (R2
= 0.87; Figure 3). The question then focuses on why Douglas County and
Academy District 20 are outliers in the first place. Both districts have uniquely high academic
outcomes overall. The demographics of these regions or the schools involved may also be more
complex than this model has been set up to account for. Unfortunately, these are purely
R² = 0.59
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
PQEM
Weighted PQEM
Z = 0.03
] = 0.79
V = 0.18

speculative reasons, and a more in-depth analysis is beyond the scope of this project.
Nevertheless, it indicates that for this model to be more successful, more factors may have to
taken into account or adjusted to increase its predictive abilities.
Figure 3: When the two strong outliers are removed, a strong correlation between the PQEM and quality of education can be
observed.
Through all of these optimization weighted iterations, an interesting trend occurs with the
weighted metrics. As correlation increases, the violent crime density weight approaches zero.
The free and reduced lunch weight appears to have some use in producing a correlation. And the
pupil-teacher ratio weight appears to have the most relevancy in the optimization calculation by
far. These are all interesting outcomes and were not expected when this project began.
The violent crimes density is a large surprise in its possible irrelevancy to the PQEM
model. Due to the complexity of issues that go into the why and how violent crimes occur in a
R² = 0.87
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
PQEM
Weighted PQEM
Z = 0.00
] = 0.74
V = 0.26

region, it may be that they are too chaotic of a variable to be a reliable predictor towards
measuring quality of education. While research shows how there is a strong negative correlation
towards violent crimes versus academic outcomes, the current attempt made at modeling it in
this initial research may be flawed in an unknown manner. In future iterations of this model it
may be necessary to reevaluate the methodology in inclusion of violent crimes to the PQEM
model.
Another interesting standout is the high weight calculated for the pupil-teacher ratio
metric. One major possibility to this high weighted value is that pupil-teacher ratios fluctuate
much less than other predictive measurements. This would indicate that it would be
advantageous to seek out other stable variables that may predict the potential for quality
education in future iterations of this model. Another possibility is that the pupil-teacher ratio is
simply a reliable indicator of the potential for quality education. If this is the case it would be
another argument for the importance of keeping the pupil-teacher ratio low so that students have
the highest chance of success in the classroom.
The fact that reducing both the pupil-teacher ratio and free and reduced lunch ratio tend
to positively correlate to higher potential for quality education, it appears there is a strong
argument that improving both of these figures will increase the potential of high academic
outcomes. While some educational researches have begun to argue that pupil-teacher ratios and
economics do not inherently indicate that academic outcomes will be poor, this mathematical
model may lay the ground work to show the importance of economics and the number of teacher
in relation to students have significant effects on academic outcomes. Therefore, I believe this
research should be continued to improve the accuracy and predictive outcomes for the PQEM
model. With the knowledge this research has provided it should be argued that, at least in the

State of Colorado, that districts need to continue being considerate of the pupil-teacher ratio and
the economic viability of their region to understand how their students will perform
academically. Knowing this information may allow resources at all levels to be used more
effectively towards positive student academic outcomes.
CONCLUSION
Educational reform to increase academic achievement has been an issue that has been
researched and discussed in the American system for decades. The purpose of this research was
to develop a new mathematical model in an attempt to understand the indicators of potential for
quality education in a region or district. Four indicators of quality education were considered
after studying years of previous research, including the pupil-teacher ratio, violent crimes
density, free and reduced lunch ratio, and teacher salaries. Unfortunately, the teacher salaries
statistics was not included in the final model for this project due to issues with the Cost of Living
Index. The results indicated a moderate to strong correlation between the weighted PQEM and
composite value measuring quality of education in the districts. Two strong outliers appeared to
effect the correlation and may indicate anomalies in the model that need to be researched further
to understand their importance in further iteration of this model. The model shows an indication
of strong importance on the pupil-teacher ratio in the ability to predict academic outcomes.
Further research may improve the understanding of this correlation, but it should be a note to
policymakers that the pupil-teacher ratio is still an important factor for academic performance.

APPENDIX A: ZERO NORMALIZATION MATHEMATICS
The purpose of using zero normalized metrics is to show positively correlating indicators as
positive numbers and negatively correlating indicators as negative numbers. The mathematics for
this version of the model are include in this section.
PUPIL-TEACHER RATIO
!"# =
!"#7'-8%1'(
!"#$%&'(
− 1 for
!"#7'-8%1'(
!"#$%&'(
≥ 1
1 −
!"#$%&'(
!"#7'-8%1'(
for
!"#7'-8%1'(
!"#$%&'(
< 1
FREE AND REDUCED LUNCH RATIO
=#> =
=#>7'-8%1'(
=#>$%&'(
− 1 for
=#>7'-8%1'(
=#>$%&'(
≥ 1
1 −
=#>$%&'(
=#>7'-8%1'(
for
=#>7'-8%1'(
=#>$%&'(
< 1
VIOLENT CRIME DENSITY
U =
U7'-8%1'(
U$%&'(
− 1 for
U7'-8%1'(
!"#$%&'(
≥ 1
1 −
!"#$%&'(
U7'-8%1'(
for
U7'-8%1'(
U$%&'(
< 1

APPENDIX B: TEACHER SALARY METHODOLOGY
Another easily measured indicator is teacher salary. This is a controversial statistic
because it has been argued that teacher salaries do not correlate to increased test scores. Basic
economic theory contradicts this notion fairly quickly (Dolton & Marcenaro-Gutierrez, 2011).
As you increase salary, within certain margins, you increase the pool of quality candidates for
the position. If teacher pay is low relative to the cost of living, then the propensity for teachers to
apply for positions in the region decreases. Therefore, despite detractors, relative teacher pay is
an important indicator for assessing the potential of quality education. There are several ways in
which relative teacher salary can be measured.
One method is to take the average of all teacher salaries in a region or district. This is a
difficult measurement because it usually requires districts to voluntarily release or report the
information. It is also a difficult measurement to verify independently. Another method simply
looks at the salary of the first year teachers. This method assumes a single-pay scale matrix,
often divided by years of experience and level of education. While this is often an easier
measurement, it may not point the most accurate picture of teacher salaries in the district.
Another possible measurement looks at the average pay-scale of masters-level educated teachers
for the first five years of experience. This method is slightly more complex than others, but looks
at an often publically published figure from teacher unions or district boards. Investigating
masters-level pay for these experience levels is important because they are the most important
when discussing quality teacher retention versus mobility and career exit from a district.
Although not all districts scale their pay on a single schedule, base-salary for this metric can be
used for all districts to simplify and unify the metric. Further investigation into modifying the
metric for more intricate complexities will be left for future research if deemed necessary.

From the mathematical standpoint the relative salary will have to be normalized. Using
the U.S. Bureau of Labor Statistics we will define a national average salary, X7'-8%1'(. Using the
Cost-of-Living Index (CLI; St Louis Robert, 1989), a relative average pay can be determined and
normalized:
X =
X7'-8%1'( ∗ (m>n/100)
X$%&'(
This normalized value can then be weighted into the PQEM.

Running Header: POTENTIAL FOR QUALITY EDUCATION METRIC 18
APPENDIX C: DATA TABLES
District N Students N Teachers
Pupil-Teacher
Ratio
N Free & Reduced
Lunch Ratio FRL
Denver Public Schools 90,150 5,245 17.19 62,826 0.697
JeffCo Public Schools 86,572 4,700 18.42 27,530 0.318
Douglas Country School District 61,465 3,297 18.64 6,515 0.106
Cherry Creek School District 54,228 2,983 18.18 14,099 0.260
Adams 12 Five Star Schools 42,230 2,092 20.19 14,632 0.346
Aurora Public Schools 40,877 2,083 19.62 29,145 0.713
Boulder Valley School District 30,546 1,688 18.10 5,498 0.180
St. Vrain Valley School District 30,195 1,616 18.69 10,870 0.360
Colorado Springs School District 11 28,407 1,686 16.85 16,334 0.575
Poudre School District 29,053 1,716 16.93 9,794 0.337
Academy District 20 24,481 1,463 16.73 3,183 0.130
Mesa County Valley School District
51 21,906 1,263 17.34 9,201 0.420
Greely-Evans School District 6 20,441 1,142 17.90 13,234 0.647
Falcon School District 49 18,880 914 20.66 5,664 0.300
Pueblo City Schools 18,034 1,048 17.21 12,966 0.719
School District 27J 16,987 803 21.15 6,428 0.378
Thompson School District 16,226 868 18.69 5,694 0.351
Littleton Public Schools 15,830 843 18.78 3,324 0.210
Harrison School District Two 11,777 688 17.12 8,362 0.710
Adams County School District 50 10,101 540 18.71 8,081 0.800

District
Violent Crimes per
100k (2010)
Violent Crimes per
100k (2011)
Violent Crimes per
100k (2012)
Mean Violent
Crimes per 100k
Denver Public Schools 564 607 616 596
JeffCo Public Schools 227 231 234 231
Douglas Country School District 119 138 92 116
Cherry Creek School District 295 421 418 378
Adams 12 Five Star Schools 701 476 263 480
Aurora Public Schools 446 438 425 436
Boulder Valley School District 216 278 248 247
St. Vrain Valley School District 599 538 564 567
Colorado Springs School District 11 471 440 455 455
Poudre School District 182 168 172 174
Academy District 20 471 440 455 455
51 263 297 314 291
Greely-Evans School District 6 397 397 433 409
Falcon School District 49 129 74 65 89
Pueblo City Schools 831 766 731 776
School District 27J 207 259 345 270
Thompson School District 191 157 210 186
Littleton Public Schools 127 130 130 129
Harrison School District Two 471 440 455 455
Adams County School District 50 220 227 260 236

District
Normalized Violent
Crime Density
Normalized Pupil-
Teacher Ratio
Normalized Free &
Reduced Lunch
HS
Graduation
Rate ACT Average
Denver Public Schools -0.517 -0.074 -0.358 0.628 17.6
JeffCo Public Schools 0.703 -0.151 0.613 0.829 21.2
Douglas Country School District 2.377 -0.165 3.840 0.914 21.7
Cherry Creek School District 0.039 -0.136 0.973 0.871 21.4
Adams 12 Five Star Schools -0.222 -0.262 0.481 0.74 19.5
Aurora Public Schools -0.111 -0.227 -0.390 0.6 16.7
Boulder Valley School District 0.588 -0.131 1.850 0.91 23.4
St. Vrain Valley School District -0.443 -0.168 0.425 0.83 20.2
Colorado Springs School District 11 -0.159 -0.053 -0.121 0.784 18.9
Poudre School District 1.258 -0.058 0.522 0.84 22.1
Academy District 20 -0.159 -0.046 2.946 0.91 22.1
51 0.348 -0.084 0.221 0.78 19.5
Greely-Evans School District 6 -0.041 -0.119 -0.262 0.78 17.4
Falcon School District 49 3.397 -0.291 0.710 0.9 19.1
Pueblo City Schools -0.975 -0.076 -0.402 0.719 18.3
School District 27J 0.453 -0.322 0.356 0.79 18.6
Thompson School District 1.112 -0.168 0.462 0.773 20.9
Littleton Public Schools 2.045 -0.174 1.443 0.92 22.5
Harrison School District Two -0.159 -0.070 -0.384 0.776 18.8
Adams County School District 50 0.667 -0.169 -0.559 0.64 16

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