# Average Class Size: A Decision Made by School Boards

Education, Featured, Policy — By Paul Tyahla on January 14, 2011 at 12:24 PM*by Mark ‘Jay’ Williams
Economics Fellow*

When policy makers use statistics to either justify or tear down a public policy, it is important that the numbers be used properly and are connected to real-world outcomes. Unfortunately, the recent debate about school budget cuts has used statistics to reach invalid conclusions.

Average classroom size is one of those descriptive statistics that is often misused. It is not a ‘natural’ variable such as the number of teachers, or the number of students. Instead, it is a ‘computed’ variable, which is derived from two or more variables that have multiple attributes.

Without the understanding of these measures, average classroom size tells the reader very little, particularly when implying a causal link where “budget cuts equal higher classroom sizes”.

For example, immediately following Governor Christie’s State of the State address, the New Jersey Education Association (NJEA) issued a press statement declaring, “He (Governor Christie) conveniently failed to mention his $1.3 billion in cuts to public education, but students, parents, and all New Jerseyans have paid steeply for them in larger class sizes, reduced programs, and higher property taxes.” That is a false conclusion, and we will explain why later in this commentary. It is not only NJEA that misuses this statistic; it is also the newspapers that report on classroom-interest stories of “’Teacher X’s math class growing from 20 to 25 students because of budget cuts.” No direct causal link has been established between budget cuts and teacher cuts, specifically at a classroom level.

According to the most-recent New Jersey Department of Education Report Card, which is for 2009, the average class size statewide was 18.8 students for 1st grade; 20.6 students in 5th grade, and 20.0 students in 9th grade. The Report Card also identifies that New Jersey had 1,346,946 students and 110,380 certificated staff, which equals 12.2 students per teacher. So how do you go from a teacher/pupil ratio of 12.2, to an average class size of 20.6?

Before delving into the economics of average class size, it should be noted that the U.S. Department of Education identified only three states as having a lower student/teacher ratio than New Jersey: The District of Columbia, North Dakota, and Vermont. So at least compared to peer states, New Jersey has the highest certificated staff levels in relationship to total students - but even here, the statistics are misleading.

First, there is a simple explanation for the difference between average class size and student/teacher ratio. A student/teacher ratio is calculated for the entire district (or state), whereas average class size is a sample statistic. Let us take a sample district with 50 teachers, 12 of whom teach English, and 400 students. Given a total population of students where N=400, then the student/teacher ratio would be N=8, and average class size for English classes is n=33.3, a very large difference. (Note the symbol change, N and n, for population and sample statistics).

Understanding the population sizes of what is being measured is important, but particularly for average class size, several additional statistics are needed.

The name ‘average’ class size itself identifies that there is an underlying statistic, in this case average=population mean (symbol ‘µ’), where the average class size for the state (or N or n population,) was 20.3 students. However, a population statistic tells nothing about a specific grade or course’s average class size; it is specifically for the entire school.

A school’s (N) average class size may be 21 students per classroom, but a 9th grade English (n) average class size might be only 13 students per classroom. In this case, the specific English class is a sample size mean (symbol ‘x?’), which is obviously different from the ‘µ’ population mean.

Let us go a little further in understanding the concept of average class size. Variance (?2) is a measure of how far a set of numbers are spread out from each other, describing how far the numbers lie from the mean. Standard deviation (?X ) identifies how much variation or dispersion there is from the mean. Technically, the standard deviation of a statistical population is the square root of its variance. I know, pretty boring, but important if we are to understand the implications of statements on public agendas.

An example of a standard deviation would be that for a sample size n=45 teachers, the ?X = 2.5 for teacher age where x? (average age)=40. The interpretation for this is that for that population of 45 teachers, 68% of these teachers would have an age of 37.5-to-42.5 (1 std. deviation of 2.5 ± 45), and 95% would have an age of 32.5-to-47.5 (3 std deviations of 2.5 ± 45).

So why does this matter?

As stated earlier, average class size is a computed statistic, and is based upon the attributes of two or more statistics.

The basics of an average class size are: (1) number of teachers for a specific subject, (2) number of students for a specific subject, and (3) number of classrooms for a specific subject. School Boards and Superintendents directly determine two of those factors - teachers and classrooms. The number of students for a specific subject could be influenced by school administration (completion of pre-requisites, guidance counseling recommendation, etc), but we will assume students choose a normal course sequence.

Teachers are another matter. School leadership determines their ‘distribution’. This includes the total number of teachers, the number of teachers for a specific subject and grade level, the age of the teachers, the salaries of the teachers, etc. When there are budget cuts or surpluses, school leadership has a greater affect on the consequences of firing and hiring than the budget cut itself.

Let us use an example of a school that has 500 students and 50 teachers, but only six of those teachers are Biology certificated. The student/teacher ratio for the school is 10:1, but for Biology Teachers it is 83.3:1. If those students can be divided into six classrooms daily, the average class size for biology is 13.8 (83.3 / 6).

Now assume this school district received a state appropriation and local tax revenue budget cut, and was required to lay off five teachers. Based upon standard collective bargaining agreements, the five newest teachers would be the first to be laid off. The new population (N) teacher ratio would be 500 students divided by 45 teachers, or a 11.1:1 student/teacher ratio, an 11% increase in student/teacher ratio on a 9% decrease in teachers (from 50 to 45).

However, one of the biology teachers may have just started, and although they are a core curriculum subject, they would be dismissed because of the last-in/first-out contract requirement. We now have five biology teachers, not six. The sample (n) average class size now would be 500 divided by 5 = 100, divided by 5 teachers, or 20 students for an average class size. Therefore, by eliminating one biology teacher, average class size almost doubled (from 13.8 to 20 students in a Biology class).

Let us take this a little further. Out of the six Biology teachers, the newest teacher may have earned $50,000 annually, while the remaining five teachers averaged $68,000 annually. However, if one of the longer-serving teachers had poorer performance than the newer teacher, the district contract would still require the lower-paid, better-performing teacher be laid off.

Even though the district could have reduced expenditures 36% more by eliminating the lower-performing teacher ($68,000 vs. $50,000=36%), existing illogical contract terms prevailed.

So what have we learned?

(1) Be careful when using computed statistics. To use them correctly, you must also know the underlying variable attributes;

(2) Average class sizes are determined by several factors, but generally, humans at the district level design the class-scheduling algorithm;

(3) The class-scheduling algorithm is based upon available resources, and humans at the district level make the resource allocation decisions (how many Biology teachers, how many Biology classes per day, etc);

(4) There is not a direct causal link between a school district budget cut and a specific school’s Biology average class size. There are several intervention points that are determined by humans at the district level. Obviously, budget cuts have ramifications, and these ramifications may travel down to a classroom determining on the resource allocations made by district leadership.

We have also addressed the issue of the population student/teacher ratio. Why is there a statewide teacher/student ratio of 12.2 students, but a statewide average class size of 20.6 students? Here are some quick answers:

(1) Certificated staff includes any employee with a certification: teachers, curriculum supervisors, guidance counselors, principals and vice principals, and certificated administrators. During budget cuts, non-instructional certificated positions should go first.

(2) Certificated staff contracts typically require one lunch period and two preparation periods per day. Therefore, on a 7-period schedule, a teacher is not available for instruction 42% of the time. During budget cuts, eliminating a prep period would increase teacher availability by 66%.

(3) Wide variances in teacher pay, specifically not based upon performance, means that the lowest paid teacher will probably be eliminated before a more senior but lower performing teacher. During budget cuts, reductions-in-force must be based upon performance.

(4) Teacher certifications for specific subjects dictate the number of specific courses that can be assigned, such as Biology certification requirements for Biology courses. During budget cuts, a minimum level of specific teacher certifications for each core curriculum course must be maintained, even before longevity.

As taxpayers, we should expect that for the next several years, the local property tax cap and the State’s decreasing aid appropriations will impact teacher and staff levels. How we determine these levels will be found directly in the decisions of school leadership. That is where the human-interest story on average class size changes can be found.

**About the Author**

Mark Jay Williams is an Economics Fellow at the Common Sense Institute, New Jersey’s newest public policy think-tank that explores and advances public policy alternatives that foster individual liberty, personal responsibility and economic opportunity. Jay has served as a senior Director within a New Jersey K12 public school district and a public state college. He research interests converges his doctoral work in econometrics and operations research in designing solutions for socially complex problems.

(Note: Mark Jay Williams was the Director of Research & Restructuring at the Atlantic City Public Schools during 1995-1996, a state-mandated position for Level III monitoring).

To download this position paper as a pdf, click here.

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