The Size of a Class: Two Viewpoints

Larry Lesser of The University of Texas at El Paso posed a problem in The Playground section of Math Horizons magazine (v 17, n 3, February 2010, p. 30), a problem that has a bearing on the oft-discussed question of the class size. As the problem implies, the view point on the size of a class may depend on who you ask, a teacher or a student.

Consider a 9-student school consisting of two 2-student classrooms and one 5-student classroom. The mean class size on a per-class basis would of course be

 
2 + 2 + 5

3
 = 3

while the mean class size on a per-student basis would be

 
(2 + 2) + (2 + 2) + (5 + 5 + 5 + 5 + 5)

9
 = 
11

3

This means that the average class size experienced by the teachers at this school, and likely advertised by the administration, is smaller than the average class size experienced by the students. The question is, if this is just one case of a more general phenomenon: if n students are distributed among k non-empty classes, must the per-student mean class size always be at least as large as the per-class mean class size?

Solution

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The Size of a Class: Two Viewpoints

Is the average class size experienced by the teachers at a school is smaller than the average class size experienced by the students?

The answer to this question is yes, absolutely, in so far as the class size varies between different classes.

Let as, s = 1, 2, ..., k be the class size at a particular school. Judging from the given example, the problem asks to compare two averages

 A1= (a1 + ... + ak) / k and
 A2= (a1² + ... + ak²) / (a1 + ... + ak).

A1 is the average number of students per class, i.e., the average number of students a teacher faces during a lesson. A2 is the average number of students in a class a student participates in. A brief derivation shows that A1 ≥ A2 as long as not all as are equal. To see that, compare instead A1A1 and A2A1, i.e.,

 M1M1= (a1 + ... + ak)² / k² and
 M2M2= (a1² + ... + ak²) / k.

where Mi denotes the i-th mean of a given set of numbers {as, s = 1, 2, ..., k}. M1 is the arithmetic mean; M2 is the quadratic mean. We know that in a general

M1 ≥ M2.

with the equality only when all the numbers are the same. It follows that A1 ≥ A2.


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