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 oftdiscussed 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 9student school consisting of two 2student classrooms and one 5student classroom. The mean class size on a perclass basis would of course be
 =  3 
while the mean class size on a perstudent basis would be
 = 

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 nonempty classes, must the perstudent mean class size always be at least as large as the perclass mean class size?
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Copyright © 19962018 Alexander BogomolnyThe 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 a_{s}, s = 1, 2, ..., k be the class size at a particular school. Judging from the given example, the problem asks to compare two averages
A_{1}  = (a_{1} + ... + a_{k}) / k and  
A_{2}  = (a_{1}² + ... + a_{k}²) / (a_{1} + ... + a_{k}). 
A_{1} is the average number of students per class, i.e., the average number of students a teacher faces during a lesson. A_{2} is the average number of students in a class a student participates in. A brief derivation shows that
M_{1}M_{1}  = (a_{1} + ... + a_{k})² / k² and  
M_{2}M_{2}  = (a_{1}² + ... + a_{k}²) / k. 
where M_{i} denotes the ith mean of a given set of numbers
M_{1} ≥ M_{2}.
with the equality only when all the numbers are the same. It follows that
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Copyright © 19962018 Alexander Bogomolny