**By André van Delft**

“The Ministry of Education study shows an average class size of 22.8 pupils in 2012.”

This is where the manipulation begins. Initially, it seems that the average is indeed 22.8 pupils per class. In reality, this is a classic example of misleading statistics.

What method has been used to calculate the average? The result is derived by dividing the total number of pupils by the total number of classes. This is incorrect. When referring to the quality of education it is of course relevant which size of class or school the pupils experience in reality. That is 24.0 pupils. This result is derived by asking each pupil how large his class is and then dividing the total sum of all the pupils answers by the number of pupils. Therefore, the number that the Ministry should publish is 24.0.

The error made by the Ministry is quite significant. It had not stated 23 but 22.8; suggesting a certain degree of accuracy by means of the digit 8 after the decimal point. The actual number should then be between 22.75 and 22.85. However, the reality deviates strongly from this result.

**Two Averages**

Consider the hypothetical case of one class with 30 pupils and one with 3 pupils. The average class size according to the Ministry would then be: (30+3) / (1+1) = 33/2 = 16.5. However, there are 30 pupils who will each say that they are in a class of 30 , therefore the number 30 must be applied in the calculation 30 times, and the number 3 must be applied 3 times. Finally, the result must be divided by the number of pupils. The average size of the classes that pupils attend is then (30×30 + 3×3) / (30+3) = 909/33 = 27.5 pupils (rounded off). Let’s call this the *weighted average* as opposed to the other*unweighted* one.

Why does the Ministry publish the incorrect, lower average? Would that be intentional; in an attempt to improve public opinion of the Government policy? Or would it be ignorance; which really should be unacceptable for a Ministry that includes the word “*Science”* in its title. I am uncertain which alternative is more distressing.

Furthermore, last year the Ministry had the class size calculation validated by independent scientists. Strange that those scientists had not reported the problem of the misleading unweighted average.

**Melkert versus Fortuyn**

*You consider there being a direct relationship between absence due to sickness and large scale schools. Let’s now look at what the average size of a school is. How many pupils are there in a primary or secondary education building. Any idea?*

*Yes I know. Primary education – about two to three hundred pupils.*

*Wrong, 160. Secondary education?*

*1000, 1200.*

*320.*

Following this debate I studied the official figures previously published by the Ministry in 2001.

The appendix on page 3 states that the average size was 163 pupils per primary school building.

A few years ago I learned that for the official 2001 Ministery figures large school conglomerates had not been individually counted as a single large school but as several small schools that were located coincidentally within a single building or complex of buildings. This way the figures had been manipulatively reduced. Therefore, when also considering the other systematic deception, Fortuyn’s estimate for secondary schools may well have been much more realistic than the official average.

BTW Pim Fortuyn was assassinated on 6 May 2002, a few weeks after the debate.

**Technical Details**

In the letter from the State Secretary of Education, Sander Dekker dated November 15th, 2012 the following bar chart was included:

The nice advantage of the bar chart is that it is possible to read back the percentages by class size.

The formula calculates the weighted average number of pupils per school or class, where the weighting factor is equal to the same number of pupils. Mathematicians see a quadratic mean looming up here.

**Conclusion**

**More reading**

“Class size paradox ” yields 17,500 Google hits.

The class size paradox has strange variants. An excerpt from a Social Networks course at the University of Maine :

In 1991, sociologist Scott Feld applied the class size paradox to social networks , demonstrating that the average person has fewer friends than his / her friends (in network terms , having more friends is called having a higher “degree”). This is because being part of your friend ‘s circle of friends is like being a member of a class at a university. The size of those circles of friends varies, there being more people in the large circles of friends than in the smaller circles of friends. The typical person will be friends with an unusually popular person.

See also the article “Why Your Friends Have More Friends Than You Do – And why your girlfriend is a whore” in Psychology Today.