561. A Fallacy When Applying The Bell Curve
The bell curve is a very useful statistical representation of the distribution of given facts. But it is dangerous to use it without careful consideration for making decisions.
In entry 39 I wrote:
'Regulations require a hierarchy of achievement, and the teachers are expected to have results distributed along a bell curve, as many As as Fs. If all the pupils have As and Bs, this is not appreciated as an indication of a good teacher with clever students. If all have Es and Fs, it is not accepted as an indication of pupils with special problems or a bad teacher.
There has to be a bell curve to put the achievements relative in a hierarchy. Rationally, this makes no sense. The same pupil with the same achievement, could by haphazard be in a class, where he gets an A for it, or in another class, where he gets an F, his achievements are distorted by others, whom he might not have even chosen.
Rationally, marking should be independent of the other pupils, measured instead on how much he has learned of what has been taught. Everybody, who has learned 100% or little less, should get an A, independent of how many others have achieved the same. That would be just.'
A recent comment to this entry contains the expression "defiance of the logic of the bell-shaped curve."
This phrase caught my attention to the problems of the fallacy of confounding segments of a bell curve with an entire one.
The bell curve is a very useful statistical representation of the distribution of given facts. But it is dangerous to use it without careful consideration for making decisions.
In entry 39 I wrote:
'Regulations require a hierarchy of achievement, and the teachers are expected to have results distributed along a bell curve, as many As as Fs. If all the pupils have As and Bs, this is not appreciated as an indication of a good teacher with clever students. If all have Es and Fs, it is not accepted as an indication of pupils with special problems or a bad teacher.
There has to be a bell curve to put the achievements relative in a hierarchy. Rationally, this makes no sense. The same pupil with the same achievement, could by haphazard be in a class, where he gets an A for it, or in another class, where he gets an F, his achievements are distorted by others, whom he might not have even chosen.
Rationally, marking should be independent of the other pupils, measured instead on how much he has learned of what has been taught. Everybody, who has learned 100% or little less, should get an A, independent of how many others have achieved the same. That would be just.'
A recent comment to this entry contains the expression "defiance of the logic of the bell-shaped curve."
This phrase caught my attention to the problems of the fallacy of confounding segments of a bell curve with an entire one.
The marking of pupils' achievements is a good example:
As long as all children attend elementary school together, it can be assumed that their intelligence is roughly distributed along the bell curve and thus also the achievements, as long as the teacher optimizes the teaching for the pupils of average intelligence.
In Germany, children are sorted into different schools usually after four years of elementary school. While I have not information about the actual fractions, when and where I went to school, one third went to the school with the lowest level, one third to the one with a medium level and one third entered the school leading to university after having passed a test.
Thus each type of school catered for a different segment of the bell curve of intelligence.
Thus each type of school catered for a different segment of the bell curve of intelligence.
Looking at the distribution of only the segment containing the highest third of the bell curve, it is a downward curve. The higher the intelligence, the fewer the cases.
The fallacy of assuming the distribution of the ability of the pupils in this segment as an entire bell curve misleads teachers to confound the median of intelligence with the average. The average intelligence in this segment of the bell curve is much closer to the lowest than to the highest.
The fallacy of assuming the distribution of the ability of the pupils in this segment as an entire bell curve misleads teachers to confound the median of intelligence with the average. The average intelligence in this segment of the bell curve is much closer to the lowest than to the highest.
By lessons optimized for the median, a predominance of bad marks can be expected by the majority below. Teaching needs to be optimized for the average pupils in this segment.
Distributing marks along an alleged bell curve for the purpose of obtaining more As and Bs in a class of this segment is not a solution but a distortion. A bell curve of marks with C as the average implies to have roughly as many Fs as As, but this cannot be the case in the upper segment of the bell curve of intelligence. It cannot be justified to make the worst student fail, if he is only slightly less intelligent than the average, while the really bright students cannot be deprived of their As.
