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Let’s ignore the trickiness that is even defining “intelligence” as a monolithic concept. Let’s ignore the many criticisms and problems arising from trying to condensate that nebulous concept into a quantifiable measure. Let’s ignore the problems of self-assessing that nebulous thing called intelligence, and comparing it to others’.

What would it look like if 90% of people really were of above average intelligence?


Let the average intelligence be a constant c. The average intelligence is, obviously, calculated by measuring everyone’s intelligence, adding up all the scores and dividing the resulting number by the number of individuals. Let’s assume we have n individuals in our population.

For simplicity’s sake, we can define then that 90% of the population scores some intelligence x > c. The other 10% have a score1 of y < c. The average is then:


c = x + x + x + x + … + y + y + y

Where there are 0.9n copies of x and 0.1n copies of y, for any population n. Then, the equation can be simplified as this:


c = 0.9nx + 0.1ny

And, with a bit of algebraic manipulation, we come to:


y =  − 9x + 10c/n

Right off the bat we can see that this equation has the form of a straight line, with a slope of  − 9 and a y-intercept of 10c/n. Let’s use the arbitrary values c = 100 and n = 50. The plot will look as follows:2 3 4


+---------------------------------------+
| This graph brought to you by gnuplot! |
+---------------------------------------+

         What if 90% of the population were of above average intelligence?      
                                                                                
   20 +---------------------------------------------------------------------+   
      |  *   +      +      +      +      +      +      +      +      +      |   
      |   A**                                        '90percent.dat' **A*** |   
      |                                                                     |   
      |      A**                                                            |   
    0 |+        *                                                          +|   
      |          A                                                          |   
      |           *                                                         |   
      |            *                                                        |   
      |             A**                                                     |   
  -20 |+               *                                                   +|   
      |                 A**                                                 |   
      |                                                                     |   
      |                    A**                                              |   
  -40 |+                      *                                            +|   
      |                        A**                                          |   
      |                                                                     |   
      |                           A**                                       |   
      |                              *                                      |   
  -60 |+                              A**                                  +|   
      |                                                                     |   
      |                                  A*                                 |   
      |                                    **                               |   
      |                                                                     |   
  -80 |+                                     A**                           +|   
      |                                                                     |   
      |                                         A**                         |   
      |                                            *                        |   
      |                                             A**                     |   
 -100 |+                                                                   +|   
      |                                                A**                  |   
      |                                                   *                 |   
      |                                                    A**              |   
 -120 |+                                                                   +|   
      |                                                       A*            |   
      |                                                         **          |   
      |                                                                     |   
      |                                                           A**       |   
 -140 |+                                                                   +|   
      |                                                              A**    |   
      |                                                                 *   |   
      |                                                                  A  |   
      |      +      +      +      +      +      +      +      +      +      |   
 -160 +---------------------------------------------------------------------+   
      0      2      4      6      8      10     12     14     16     18     20  
                        Intelligence score of the upper 90%                     

But wait! The value of x is not arbitrary! We know, from the premises of this problem, that x > c, which in our arbitrary example means x > 100. Therefore, we must travel a bit through the graphic to see the real deal:

         What if 90% of the population were of above average intelligence?      
                                                                                
  -880 +--------------------------------------------------------------------+   
       |      +      +      +      +      +     +      +      +      +      |   
       |  A**                                        '90percent.dat' **A*** |   
       |     *                                                              |   
       |      A**                                                           |   
  -900 |+                                                                  +|   
       |         A*                                                         |   
       |           **                                                       |   
       |                                                                    |   
       |             A**                                                    |   
  -920 |+                                                                  +|   
       |                A**                                                 |   
       |                   *                                                |   
       |                    A**                                             |   
  -940 |+                                                                  +|   
       |                       A**                                          |   
       |                          *                                         |   
       |                           A**                                      |   
       |                                                                    |   
  -960 |+                             A**                                  +|   
       |                                 *                                  |   
       |                                  A                                 |   
       |                                   *                                |   
       |                                    *                               |   
  -980 |+                                    A**                           +|   
       |                                                                    |   
       |                                        A**                         |   
       |                                           *                        |   
       |                                            A**                     |   
 -1000 |+                                                                  +|   
       |                                               A**                  |   
       |                                                  *                 |   
       |                                                   A**              |   
 -1020 |+                                                                  +|   
       |                                                      A*            |   
       |                                                        **          |   
       |                                                                    |   
       |                                                          A**       |   
 -1040 |+                                                                  +|   
       |                                                             A**    |   
       |                                                                *   |   
       |                                                                 A  |   
       |      +      +      +      +      +     +      +      +      +      |   
 -1060 +--------------------------------------------------------------------+   
      100    102    104    106    108    110   112    114    116    118    120  
                        Intelligence score of the upper 90%                     

There it is! It’s clear from the graphic itself that the only way for 90% of the population to have above average intelligence is for the other 10% to have abysmal scores.

There's another, larger discussion about the perils of using the average (arithmetic mean) alone in statistics, the usage of median, mode and proper context for statistic results, and the mathematical and social education of that «90%»; but is left to the reader as an exercise.


THE IRON NODER CHALLENGE XII: WE'LL RUST WHEN WE'RE DEAD


  1. I’m assuming these population slices have the exact same intelligence score.

  2. Astute readers will realize by now that the main shape of the plot is the same for any realistic values of c and n, since they are both positive.

  3. Even more astute readers will realize that given the fact that, for any values of c and n, the y-intercept is just a different constant value.

  4. It doesn’t take a particularly astute reader to know that the notion of «90% of the population is above average» leads to mathematically interesting but unrealistic results.