The Bad Mathematics of Psychometrics
Are you Driving, Amiable, Analytical or Expressive?
I do not like psychometrics.
At best they are fun surveys you can take in between clicking through buzzfeed articles. At worst, they are kafkaesque personality tests based off of shaky pseudoscience. I learned some job recruiters ask candidates to take these surveys in anticipation of finding the best candidates for a job. To me, that’s the equivalent of picking candidates based on their horoscope. “Cathy, we need more ISFJ in our salesfloor.”
It’s no secret that the MyersBriggs test is entirely entertainment. Participants are put into one of two states for four separate categories, making different types people can fall into. All of the results have mostly positive connotations. This is a small enough pool of different “types” people can get behind. It’s like horoscopes for the slightly less gullible.
We took a different psychometric test in our projects class. We took the Social Style test, a product of the TRACOM Group, a social sciences company based out of Centennial, Colorado. It was presented to us (the students soontobe graduates soontobe code monkies) like something we prepare for when entering the workforce. A personality test!
What got under my skin with this test was how your score were calculated. An example of the test is here. Basically, you answer 15 questions for two sections: Assertiveness and Responsiveness. Each question you give an integer score from 1 to 4. You then sum up all of your assertiveness scores and all of your responsiveness scores, divide each by 15, and then plot yourself on a 2D graph. If you’re curious, I scored a on the social styles test.
A bad mathematician will think that plotting these scores on a 2d graph of means that a broader typing of people allows for individuals to be compared on two axes, without pidgeonholing everyone into only one of sixteen categories, a la MyersBriggs.
It should be clear why this is bad mathematics. For each axis, there are or roughly a billion different ways to take the survey for each axis. But there are only so many scores you can get with the test. In fact, before dividing by 15, the maximum score you can get on an axis is a 60, and the minimum is a 15. Thus, there are 46 unique scores on each axis you can get (Remember the scores are inclusive, so counting from 15 to 60, you will count 46 numbers). These 46 different scores are squeezed between 1 and 4 on the scatterplot. On a 2d graph, the Social Styles test gives unique types of people.
This is not exactly every real number between 1 and 4. Remember when I said I scored ? I lied. That score is not even possible with this test. I would need respective undivided scores of and to get , which is impossible. It’s very easy to be misled by this test. There are only 46 unqiue points on the axis people can actually score on this test.
A good mathematician will look at this test and immediately see that it is skewed towards participants landing in the middle, and not at the ends of the test. Most people taking the Social Styles test will wind up in the middle of the quadrant. Why though?

To score a 15, you will need to answer 1 for all questions. There is only ways to score a 15.

To score a 16, you will have to answer 1 for 14 questions, and a 2 for one question. ways to score a 16.

To score a 17, you will have to answer 1 for 14 questions, and a 3 for one question, or 1 for 13 questions, and 2 for two questions. Then you have ways to score 17.
As we get up closer to 30 and 31, there will be more ways to take the test and answer it in a way that will give us that score. The result is most participants will land in the center if you take a random enough sample. And as we aim to score higher, with 58, 59, and 60 scores, we wind up with less ways of taking the test to give us that score.
I think personality tests and psychometrics are pseudoscience. But the Social Styles test has a bell curve baked into it. Trying to analyze largescale results of this test requires fighting a skew that puts most candidates towards a central score. If the tests participants answered randomly, the results would be exactly the same. Bad mathematics were baked into this test.
What makes the Social Styles test interesting is the quadrants people land in, labeling them as Driving, Amiable, Analytical or Expressive. Because a bell curve is baked into the test, the results of the test are skewed. When you try to show a scatterplot of peoples scores, it will come up meaningless.