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## Monday, 4 January 2016

### The unknown reality behind IQ tests

The unknown reality behind IQ tests

For most people their carrier depends on clearing public exams. There is a lot of competition out there, for getting the job they want.  Usually, exams conducted nationwide are used in shortlisting the candidates appearing for the job interview. It could be a corporate company or a government organization, the number of people willing to take up a post is alarmingly high in the third world countries. If you observe closely, people willing to get a post in these organizations don’t choose their jobs based on personal interests, but rather take them up through others recommendations and social pressures. So the corporate companies and the government organizations need a constant and intact method for screening the huge number people seeking jobs, and shortlist them to a very small number.

Now in most exams that help in shortlisting, you may find at least one IQ test paper in it. Most IQ test questions have a series of elements, either numerical or image, and have one or more blank regions which are to be filled by the candidates. In this article, I am going to talk about the truth between these kinds of problems specifically and go on to address the general issue with using exams as a tool for screening candidates. Let’s first address the numerical IQ tests where the question is a series of numbers, and it asks the candidate to fill the blank square following the pattern. When the question is like the following:

Fill up the blank square based on the pattern shown by the series
$2, 4, 6, 8, 10, 12, 14, 16, ?$

The candidate will immediately come up with the answer 18. But is it really the answer? Can there be other answers that fit into the blank square? Before we answer the above questions, let’s observe the question more closely. Note that the questioner didn’t mention the series to be an arithmetically progressive series. So it is not defined to be an arithmetic series. Now let’s deduce how we arrived at the answer 18. The first number is 2, the next number if 4 and the number following 4 is 6 and then we have 8 and then 10, 12, 14 and finally 16. This clearly shows that as we go along the series, the succeeding number is 2 greater than preceding one. So people obviously conclude with the answer 18 as it succeeds 16 by 2. So the mathematical expression expressing the series is $f(t) = 2t$, where $t$ is the term number.

So the candidate’s “logic” is $f(t) = 2t$. But let me claim the answer to be 200 rather than 18. On a first glance it feels like it doesn’t make sense and it’s “logically incorrect”. So let me give a “logical” explanation for this. Let’s modify $f(t)$ to become $f(t) = [t/9] \times 182 + 2t$, where $[.]$ represents the greatest integer of the expression. So we have, $f(1) = [1/9] \times 182 + 2 = 2, f(2) = 4, … , f(8) = 16$ and finally, $f(9) = 200$.  So we have another “logic” for this. Before I go any further with this, let me explain why I enclosed the word logic between quotes. Using the word logic here doesn’t make sense. Logic is the reasoning done to arrive at an answer based on strict constrains, that are assumed to hold true. So guessing the pattern followed by a series of numbers is not a logical reasoning.

One might argue that using the operator $[.]$ is not acceptable (though there are IQ test questions involving the greatest integer operator) and only the fundamental arithmetic operators are (neither do the questions given in IQ tests signify the presence of such constrains, rather asking as to just assume that there exist just one answer to each question). So for the readers convenience, let me give an example using just the basic arithmetic operators (algebraic: $+, -, \times, /$). But remember that using the greatest integer operator fits in well, as IQ questions simply present only a set of terms and don’t declare any constrains other than saying, “Find the number following the pattern presented by the series’”.

Let’s take for example $g(t) = \sum_{i=0}^{9} a_it^{i}$. So, $g(t)$ is a polynomial of degree 10. And $a_i$ represent the constants. Now if we relate, $g(1) = 2, g(2) = 4, … , g(9) = 200$ and solve for the values of $a_i$, we get a solution for each of the coefficients. Now we have a defined polynomial function such that, $g(1) = 2, g(2) = 4, ..., g(9) = 200$. Now, let me not go into the process of forming the polynomial, because the calculations involved require a lot of number crunching and the fact that we will have a solution (or a set of solutions if the degree of the polynomial is greater than 9) for $g(t)$ is quite easy to prove.

Let’s see the proof (You can skip the proof if you want)

Lemma 1: Let $g(t)$ be a polynomial of degree $n$. Then, the polynomial $g(t)$ can relate to any set  of $n$ number of randomly chosen numbers (assuming the numbers are $b_1, b_2, …, b_m$ where, $m \leq n$) such that $g(1) = b_1, g(2) = b_2, …, g(m) = b_m$. This relation can be formed by selectively choosing the coefficients of the polynomial.

Proof:

The polynomial $g(t)$ can be expressed as,

$g(t) = \sum_{i=0}^{n}a_it^i$ Now,

$g(1) = \sum_{i=0}^{n}a_i1^i = b_1$,

$g(2) = \sum_{i=0}^{n}a_i2^i = b_2$

$,…,$

$g(m) = \sum_{i=0}^{n}a_im^i = b_m$

So we have $m$ equations and $n+1$ unknown terms, so, we have the following cases.

Case 1: $m = n + 1$

For this case, we will have $m$ equations and $m$ unknowns, so we can find the value of each coefficient.

Case 2: $m < n +1$

Let $c = n+1 – m$. So we may be able to eliminate $m$ unknowns, and we will have a linear equations involving $c$ unknowns leftover. We have the liberty to choose the values for each of the $c-1$ unknowns arbitrarily. Now, the last unknown would get a definite value based on the $c-1$ values we had chosen randomly. Now, we may proceed further to find the values of each other coefficients in the polynomial. Note that with $c$ known coefficients, we can obtain the coefficient in the place $c+1$. And proceeding further, all the coefficients can be revealed.

A brief conclusion for this proof is required. You might have observed that the second case helps you choose some coefficients yourself. What does this signify in the context of this article? When there are more constant coefficients than there are terms in the series (that is when the degree is greater than the number of random values to relate in the series), then there is room for relating more terms than the number of terms that have to be related. A $n$ term polynomial can be related to $n$ random terms of a series. But instead, if it is related to $m$ terms, where $m < n$ there is room more terms to be related. So there are $n-m$ blank squares which can hold $n-m$ more terms. So the values you choose randomly determine $g(m+1), g(m+2) …, g(n)$ in random.

And without providing the proof, we can also say that g(h(t)) can be used in place of g(t) for the same kind of relation as above, given that $h(t)$ is strictly a one to one relation.

So, from lemma 1, we have seen that we may relate random values and call it a series. Unless we have a definite solution for a series, why is it asked in IQ tests? Why do they include such questions knowingly in tests that will determine your future carrier? Before I answer these questions, let me claim that the above method can be extended to most IQ questions involving numerals and series. There are some questions which are star shaped, and have numbers in some of the loops and ask you to fill up the rest. This can be translated to the above case, and again lemma 1 can be applied to show that any value is possible.

Let’s show that this also holds for visual patterns. In these kind of questions, the candidate will be presented with a series of images following a pattern and the next image following the series will be asked. This is same as the numerical questions, with the only difference being that images are used instead of numbers. We may form a language to express the orientation of the image algebraically or numerically. So there are again infinite possible such “languages”, and each will hold infinite possible outcomes same like the numerical case.

Hence, we have come to the point where IQ test seems totally meaningless. Then why is it used so widely? Let’s see the actual secret behind having IQ tests now. The motive of IQ tests is not to get answers from the candidates but to group the candidates in several categories. Let’s see how are IQ test used in such classification.

Let’s say that a corporate company needs employees with the following qualifications:
1. The candidate is required to have a good knowledge and experience in subjects like mathematics
2.  The candidate is required to be able to analyze common trends in business or technological related issues as accurately as possible.
3. The candidate is required to have the habit of foreseeing outcomes for events with optimal accuracy.
4. The candidate is required to be consistent and sincere in his work.

Of course, IQ tests don’t always prove so accurate in finding employees with the above qualities. But its power to screen a large number of candidates is quite alarming. First note that people who have good experience with mathematics and study that subject well will naturally come up with IQ solutions as they would have the habit of trying various combinations in finding their answer. There is an exception to this. Not all view the same subject material the same way. But a group of people good at mathematics would show some pattern and characteristics in common. That common character gets revealed in IQ tests. IQ tests just gives you a set of numbers and asks you to guess another one yourself. People who have been in the area of mathematics will have a common answer to it, because of the similarity they share with each other.

The outliers who think different may or may not be classified correctly by this IQ test. But, the companies don’t need them. They don’t need extra-intelligent people. Most business organisations just look for people who can do the task given to them perfectly and on-time.

Now let’s see how the second necessity is met by IQ tests. People who have the habit of guessing a number seeing a series of numbers can be assumed to be suitable in assessing business trends and outcomes. At the same time, toppers who train themselves to solve such problems rigorously will be confined to just answering the questions well. So they act as a “White Noise” in the IQ test method.  The third point is a direct follow-up of the second.

And as for the fourth point; the exam system is a gamified system used to screen out people for employment. The exam is the game field and the candidates are the players. The candidates score is their win or lose and the companies just take in the ones who win. The external pressure and the social opinion about scoring marks uplifts the importance of scoring marks in people’s mind.
Watch closely. The college subjects that are most emphasized are the ones requited by multi-billion dollar companies’ for their employment. The society has a constant path which ought to be taken by every human who is born. The society further enforces people to stay on line, and in-turn fuel the huge corporate to function.

Let me now conclude this article saying that, IQ tests are just means for classifying people based or certain categories. They are the labels given by the corporate. They are not designed to assess a single student, but it is a system devised to assess lakhs of people and shortlist them. The score IS NOT significant on the candidate level, but is only significant on the whole. IQ scores should not be used to asses a single person, but should only be used to assess the average behavior of a well performing candidate.

IQ test is not your tool, it is the corporate's tool.