An IQ is a number that indicates how good a person is at solving certain classes of problems using different faculties of their mind. Such faculties typically include
Of course, this list may seem a bit arbitrary, but it seems to offer a good estimate of the intelligence of a human being.
An IQ test is performed by measuring the faculties in isolated tests and then comparing each result to the results of a sample (a group of randomly chosen people). A statistical sample looks like this:
Each bar in the sample indicates the number of people that achieved one particular score in a specific part of the test. When the sample is big enough, the tops of the bars will form a particular shape known as the bell curve, because it is shaped like a bell.
At that point, a value called the standard deviation of the sample can be computed and the bars can be removed. The mean plus/minus one standard deviation is the area under the center of the bell curve that covers about 68% of the sample. [f1] This is what statisticians consider to be "standard".
The score a person made in a test is looked up in a normal distribution table, resulting in a percentile, i.e. a value indicating how many percent of the population would not achieve the same score. E.g., a value in the 75th percentile would indicate that the test subject scored higher than 75% of the population, but lower than 25% of the population.
The scores of the isolated partial tests are then weighted, because some tests have a higher variance than others. The variance of a distribution indicates how "broad" its bell curve is. Tests with a large variance are considered to be "harder" than those with a small variance. Spatial thinking, for instance, is considered to be hard while verbal skills are considered to be more mundane. "Hard" test results get a higher weight than not-so-hard ones.
The last step in the computation of the IQ is to take the average of the weighted percentages and then turn the resulting percentage back into a score by reverse lookup in the normal distribution table. That score is called the cumulative IQ, or just IQ.
Note that the result is also adjusted for age, gender, and education because, for some reason, we like the idea that IQ is stable while you age and independent of gender and education.
While most of the mathematical model behind IQ testing is solid, there is, of course, a lot of surface for criticism, most blatantly: