In this video I go over an example on normal distributions and this time look at the very well known application that follows a normal distribution probability density function, the Intelligence Quotient (IQ) scores. IQ tests have been widely used for about a century now and they, like many phenomena, are normally distributed. In the example I show that about 2/3rd of people (or 68%) have an IQ between 85 and 115 while less than 0.4% have an IQ higher than 140.
I also go over a brief history of the IQ tests as well its uses and drawbacks. IQ tests focus mainly on cognitive abilities such as memory, attention, speed, and pattern recognition. This is a very useful video if you are interested in learning more about the popular IQ testing concept so make sure to watch this video!
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Applications of Integrals: Probability: Normal Distributions: Example 1
Intelligence Quotient (IQ) scores are distributed normally with mean 100 and standard deviation 15 and the probability density function is shown in the graph below.
a) What percentage of the population has an IQ score between 85 and 115?
b) What percentage of the population has an IQ above 140?
Solution
Recall from my earlier video titled "Can we integrate all continuous functions?", the function y= ex2 doesn't have an elementary antiderivative, so we can't evaluate the integral exactly.
But we can calculate it numerically using a calculator (or manually using the Midpoint Rule or Simpson's Rule) to estimate the integral:
So about 68% of the population has an IQ between 85 and 115, that is, within one standard deviation of the mean.
Thus, about 0.4% of the population has an IQ over 140.
Brief Mainstream History of the IQ Score
- The term Intelligence Quotient (IQ) was coined by the German psychologist and philosopher William Stern for a scoring method for intelligence tests in his 1912 book.
- Currently the mean (and median) is defined as IQ 100 and each standard deviation up or down is defined as 15 IQ points but this was not always so historically.
- IQ Scores have been shown to be associated with such factors as health, mortality, parental social status, and to a significant degree, biological parental IQ.
- While the inheritance of IQ has been investigated for almost a century, the significance and mechanisms of inheritance are widely debated.
- IQ Scores are used for educational placement, assessment of intellectual disability, and evaluating job applicants.
- IQ scores for many populations have been rising at an average rate of about 3 IQ points per decade since the 20th century.
- This phenomenon is called the Flynn effect.
- They are aimed to test a person's cognitive abilities such as memory, attention, and speed.
- Standardized tests focus more on knowledge of the specific topic at hand, thus even if students were to improve their test scores, they don't necessarily improve their IQ scores as well.
- An example of one type of IQ test question is based on Raven's Progressive Matrices which is shown below.
- These matrices were first developed by John C. Raven in 1936.
- The concept is to find the missing element that completes a pattern.
Reveal spoiler
- About 5% of the population have scores above 125
- About 5% of the population have scores below 75
- About 2/3rd of the population (68%) have between 85 and 115.
- Although IQ is the most widely used method of measuring intelligence, there are still many criticisms and fallbacks such as:
- It is neglects some areas of intelligence such as creativity and social intelligence
- Intelligence is an abstract concept so defining it as a single number is an over-simplification
- The results of IQ tests have been often assessed more so for political reasons rather than simply scientific reasons.
- IQ classifications such as "superior" or "average", etc., vary with different publishers.