Standard Deviation, IQ vs. Salaries

The standard deviation of a probability distribution is defined as the square root of the variance ,

			(1)
			(2)

where is the mean, is the second raw moment, and denotes an expectation value. The variance is therefore equal to the second central moment (i.e., moment about the mean),

(3)

The square root of the sample variance of a set of values is the sample standard deviation

(4)

The sample standard deviation distribution is a slightly complicated, though well-studied and well-understood, function.

However, consistent with widespread inconsistent and ambiguous terminology, the square root of the bias-corrected variance is sometimes also known as the standard deviation,

(5)

The standard deviation of a list of data is implemented as StandardDeviation[list] starting in Mathematica Version 5.0.

Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a quantity from a given baseline.

The standard deviation arises naturally in mathematical statistics through its definition in terms of the second central moment. However, a more natural but much less frequently encountered measure of average deviation from the mean that is used in descriptive statistics is the so-called mean deviation.

The variate value producing a confidence interval CI is often denoted , and

(6)

The following table lists the confidence intervals corresponding to the first few multiples of the standard deviation.

range	CI
	0.6826895
	0.9544997
	0.9973002
	0.9999366
	0.9999994

To find the standard deviation range corresponding to a given confidence interval, solve (?) for , giving

(7)

CI	range
0.800
0.900
0.950
0.990
0.995
0.999

REFERENCES:

Kenney, J. F. and Keeping, E. S. "The Standard Deviation" and "Calculation of the Standard Deviation." �6.5-6.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 77-80, 1962.

CITE THIS AS:

Eric W. Weisstein. "Standard Deviation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/StandardDeviation.html

IQ Scores: IQ Score Interpretation

IQ scores are often misunderstood. Learn the basics of IQ score interpretation in this article.

Intelligence testing began in earnest in France, when in 1904 psychologist Alfred Binet was commissioned by the French government to find a method to differentiate between children who were intellectually normal and those who were inferior. The purpose was to put the latter into special schools. There they would receive more individual attention and the disruption they caused in the education of intellectually normal children could be avoided. 

This led to the development of the Binet Scale, also known as the Simon-Binet Scale in recognition of Theophile Simon's assistance in its development. The test had children do tasks such as follow commands, copy patterns, name objects, and put things in order or arrange them properly. Binet gave the test to Paris schoolchildren and created a standard based on his data. Following Binet’s work, the phrase “intelligence quotient,” or “IQ,” entered the vocabulary.  

Lewis M. Terman worked on revising the Simon-Binet Scale. His final product, published in 1916 as the Stanford Revision of the Binet-Simon Scale of Intelligence (also known as the Stanford-Binet), became the standard intelligence test in the United States for the next several decades. By the 1920s mass use of the Stanford-Binet Scale and other tests had created a multimillion-dollar testing industry. 

Despite the fact that the IQ test industry is already a century old, IQ scores are still often misunderstood. Comments like, “What do you mean my child isn’t gifted — he got 99 on those tests! That’s nearly a perfect score, isn’t it?” or “The criteria you handed out says ‘a score in the 97th percentile or above.’ Susan got an IQ score of 97! That meets the requirement, doesn’t it?” are not unusual and indicate a complete misunderstanding of IQ scores.

Understanding IQ Scores

IQ stands for intelligence quotient. Supposedly, it is a score that tells one how “bright” a person is compared to other people. The average IQ is by definition 100; scores above 100 indicate a higher than average IQ and scores below 100 indicate a lower that average IQ. Theoretically, scores can range any amount below or above 100, but in practice they do not meaningfully go much below 50 or above 150. 

Half of the population have IQ’s of between 90 and 110, while 25% have higher IQ’s and 25% have lower IQ’s: 

Apparently, the IQ gives a good indication of the occupational group that a person will end up in, though not of course the specific occupation. In their book, Know Your Child’s IQ, Glen Wilson and Diana Grylls outline occupations typical of various IQ levels: 

IQ Expressed in Percentiles

IQ is often expressed in percentiles, which is not the same as percentage scores, and a common reason for the misunderstanding of IQ scores. Percentage refers to the number of items which a child answers correctly compared to the total number of items presented. If a child answers 25 questions correctly on a 50 question test he would earn a percentage score of 50. If he answers 40 questions on the same test his percentage score would be 80. Percentile, however, refers to the number of other test takers’ scores that an individual’s score equals or exceeds. If a child answered 25 questions and did better than 50% of the children taking the test he would score at the 50th percentile. However, if he answered 40 questions on the 50 item test and everyone else answered more than he did, he would fall at a very low percentile — even though he answered 80% of the questions correctly. 

On most standardized tests, an IQ of 100 is at the 50th percentile. Most of our IQ tests are standardized with a mean score of 100 and a standard deviation of 15. What that means is that the following IQ scores will be roughly equivalent to the following percentiles: 

An IQ of 120 therefore implies that the testee is brighter than about 91% of the population, while 130 puts a person ahead of 98% of people. A person with an IQ of 80 is brighter than only 9% of people, and only a few score less than 60.

It is necessary to be very cautious in using a descriptive classification of IQ’s. The IQ is, at best, a rough measure of academic intelligence. It certainly would be unscientific to say that an individual with an IQ of 110 is of high average intelligence, while an individual with an IQ of 109 is of only average intelligence. Such a strict classification of intellectual abilities would fail to take account of social elements such as home, school, and community. These elements are not adequately measured by present intelligence tests. Furthermore, it would not take account of the fact that an individual may vary in his test score from one test to another. 

There appear to be two different definitions of the standard error.

The standard error of a sample of sample size is the sample's standard deviation divided by . It therefore estimates the standard deviation of the sample mean based on the population mean (Press et al. 1992, p. 465). Note that while this definition makes no reference to a normal distribution, many uses of this quantity implicitly assume such a distribution.

The standard error of an estimate may also be defined as the square root of the estimated error variance of the quantity,

(Kenney and Keeping, p. 187; Zwillinger 1995, p. 626).