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Standard Deviation

IQ vs. Salaries, Test Scores, Driving Safety, and Alcohol Consumption


You have a fifty percent chance of scoring higher than the mean.

You have one chance out of six of scoring one standard deviation higher than the mean.

You have one chance out of 44 of scoring two standard deviations higher than the mean.

You have one chance out of 714 of scoring three standard deviations higher than the mean.

You have one chance out of 31,546 of scoring four standard deviations higher than the mean.

You have one chance out of 3,333,333 of scoring five standard deviations higher than the mean.

You have one chance out of 506,797,346 of scoring six standard deviations higher than the mean.

You have one chance out of 3,506,800,000 of scoring seven standard deviations higher than the mean.

You have one chance out of one quadrillion of scoring 8 standard deviations higher than the mean.

iow, IF there were a BILLION Africans in the world, which of course there are NOT, not a SINGLE one of them would have scored 784 and thus demonstrated an IQ of 132.


One Standard Deviation is 3.7 IQ Points

In 2007, in TIMSS math, 4,972 students from Hong Kong scored 584, with a Standard Deviation of 68, which is 9.5 SD higher than the 5,101 students from Ghana who scored 261. 

According to Professor Lynn, Ghana has an IQ of 71, which is 35 IQ points lower than Hong Kong at 106.

Thus 35 IQ points = 9.5 SD, statistical proof that the actual SD for IQ scores is 1SD = 3.7 IQ points.

This means that the 106 - 64 = 42 point gap between Hong Kong and Senegal is 11.4 SD, whereas a Sengalese has one chance out of 506,797,346 of scoring only 5 standard deviations higher than the mean for Senegal of 64, which is an IQ of only 106, not 137.

For a Senegalese to have an IQ of 137, he would have to score TWENTY STANDARD DEVIATIONS higher than the average of 64 for Senegal.

This is pure math. We don't need to engage in character assassination against Senegal to make this point.

Another way to look at this is Exhibit 2.2 on pg. 71 which shows that NONE of the students in Botswana or Ghana (nor even Northern Africa, in Algeria, and Morocco) scored higher than 550, compared to 71% in Taipei, Korea, or Singapore, 61% in Hong Kong and Japan, and half in Massachusetts:

Needless to say, none in Africa scored higher than 625 compared to almost half in Taipei, Korea, and Singapore, and one third in Japan and Hong Kong.

625 equals an IQ of 106 so any claim of a Senegalese to have an IQ of 137 is 4.2 SD higher than Japan, an IQ that NONE of the Japanese students exhibited.

We know that the real serious education in most Asian nations starts AFTER the 8th Grade, which is the grade tested by both PISA and TIMSS and what the above standard deviation calculations are based on. Since no Asian nation has participated in any of the 12th grade tests as we have. Both the race and gender gaps get much bigger between 8th and 12th grade.

American Blacks go BACKWARDS between 8th and 12th grade, and yet we already know that on IAEP at the 8th grade level, Blacks in D.C. score one standard deviation lower than Mozambique.

One Standard  Deviation = 3.1 IQ Points

On TIMSS Math in 2003, the Netherlands, with an average IQ of 103, scored 540, and South Africa, with an average IQ of 72, scored 264. Since the standard deviation for TIMSS  scores was 55, this 276 point gap is 10.0 standard deviations. And since this 10.0 standard deviations is equivalent to a 31 point gap in IQ scores, the standard deviation for IQ scores is 31 / 10 or 3.1 IQ points.

540 - 264 = 276

/ (55 /2) = 10.04

103 - 72 = 31

31 / 10.04 = 3.1



Possible Scenario?


One Third of India  Earn Less than 22 US Cents per DAY

30.9% on the following Normal Curve score lower than one half of a standard deviation lower than the average.  The UN Millennium Study found that this many Indians earn less than 22 cents per day, or $80 per year, and that another third earn more than 23 cents per day, or $84 per year.







Drexel Math Forum

"This is just to inform that I am really, REALLY keen on knowing this earth-shaking
piece of information."


Rather than taking a direct and honest approach at answering your question, I'm going to take an indirect and
less honest approach which MIGHT not be censored by the education mafia.

If you look at the 2007 TIMSS math scores for 4,208 students in Botswana, 5,294 students in Ghana, and 3,407
students in Hong Kong who participated in this study, you will see that only 6 students in Botswana scored
higher than 595 and only 7 students in Ghana scored higher than 585. If you take a detailed look at the
Gaussian Distribution of these scores, you will see that NONE of the 4,208 students in Botswana and NONE of
the 5,294 students in Ghana scored higher than 597.

So how does this equate to "IQ" that you are asking about?

The AVERAGE score for the 3,470 students in Hong Kong who took this test was 607, and NONE of them scored
lower than 401 (37 points higher than the AVERAGE for Botswana of 364 and 92 points higher than the AVERAGE
for Ghana of 309).

Professor Lynn graciously puts Hong Kong's average IQ at 107. But he also puts the average IQ of Japan at
only 105 and of Korea at 106 even though those two countries score higher than Hong Kong on many of these
international tests. So it would be conservative [or perhaps more reasonable] to estimate that a TIMSS math
score of 607 is equivalent to an IQ of 105, a score which NO student in either Botswana or Ghana achieved.

Does THAT answer your question?

From a Hong Kong perspective, a TIMSS score of 784 is equivalent to an IQ of 132. Ghana's score of 309 is
4.4 standard deviations (4.4 SD) lower than Hong Kong's average score, and Botswana's score of 364 is 3.6 SD
lower. In order for a student in Hong Kong to score higher than 784 and thus demonstrate an IQ higher than
132, he would have to score 2.6 SD higher than Hong Kong's average, a score which 34 Hong Kong students DID

As they scored MORE than 7 SD higher than Ghana and 6.2 SD higher than Botswana, can YOU tell us, GS, exactly
how many Ghanans and Botswanans YOU would expect to have an IQ higher than 132?

Standard Deviation for Salaries Only 10%

Each minor vertical gridline on the following graph of educator's salaries around the world represents two standard deviations. Thus, Israeli salaries are four standard deviations lower than Italy's but 8 standard deviations higher than Brazil's. Similarly, salaries for Israeli teachers are 22 standard deviations lower than Germany and 30 standard deviations lower than the US. The highest paid teachers in Israel earn $1,360 per month and the lowest paid teachers in the US earn $3,082, almost a three fold difference.  We should note that Israel's own internal government reports claim that Israeli teachers earn almost twice as much as this.  If given a choice between which is more accurate, a government report issued by any nation, but in particular Israel's, or a searchable worldwide database like this, I'll opt for the latter.



Ditto for the following gridlines on the graph of brain sizes in cubic centimeters, which shows that the average brain size of African women is six standard deviations lower than East Asian women, 16 standard deviations lower than Europeans, and 28 standard deviations lower than East Asian men.



However, the minor gridlines on the following graph of worldwide IQ's as presented by Lynn et. al., with a standard deviation of 15, equals about one fourth of a standard deviation. Therefore the gap between Israel and Kenya is only three quarters of a standard deviation, between Israel and the US is only three quarters of a standard deviation, and between Israel and Korea is 1.25 standard deviations.




This would suggest that there's a poor correlation between IQs and incomes. The problem is that there's a strong correlation between them. It takes an amazingly small correction to Lynn's estimated IQ by country to achieve linearity. So why would it seem that there are teachers in Israel with IQs as high as American teachers who earn less than one third less, at the same time that there are teachers in the US with IQs as low as Israel who earn three times more?



The idea behind standardized tests and IQ tests is to try to approximate real life situations. But this indicates they do a poor job of it, because there is no actual data based on observations of real life which have the race and sex overlaps that these tests have.


Top Ten Percent Score One Standard Deviation Higher than the Average


90th Percentile 1995

France 558

Switzerland 483

Canada 473

Sweden 487

Germany 489

Czech Republic 343

Australia 496

Austria 487

Italy 432

United States 383

Denmark 526

Slovenia 577


Top 10%, Vs. 90th Percentile 1995


France 612, 533

Switzerland 575, 539

Canada 567, 520

Sweden 564, 552

Germany 550, 495

Czech Rep 485, 466

Australia 589, 525

Austria 537, 524

Italy 520, 477

US 485, 383

Denmark 582, 549

Slovenia 629, 513



horizontal rule

90th Percentile 2008

Armenia 562

Iran 629

Lebanon 622

Netherlands 610

Norway 546

Philippines 494

Russia 677

Slovenia 567

Sweden 544

US (from 1995) 383


Standard Deviation

The standard deviation sigma of a probability distribution is defined as the square root of the variance sigma^2,

sigma = sqrt(<x^2>-<x>^2)(1)
= sqrt(mu_2^'-mu^2),(2)

where mu==x^_==<x> is the mean, mu_2^'==<x^2> is the second raw moment, and <f> denotes an expectation value. The variance sigma^2 is therefore equal to the second central moment (i.e., moment about the mean),


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


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,


The standard deviation s_(N-1) 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 x_(CI), and


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


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

0.800 +/-1.28155sigma
0.900 +/-1.64485sigma
0.950 +/-1.95996sigma
0.990 +/-2.57583sigma
0.995 +/-2.80703sigma
0.999 +/-3.29053sigma


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.


Eric W. Weisstein. "Standard Deviation." From MathWorld--A Wolfram Web Resource.

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: 

Descriptive Classifications of Intelligence Quotients




% of Population


Very superior






High average






Low average





Below 70

Extremely low


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: 


Top Civil Servants; Professors and Research Scientists.


Physicians and Surgeons; Lawyers; Engineers (Civil and Mechanical)


School Teachers; Pharmacists; Accountants; Nurses; Stenographers; Managers.


Foremen; Clerks; Telephone Operators; Salesmen; Policemen; Electricians.


Machine Operators; Shopkeepers; Butchers; Welders; Sheet Metal Workers.


Warehousemen; Carpenters; Cooks and Bakers; Small Farmers; Truck and Van Drivers.


Laborers; Gardeners; Upholsterers; Farmhands; Miners; Factory Packers and Sorters.


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.

Be Cautious!

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. 



Standard Error

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

The standard error of a sample of sample size n is the sample's standard deviation divided by sqrt(n). 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 sigma^^^2 of the quantity,


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

standard deviation

Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $32,000 and a standard deviation of $3000. If a teacher is selected at random, find the probability that he or she makes more than $36,000.

3 Answers

You will need to use the z-score formula. 
Let  X = 36000
Z = (X - μ ) / σ
Step 1: Insert given information and calculate.

Z = (36000 - 32000) / 3000 = 1.33
simplify the problem to 
Z=(36-32)/3   by reducing by a factor of 1,000
Step 2: look at a standardized normal distribution table to find the probability for P(Z<1.33),
P(Z<1.33) =0.9082 for those that make less than $36,000
To find the probability for those that make more than $36000 we have to subtract those that make less than $36000 from the whole sample (1).
P ( Z>1.33 )=1−P ( Z<1.33 )=1−0.9082=0.0918
The above explanation is great, however, since you are looking for the probability that x > 36,000, you need to subtract that value from 1. This is because the z table tells you the probability of scoring below a certain z score, but we are looking for the probability that we score above it. 
1-.9082=.0917 so the probability of a teacher making above $36,000 is about 9.17%
So, I'm not sure if this is completely necessary, but it might be easier to look at the standard normal distribution.  We use a translation to get from normal to standard normal.  It condenses our probability function now to a normal distribution centered on 0 with SD of 1.  So, we are going to go from P(X>36,000) to the translation P(X>(36,000-mean)/3000), because we often refer to a standard normal chart that displays the probabilities for given values.  I'm sure there is one in the book you're using.  Now, P(X>1.33333).  Which, if you look that up, is 90.82%.  This makes sense since 36,000 is almost two standard deviations out from the mean.   










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