This page provides worked examples based on the discussions of 140 and 165 IQ in intelligence and madness, and also for universitylevel student IQs of 110 and 126...
Probability, percentage, proportion and fractions
Say one in a hundred fish are blue, this can be expressed and written numerically in several manners.
 The probability that a fish is blue is 1:100.
 The percentage of blue fish is 1%.
 The proportion of blue fish to other colours is one in a hundred.
 Blue fish are a 1/100 fraction of a group of fish, or 0.01 in decimals.
All these descriptions are expressions of the same situation  one blue fish and ninetynine other fish.
There is more discussion at fractions,
decimals, percentages and ratios 1.
Mean, median and mode
The mean, commonly known as the average, is calculated by adding together all the values in the group of numbers, and then dividing that total by the number of items in the group. The result is the mean, or average, value.
In the group 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the mean is
(1+2+3+4+5+6+7+8+9+10 = 55)
55 ÷ 10 = 5.5
The median and mode are explained in the section on nonparametric statistics.
The relationship of standard deviation to IQ
A standard deviation describes numerically now much more (or less) an IQ level differs, or deviates from the average level of 100 IQ.
Calculating how many standard deviations a value is from the average
Using a standard deviation of 16, 1 sd is calculated thus:
116  100 = 16 (deviation from average IQ)
16 ÷ 16 = 1 (1 sd)
 The sd for 140 IQ is
 140 ÷ 100 = 40
40 ÷16 = 2.5 sd
The calculations on this page are worked examples only for the basis of illustration.
Statistics based on the normal curve are known as parametric statistics.
A sample of under 30 items (candidates, children...) is not usually regarded as valid for parametric statistics.
The examples given below have less than 30 items. They are strictly to enable the reader to understand the logic.
Nonparametric statistics are used for smaller samples and those not meeting the continuous status required by parametric statistics, but I am not dealing with nonparametric stats here. ( The mean is the standard parametric statistic.)
A standard deviation commonly has both positive and negative components; that is, more or less than the average value of 0 (zero) deviation. When added together as the first stage of finding the sd, there will be negative and positive deviation values that cancel each other out :
(+2) + (2) = 0 sd.
To avoid this anomaly, the deviations are squared (multiplied by themselves) before adding, eliminating cancelling out effects :
(+2)² + (2)² = (4 + 4) sd.
Because the result is a magnitude larger than the true magnitude of the deviations, the square root is then found :
√((+2)² + (2 )²) = √8 = 2.83 sd.
This result (2.83) is known as the standard deviation, sometimes called the root mean square.
Fully worked example for calculating the standard deviation
The standard deviation is the average of the deviations from the mean (average) of individual population members .
So the standard
deviation is the average of an average.
Find the mean
A mean is the sum of the numbers (measurements...) divided by the number of numbers (n) : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
This population has ten numbers : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
(1+2+3+4+5+6+7+8+9+10 = 55)
55 ÷ 10 = 5.5
The mean is 5.5.
Calculate the deviations
Now subtract the mean from each number to find by how much each number deviates from the mean (of 5.5):
1  5.5 = 4.5 2  5.5 = 3.5 3  5.5 = 2.5 4  5.5 = 1.5 5  5.5 = 0.5
6  5.5 = 0.5 7  5.5 = 1.5 8  5.5 = 2.5 9  5.5 = 3.5 10  5.5 = 4.5
Square those deviations
As you see, several deviations have negative values. Because these results are to be added together to find the overall mean deviation (average), something has to be done to prevent the positives and negatives cancelling each other out. That something is squaring all the results: a negative number squared is positive.
(4.5)² = 20.25 (3.5)² = 12.25 (2.5)² = 6.25 (1.5)² = 2.25 (0.5)² = 0.25
(0.5)² = 0.25 (1.5)² = 2.25 (2.5)² = 6.25 (3.5)² = 12.25 (4.5)² = 20.25
Add these results together to find their sum (∑)
20.25 + 12.25 + 6.25 +2.25 + 0.25 + 0.25+ + 2.25 + 6.25 + 12.25 + 20.25 = 82.50
Divide by the number in the population, n, to find its average
82.50 ÷ 10 = 8.25.
8.25 is known as the variance. The variance is sometimes quoted in papers, I know not why.
Next find the square root of the variance.
Because after squaring, each result is now a magnitude bigger than the original values, a correction is made by finding the square root of the average:
√8.25 = 2.87 This is the standard deviation of the population 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
This calculation can be laid out in columns, as a table, which is helpful when the population is larger. As the sample becomes even larger, it is common to group the individual measurements in deciles, or more groups. This sacrifices some 'accuracy' to gain easier calculation.
standard deviation calculation 
number 
deviation
(number  mean) 
deviation² 
∑(deviation²) 
average ÷ n (variance) 
standard deviation
(√variance) 
1 
4.5 
20.25 
82.50 
8.25 
2.87 
2 
3.5 
12.25 
3 
2.5 
6.25 
4 
1.5 
2.25 
5 
0.5 
0.25 
6 
0.5 
0.25 
7 
1.5 
2.25 
8 
2.5 
6.25 
9 
3.5 
12.25 
10 
4.5 
20.25 
There are also free apps/pages on the net that will do all the calculating for you. However, putting numbers into an electronic black box may not help a person's understanding.
Finding the proportion of a IQ cohort in a population
[The quoted rose paragraphs below have been copied from section 36 of intelligence and madness. Here is shown how the calculations are done.]
"By the time the 140 IQ level is reached, the normal curve [black line on graph at right plotting population against IQ, and fullsize graph at Burt page] predicts 1:165 (one in a hundred and sixtyfive), whereas in reality [red and green lines], the number of people is about 1:70 (one in seventy)."
165 is read from the population axis where the IQ value of 140 crosses the normal distribution/black curve [marked by the lower blue box on the minigraph at the right].
Note that, coincidently, this 165 has the same pattern of numbers as the 165 IQ that is also being discussed below. They are not connected, but could try to confuse the unaware.
70 is read from the population axis at 140 IQ on the red 'reality' curve [marked by the lower black box].
"By the level of 165 IQ, the normal curve predicts 1:40,000 (one in forty thousand), whereas in reality that number is nearer to 1:3000 (one in three thousand)."
40,000 is read from the population axis where the IQ value of 165 IQ crosses the normal distribution/black curve [upper blue box].
Note that, coincidently, 165 IQ has the same pattern of numbers as the 165 population figure discussed earlier. They are not the same figure, but could try to confuse the unaware.
3000 is read from the population axis at 165 IQ on the red 'reality' curve [upper black box].
Finding the fractions
[The quoted rose paragraphs below have been copied from section 36 of intelligence and madness. Here is shown how the calculations are done.]
"With a population of about 70 million in Britain, this suggests that, under present conditions (see also statistics and intelligence) there are about 23,300 Britons at the 165 IQ level."
Looking back at the previous section,
in reality [red and green 'reality' curves] the number of people in the UK with an IQ of 165 is 1:3,000 [upper black box].
70,000,000 ÷ 3,000 = 23,300
"The normal curve predicts only about oneseventysixth of this number (1,750 people). "
From the previous section,
according to the standard (theoretical) normal curve, the number of people in the UK at 165 IQ = 1:40,000 [black curve]
70,000,000 ÷ 40,000 = 1750
1750 ÷ 23,300 = 1/76 of the number found in reality.
"Nationwide, this means the reality figure is about 300 people for every
year level."
23,300 ÷ 80 = 291 (where 80 is the average life expectancy in Britain).
The relationship between standard deviation (sd), intelligence quotient (IQ) and population
The graph below illustrates only the upper half of a bellcurve, the half that related to people of average and above average IQ.
The IQ distribution graph above is using 140 IQ and 16 sd
only the upper part of the distribution shows, that is those IQs above the average (100 IQ)
To find what percentage of a population is at any level of IQ, a table of normal curve equivalents or standard normal distribution table, is used.
The standard distribution table is symmetrical about the centre. In the half standard distribution table below, the boxed numbers are those used for the worked example
below where the calculations are described, to determine the percentage of a population at a particular deviation [Z] above the average [0].
red: finding .94 sd purple: finding 2.5 sd
light green: calculating 126 IQ orange: calculating 110 IQ blue: calculating 140 IQ
Click for an unmarked version of the standard normal distribution table to print out and use. [Opens in new tab/window.]
It must be remembered that the numbers in the standard normal deviation table refer to the area under the curve (see small graph accompanying the table above) that extends from the average [0] to any value. The numbers do not refer to a value at a particular point on the table above.
So when Z = 0, this is where there is no deviation from the mean, so the area is also 0.
When Z = .94, the standard deviation at 115 IQ, the area under the graph is 32.64% of the whole (see the graph with the table above). You can, of course, also be interested in how many people are above that level that, with this table, would 50%  32.64% = 17.36%
Remember that the table refers to half the total distribution, thus you might want the area between 1 standard deviation above and below .94. That will be IQ 85 to 115. That area will be 32.64 x 2 = 65.28%
At Z = 3.99 (and for deviations from the mean greater than that), the area under the graph is virtually 50%. This includes practically all the population.
Using the standard distribution table for calculations
[The quoted phrases and paragraphs below have been copied from section 36 of intelligence and madness. This section shows how the calculations are done.]
Finding a proportion of population from a standard deviation
" the area under the graph between .94 and 2.5 sigma (115 and 140 IQ) ... is 16.74% "
A standard deviation of 16 is being used.
115 IQ is equivalent to 15/16 sd, or (15 ÷ 16) sd, or .94 sd.
For .94 sd, referring to the red boxes in the standard deviation table,
 first, look for 0.9 in the Z column, then run your
finger across to the 4 column.
 Note that the number there is .3264.
140 IQ is equivalent to 40/16 sd, or (40 ÷ 16) sd, or 2.5 sd. Using a standard deviation of 16,
For 2.5 sd, referring to the purple boxes,
 look for 2.5 in the Z column.
 Note that the number there is .4938.
These numbers represent the probabilities that a member of a population (in this case, a person) will have either 115 or 140 IQ.
Thus the numbers are equivalent to percentages of population.
That is,
.3264 is equivalent to 32.64% [.3264 32.64%]
.4938
is equivalent to 49.38% [.4938 49.38% ]
49.38%  32.64% = 16.74%
Determining the standard deviation at very high IQ levels
[The quoted rose paragraph below have been copied from section 36 of intelligence and madness. Here is shown how the calculations are done.]
The normal curve and its standard distribution in a population are theoretical concepts.
Referring again to the small graph at right plotting population against IQ [fullsize graph at Burt page].
The graph compares graphically the theoretical normal curve predictions (in black) to real world results for above about 140 IQ from different IQ tests (in red and green). The red and green curves diverge ever more from the black normal curve from about 140 IQ.
However, for those that wish to understand how a standard deviation for 165 IQ was calculated, and how those between 140 and 165 IQ are only 0.61% of the overall population, here follows those calculations.
"the area under the graph between 2.5 and 4.06 sigma (140165 IQ) ... is 0.61%"
From the standard distribution table above, with higher values of Z the deviation values become closer and closer to .5. This because there are few people with very high IQs, so there is very little range of deviation value above 3.5 sd. The probabilities of different levels of very high IQs would only become evident with more than four figures after the decimal point. (There are standard distribution tables available that are constructed using a greater number of figures. There are also complex formulae for doing this type of calculation.)
Thus, the actual percentages for high IQ cannot be determined from our limited s.d. table. They have to be calculated another way.
Simplified calculation of very high IQ
Look back at the graph showing the upper part of the distribution curve, illustrating those IQs above the average (100 IQ). [Small version here to the right.]
The lilac box below that graph provides the necessary data used
for determining the population percentage at 4.06 sd. This is beyond the maximum sd of 3.99 shown on the table.
140 IQ = 115 IQ + 25 IQ
140 IQ = 165 IQ  25 IQ
25/16 = 1.56 sd
25 IQ points is equivalent to 1.56 sd.
Therefore,
165 IQ is (2.5 + 1.56) sd = 4.06 sd from 0 (that is the average IQ, 100) in the sd table
Finding IQ from a population proportion and its standard deviation
[The quoted rose paragraphs below have been copied from sections 42 to 44 of intelligence and madness, and rearranged a little. This section shows how the calculations are done.]
I'll assume for the moment that it is those of highest IQ who go to university and the like. This is not so in fact. In reality, it is those from the more educated homes, those with relevant networks, those who are encouraged to study and grind hard!
Only 50 or 60 years ago, about 10% of the population was educated to university level
These figures are shown graphically in this linked graph of university attendance.
Given the assumption that 10% go to uni, their average IQ will be around 126 IQ,
Calculating 126 IQ from a university attendance of 10% of the general population
When 10% of the general population go to university, their average IQ is at the level of half of them (half of those going are above the average IQ for that cohort, and half are below that average). This amounts to 50% of the top 10% of the general population, that is:
10% of 50% = 5%.
Now to use the standard distribution table and see how 126 IQ is calculated.
5% is equivalent to .05. This describes the top 5% of the general population, so on the standard distribution table this cohort is at the .45 level, not the .05 level.
Looking at the standard distribution table, find .45 in the body of the table. In this case, the nearest numbers are .4495 and .4505 [marked in light green]. Follow that line to the left of the table to find the sd 1.6, and then follow the columns from the green boxes to the table's top to make the sd figure more precise: 1.64 and 1.65.
The average of these two sds is 1.645.
Using a standard deviation of 16,
1.645 x 16 = 26.3
This number, 26.3, is how much higher the average [mean] IQ of university students is than the general population's mean IQ of 100.
100 + 26.3 = 126.3 IQ, which we are rounding to 126 IQ for ease of use.
but only about 1 in 7 of these will be above 140 IQ.
The top 10% go to university, their lowest IQ can be found using the standard distribution table. The top 10% (.3997 to .4015 probability) have a sd of 1.28 to 1.29. [These values are marked in blue.]
The average of these two sds is 1.285.
16 sd x 1.285 sd = 20.56.
This number, 20.56, is how much higher the average [mean] IQ of university students is than the general population's mean IQ of 100.
100 +20.56 = 120.56 IQ, which we round to 120 IQ for ease of use.
Referring to the graph plotting IQ against population (small version on the right),
a person with an 120 IQ is about 1 in 10 of the population (light green box).
a person with an 140 IQ is about 1 in 70 of the population (black box).
Britain has a population of about 70 million. Of those,
at 120 IQ, there are 70,000,000
x 1/10 = 7,000,000 people;
at 140 IQ, there are
70,000,000
x 1/70 = 1,000,000 people.
[Remember that this graph compares graphically the theoretical normal curve predictions (in black) to real world results for above about 140 IQ from different IQ tests (in red and green). The red and green curves diverge ever more from the black normal curve from about 140 IQ.]
Calculating 110 IQ from a university attendance of 50% of the general population
... now it is approaching 50%!
—
When 50% go, the average IQ will be about 110 IQ,
When 50% of the general population go to university, their average IQ is at the level of half of them (half of those going to uni are above the average IQ for that cohort, and half are below that average). This amounts to 50% of the upper 50%.
50% of 50% = 25%.
Now to use the standard distribution table and see how 110 IQ is calculated.
25% is equivalent to 0.25. Looking at the standard distribution table, locate .25 in the body of the table.
In this case, the nearest numbers are .2486 and .2518 [marked by orange boxes]. Follow that line to the Z column on the table's left to find the sd .6. Then follow the columns from the orange boxes to the table's top to make the sd figure more precise: .67 and .68.
The average of these two sds is .675.
Using a standard deviation of 16,
.675 x 16 = 10.8
This number, 10.8, is how much higher is the average [mean] IQ of university students than the general population's mean IQ of 100.
100 + 10.8 = 110.8 IQ, which we are rounding to 110 IQ for ease of use.
Click for an unmarked version of the standard normal distribution table to print out and use. [Opens in new tab/window.]
 Nonparametric statistics are used for smaller samples and those not meeting the standards of Section 4 Continuous, but I am not dealing with that here. Only continuous data is regarded as fully adequate for parametric statistics, those based on the bell/normal/Gaussian curve.
It becomes more complicated. IQ tests are based on answering questions right or wrong, and each question is considered weighted equally in representation of IQ levels. Thus the data, strictly, only conforms to Section 2 Ordered. However, it is hoped that these weaknesses in the data are swallowed up in the averaging of large numbers of measurements.
Other types of averages are The median and the mode.
The median is the 'middle value' in a group of numbers, listed in value order.
When there is an odd number of values, the median is the centre number in the list.
In the group 1, 2, 3, 4, 5, 6, 7, 8, 9,the median is 5.
When there is an even number of values, the median is the average of the two 'middle values'.
In the group 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the median is
(5+6) ÷ 2 = 5.5
The mode is the most commonly appearing number in a group of numbers.
In the group 1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 9, 10, the mode is 8.
There are other averages, but they are not discussed here. There are books on nonparametric sampling such as the one in the bibliography.
2. 
These four sections are arranged in increasing alleged precision of measurements.
1. Quality
E.g. apple/orange apple/elephant/tree green/blue apple/blue.
The mathematical comparator or axiom proposes ‘separation’ into ‘objects’.
2. Ordered
E.g. Places in a foot race, exits on a motorway (freeway), sizes of planets in the local solar system, human heights.
The mathematical comparator is greater than (>), or less than (<). The arithmetic expression is
1^{st}, 2^{nd}, 90^{th} etc. Such ‘numbers’ cannot be reliably ‘added’.
Consider arranging several animals and a planet in order of ‘size’, then adding the second planet to three elephants
(there are a lot of elephants around this place, even greenpainted ones). In team races, often the positions of the first six competitors
are added and the team with the lowest total wins. However, at the Olympics, they do not allow you to trade three second prizes for one
first prize. As the Americans say, “Second is another word for loser”.
3. Even step ordered
E.g. Palings on a fence; the arithmetic expressions 1,2,3, … 90 … 247, etc.
This mathematical comparator is often regarded as having ‘equal’
(‘=’) steps. ‘Equality’ means, “I don’t care about differences
4. Continuous
E.g. Measuring stick, clock, no limit to subdivision, the measurement
of continuous space and time, the arithmetic expressions 1.84, 1.333 recurring, 247, 247.838, 4701.367294, etc.
This is not the counting of supposedly separated objects, but is comparing one object with another, e.g. by using a length of stick
with marks on called a ruler.
It is commonplace to confuse the numbers used to count separated ‘objects’
with the numbers used to indicate a place upon a ruler or clock. Note that the number 247, in Section 3.above, looks very like the number
247 in this section, but they are being used in very different ways. In Section 3., ‘the’ number is being used to count objects;
in Section 4., to mark a position.
[List taken from Comparing predicates, relational strengths] 
 standard
Standard, standardised, average
 cohort
A group of people with a shared characteristic.
 covariance
Covariance is often confused with variance. Covariance is the square of the correlation between two variances. It measures how closely, or otherwise, two sets of data vary together. Remember correlation does not imply causation.
For instance, a road has white dashes marked along it, the dashes being of similar length and spaced at regular intervals.
Alongside the road have been planted plane trees at different regular intervals, the trees having all grown to a similar girth. But, although a correlation can be made between the spacing of the road markings and the average girth of the trees, the dimensions of the markings did not cause the girth of the trees, nor visa versa.
 correlation
Correlation shows whether there is any relationship between two disparate sets of data.
Correlations that can be illustrated by a twodimensional graph, one with a x and a yaxis, using a line drawn through the graph, are called linear correlations.
Correlation is positive when the values of the two data sets increase together, and negative when one value decreases as the other increases.
Correlation can be calculated by hand, laying out the values in a table (often Pearson's correlation coefficient (r) is used).
A subsidiary page on correlation is under preparation.
 population
In statistics, the word 'population' does not just apply to a collection of people. It can refer to quantities, such the number of elephant's teeth or the height of mountains.
 The righthand scale [yaxis] of this graph is a logarithmic scale. The IQ scale [xaxis] is a normal arithmetic scale.
Looking back at the standard normal distribution table, you can see that it only extends to 4 standard deviations [column z], whereas the graph of IQ curves extends to nearly 6 standard deviations. There are more extensive normal distribution tables available at considerable expense, usually found in specialist libraries. The tables can also be calculated from basic formulae, nowadays the calculations are often run as computer programs.
 "virtually 50%"
The
standard normal distribution table used is compiled with only four figures after the decimal point. Therefore, values that differ by smaller amounts than 0.0001 (1:1,000) will not be listed separately, or show as a distinguishable area under the graph. This is a visual depiction of how few there are with higher levels of IQ.
bibliography
Nonparametric statistics for behavioural science
by Sidney Seigel, (1st edition 1956)
2nd edition by Sidney Seigel and N. John Castellan Jr. (1988) 

McGrawHill Series in Psychology, 1st edn, hbk, 1956
*
McGrawHill Humanities/Social Sciences/Languages; 2nd edn, 1988
ISBN10: 0070573573
ISBN13: 9780070573574
amazon.com
amazon.co.uk 


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