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61-485: Cattell built into the CFIT a standard deviation of 16 IQ points. Crystallized intelligence (Gc) refers to that aspect of cognition in which initial intelligent judgments have become crystallized as habits. Fluid intelligence (Gf) is in several ways more fundamental and is particularly evident in tests requiring responses to novel situations. Before biological maturity individual differences between Gf and Gc will be mainly

122-560: A continuous real-valued random variable X with probability density function p ( x ) is σ = ∫ X ( x − μ ) 2 p ( x ) d x ,  where  μ = ∫ X x p ( x ) d x , {\displaystyle \sigma ={\sqrt {\int _{\mathbf {X} }(x-\mu )^{2}\,p(x)\,\mathrm {d} x}},{\text{ where }}\mu =\int _{\mathbf {X} }x\,p(x)\,\mathrm {d} x,} and where

183-614: A built-in bias. See the discussion on Bessel's correction further down below. or, by using summation notation, σ = 1 N ∑ i = 1 N ( x i − μ ) 2 ,  where  μ = 1 N ∑ i = 1 N x i . {\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\text{ where }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.} If, instead of having equal probabilities,

244-623: A class of eight students (that is, a statistical population ) are the following eight values: 2 ,   4 ,   4 ,   4 ,   5 ,   5 ,   7 ,   9. {\displaystyle 2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9.} These eight data points have the mean (average) of 5: μ = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 8 = 40 8 = 5. {\displaystyle \mu ={\frac {2+4+4+4+5+5+7+9}{8}}={\frac {40}{8}}=5.} First, calculate

305-724: A correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by ⁠ s / c 4 ⁠ , where the correction factor (which depends on N ) is given in terms of the Gamma function , and equals: c 4 ( N ) = 2 N − 1 Γ ( N 2 ) Γ ( N − 1 2 ) . {\displaystyle c_{4}(N)\,=\,{\sqrt {\frac {2}{N-1}}}\,\,\,{\frac {\Gamma \left({\frac {N}{2}}\right)}{\Gamma \left({\frac {N-1}{2}}\right)}}.} This arises because

366-798: A finite data set x 1 , x 2 , ..., x N , with each value having the same probability, the standard deviation is σ = 1 N [ ( x 1 − μ ) 2 + ( x 2 − μ ) 2 + ⋯ + ( x N − μ ) 2 ] ,  where  μ = 1 N ( x 1 + ⋯ + x N ) , {\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\left[(x_{1}-\mu )^{2}+(x_{2}-\mu )^{2}+\cdots +(x_{N}-\mu )^{2}\right]}},{\text{ where }}\mu ={\frac {1}{N}}(x_{1}+\cdots +x_{N}),} Note: The above expression has

427-585: A function of differences in cultural opportunity and interest. Among adults, however, these discrepancies will also reflect differences with increasing age because the gap between Gc and Gf will tend to increase with experience which raises Gc, whereas Gf gradually declines as a result of declining brain function. The Culture Fair tests consist of three scales with non-verbal visual puzzles. Scale I includes eight subtests of mazes, copying symbols, identifying similar drawings and other non-verbal tasks. Both Scales II and III consist of four subtests that include completing

488-441: A height within 6 inches of the mean ( 63–75 inches ) – two standard deviations. If the standard deviation were zero, then all men would share an identical height of 69 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule , or the empirical rule, for more information). Let μ be

549-464: A line that is out of place. A dotted antisigma ( antisigma periestigmenon , Ͽ ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority. In Greek inscriptions from the late first century BC onwards, Ͻ was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros son of Dionysodoros would be written Διονυσόδωρος Ͻ ( Dionysodoros Dionysodorou ). In Unicode ,

610-407: A mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if

671-435: A modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population). Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values subtracted from their average value. The marks of

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732-611: A separate letter in the Greek alphabet, represented as Ϻ . Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians . According to one hypothesis, the name "sigma" may continue that of Phoenician samekh ( [REDACTED] ), the letter continued through Greek xi , represented as Ξ . Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from

793-467: A sequence of drawings, a classification subtest where respondents pick a drawing that is different from other drawings, a matrix subtest that involves completing a matrix of patterns, and a conditions subtest which involves which, out of several geometric designs, fulfills a specific given condition. The Cattell Culture Fair Intelligence Test (like the Raven's Progressive Matrices) is not completely free from

854-779: Is defined as σ ≡ E ⁡ [ ( X − μ ) 2 ] = ∫ − ∞ + ∞ ( x − μ ) 2 f ( x ) d x , {\displaystyle \sigma \equiv {\sqrt {\operatorname {E} \left[(X-\mu )^{2}\right]}}={\sqrt {\int _{-\infty }^{+\infty }(x-\mu )^{2}f(x)\,\mathrm {d} x}},} which can be shown to equal E ⁡ [ X 2 ] − ( E ⁡ [ X ] ) 2 . {\textstyle {\sqrt {\operatorname {E} \left[X^{2}\right]-(\operatorname {E} [X])^{2}}}.} Using words,

915-429: Is a very technically involved problem. Most often, the standard deviation is estimated using the corrected sample standard deviation (using N  − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using N ) yields lower mean squared error, while using N  − 1.5 (for

976-448: Is an unbiased estimator for the population variance, s is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples ( N less than 10). As sample size increases, the amount of bias decreases. We obtain more information and

1037-403: Is known as Bessel's correction . Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by n would underestimate the variability. If the population of interest is approximately normally distributed,

1098-476: Is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts. A dotted lunate sigma ( sigma periestigmenon , Ͼ ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line marked as such is at an incorrect position. Similarly, a reversed sigma ( antisigma , Ͻ ), may mark

1159-420: Is suited for all but the smallest samples or highest precision: for N = 3 the bias is equal to 1.3%, and for N = 9 the bias is already less than 0.1%. A more accurate approximation is to replace N − 1.5 above with N − 1.5 + ⁠ 1 / 8( N − 1) ⁠ . For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of

1220-412: Is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing

1281-773: Is the p -th quantile of the chi-square distribution with k degrees of freedom, and 1 − α is the confidence level. This is equivalent to the following: Pr ( k s 2 q 1 − α 2 < σ 2 < k s 2 q α 2 ) = 1 − α . {\displaystyle \Pr \left(k{\frac {s^{2}}{q_{1-{\frac {\alpha }{2}}}}}<\sigma ^{2}<k{\frac {s^{2}}{q_{\frac {\alpha }{2}}}}\right)=1-\alpha .} Sigma Sigma ( / ˈ s ɪ ɡ m ə / SIG -mə ; uppercase Σ , lowercase σ , lowercase in word-final position ς ; ‹See Tfd› Greek : σίγμα )

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1342-464: Is the eighteenth letter of the Greek alphabet . In the system of Greek numerals , it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation . When used at the end of a letter-case word (one that does not use all caps ), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from

1403-646: Is the extent to which the Culture Fair Intelligence Test correlates with other tests of intelligence, achievement, and aptitude . The intercorrelations between the Culture Fair Intelligence Test and some other intelligence tests have been reported, as shown in the Table below. The most widely used individual tests of cognitive abilities, such as the current editions of the Wechsler Adult Intelligence Scale and

1464-492: Is the mean of these values: σ 2 = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 8 = 32 8 = 4. {\displaystyle \sigma ^{2}={\frac {9+1+1+1+0+0+4+16}{8}}={\frac {32}{8}}=4.} and the population standard deviation is equal to the square root of the variance: σ = 4 = 2. {\displaystyle \sigma ={\sqrt {4}}=2.} This formula

1525-524: Is unbiased if the variance exists and the sample values are drawn independently with replacement. N  − 1 corresponds to the number of degrees of freedom in the vector of deviations from the mean, ( x 1 − x ¯ , … , x n − x ¯ ) . {\displaystyle \textstyle (x_{1}-{\bar {x}},\;\dots ,\;x_{n}-{\bar {x}}).} Taking square roots reintroduces bias (because

1586-537: Is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by 7 (which is n − 1) instead of 8 (which is n ) in the denominator of the last formula, and the result is s = 32 / 7 ≈ 2.1. {\textstyle s={\sqrt {32/7}}\approx 2.1.} In that case,

1647-488: The Stanford–Binet Intelligence Scale , report cognitive ability scores as "deviation IQs" with 15 IQ points corresponding to one standard deviation above or below the mean. Standard deviation In statistics , the standard deviation is a measure of the amount of variation of the values of a variable about its mean . A low standard deviation indicates that the values tend to be close to

1708-867: The confidence interval or CI. To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of N = 2 has only one degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows : Pr ( q α 2 < k s 2 σ 2 < q 1 − α 2 ) = 1 − α , {\displaystyle \Pr \left(q_{\frac {\alpha }{2}}<k{\frac {s^{2}}{\sigma ^{2}}}<q_{1-{\frac {\alpha }{2}}}\right)=1-\alpha ,} where q p {\displaystyle q_{p}}

1769-411: The expected value (the average) of random variable X with density f ( x ) : μ ≡ E ⁡ [ X ] = ∫ − ∞ + ∞ x f ( x ) d x {\displaystyle \mu \equiv \operatorname {E} [X]=\int _{-\infty }^{+\infty }xf(x)\,\mathrm {d} x} The standard deviation σ of X

1830-442: The mean (also called the expected value ) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or Std Dev , and is most commonly represented in mathematical texts and equations by the lowercase Greek letter σ (sigma), for

1891-599: The standard deviation of the sample (considered as the entire population), and is defined as follows: s N = 1 N ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} where { x 1 , x 2 , … , x N } {\displaystyle \{x_{1},\,x_{2},\,\ldots ,\,x_{N}\}} are

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1952-524: The 8th century BC. At that time a simplified three-stroke version, omitting the lowermost stroke, was already found in Western Greek alphabets, and was incorporated into classical Etruscan and Oscan , as well as in the earliest Latin epigraphy (early Latin S ), such as the Duenos inscription . The alternation between three and four (and occasionally more than four) strokes was also adopted into

2013-476: The Culture Fair Intelligence Test on the fluid intelligence factor indicates that the CFIT does, in fact, have a reasonably high direct concept validity with respect to the concept of fluid intelligence. The Culture Fair Intelligence Test was found to load more highly on a "General Intelligence" factor than on an "Achievement" factor, which is consistent with the concept that the CFIT is a measure of "fluid" rather than "crystallized" intelligence. Convergent Validity

2074-459: The above variations of lunate sigma are encoded as U+03F9 Ϲ GREEK CAPITAL LUNATE SIGMA SYMBOL ; U+03FD Ͻ GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL , U+03FE Ͼ GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL , and U+03FF Ͽ GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL . Sigma was adopted in the Old Italic alphabets beginning in

2135-497: The approximation: σ ^ = 1 N − 1.5 − 1 4 γ 2 ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5-{\frac {1}{4}}\gamma _{2}}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} where γ 2 denotes

2196-411: The bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation. If the biased sample variance (the second central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of

2257-1155: The deviations of each data point from the mean, and square the result of each: ( 2 − 5 ) 2 = ( − 3 ) 2 = 9 ( 5 − 5 ) 2 = 0 2 = 0 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 5 − 5 ) 2 = 0 2 = 0 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 7 − 5 ) 2 = 2 2 = 4 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 9 − 5 ) 2 = 4 2 = 16. {\displaystyle {\begin{array}{lll}(2-5)^{2}=(-3)^{2}=9&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(7-5)^{2}=2^{2}=4\\(4-5)^{2}=(-1)^{2}=1&&(9-5)^{2}=4^{2}=16.\\\end{array}}} The variance

2318-399: The difference between 1 N {\displaystyle {\frac {1}{N}}} and 1 N − 1 {\displaystyle {\frac {1}{N-1}}} becomes smaller. For unbiased estimation of standard deviation , there is no formula that works across all distributions, unlike for mean and variance. Instead, s is used as a basis, and is scaled by

2379-493: The estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by s (possibly with modifiers). Unlike in the case of estimating the population mean of a normal distribution, for which the sample mean is a simple estimator with many desirable properties ( unbiased , efficient , maximum likelihood), there is no single estimator for the standard deviation with all these properties, and unbiased estimation of standard deviation

2440-454: The findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant" , a safeguard against spurious conclusion that is really due to random sampling error. When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to

2501-461: The influence of culture and learning. Some high-IQ societies , such as The Triple Nine Society , accept high scores on the CFIT-III as one of a variety of old and new tests for admission to the society. A combined minimum raw score of 85 on Forms A and B is required for admission. The tests are used by many including Mensa and Intertel , which offer a place in their society to anyone scoring in

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2562-636: The integrals are definite integrals taken for x ranging over the set of possible values of the random variable  X . In the case of a parametric family of distributions , the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters μ and σ , the standard deviation is ( e σ 2 − 1 ) e 2 μ + σ 2 . {\displaystyle {\sqrt {\left(e^{\sigma ^{2}}-1\right)e^{2\mu +\sigma ^{2}}}}.} One can find

2623-423: The normal distribution) almost completely eliminates bias. The formula for the population standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by s N , is known as the uncorrected sample standard deviation , or sometimes

2684-418: The observed values of the sample items, and x ¯ {\displaystyle {\bar {x}}} is the mean value of these observations, while the denominator  N stands for the size of the sample: this is the square root of the sample variance, which is the average of the squared deviations about the sample mean. This is a consistent estimator (it converges in probability to

2745-417: The population excess kurtosis . The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data. The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by

2806-466: The population standard deviation, or the Latin letter s , for the sample standard deviation. The standard deviation of a random variable , sample , statistical population , data set , or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust , than the average absolute deviation . A useful property of the standard deviation

2867-421: The population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator , as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/ N , and thus is most significant for small or moderate sample sizes; for N > 75 {\displaystyle N>75}

2928-456: The population's standard deviation, the result is s N = 1 N ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} Here taking the square root introduces further downward bias, by Jensen's inequality , due to

2989-431: The result of the original formula would be called the sample standard deviation and denoted by s {\textstyle s} instead of σ . {\displaystyle \sigma .} Dividing by n − 1 {\textstyle n-1} rather than by n {\textstyle n} gives an unbiased estimate of the variance of the larger parent population. This

3050-605: The root of σίζω ( sízō , from Proto-Greek *sig-jō 'I hiss'). In handwritten Greek during the Hellenistic period (4th–3rd century BC), the epigraphic form of Σ was simplified into a C-like shape, which has also been found on coins from the 4th century BC onward. This became the universal standard form of sigma during late antiquity and the Middle Ages. Today, it is known as lunate sigma (uppercase Ϲ , lowercase ϲ ), because of its crescent -like shape, and

3111-417: The same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population. In science , it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in

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3172-735: The sampling distribution of the sample standard deviation follows a (scaled) chi distribution , and the correction factor is the mean of the chi distribution. An approximation can be given by replacing N  − 1 with N  − 1.5 , yielding: σ ^ = 1 N − 1.5 ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} The error in this approximation decays quadratically (as ⁠ 1 / N ⁠ ), and it

3233-653: The square root is a nonlinear function which does not commute with the expectation, i.e. often E [ X ] ≠ E [ X ] {\textstyle E[{\sqrt {X}}]\neq {\sqrt {E[X]}}} ), yielding the corrected sample standard deviation, denoted by s: s = 1 N − 1 ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} As explained above, while s

3294-713: The square root's being a concave function . The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question. An unbiased estimator for the variance is given by applying Bessel's correction , using N  − 1 instead of N to yield the unbiased sample variance, denoted s : s 2 = 1 N − 1 ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}.} This estimator

3355-497: The standard deviation is the square root of the variance of X . The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because

3416-423: The standard deviation of an entire population in cases (such as standardized testing ) where every member of a population is sampled. In cases where that cannot be done, the standard deviation σ is estimated by examining a random sample taken from the population and computing a statistic of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an estimator , and

3477-560: The standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 69 inches , with a standard deviation of around 3 inches . This means that most men (about 68%, assuming a normal distribution ) have a height within 3 inches of the mean ( 66–72 inches ) – one standard deviation – and almost all men (about 95%) have

3538-399: The tails diminish quickly enough. The Pareto distribution with parameter α ∈ ( 1 , 2 ] {\displaystyle \alpha \in (1,2]} has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The Cauchy distribution has neither a mean nor a standard deviation. In the case where X takes random values from

3599-458: The top 2% and in the top 1% IQ scores respectively. Direct concept validity (sometimes called construct validity ) refers to the degree to which a certain scale correlates with the concept or construct (i.e., source trait) which it purports to measure. Concept validity is thus measured by correlating the scale with the pure factor and this can only be carried out by performing a methodologically sound factor analysis . The relatively high loading of

3660-684: The values have different probabilities, let x 1 have probability p 1 , x 2 have probability p 2 , ..., x N have probability p N . In this case, the standard deviation will be σ = ∑ i = 1 N p i ( x i − μ ) 2 ,  where  μ = ∑ i = 1 N p i x i . {\displaystyle \sigma ={\sqrt {\sum _{i=1}^{N}p_{i}(x_{i}-\mu )^{2}}},{\text{ where }}\mu =\sum _{i=1}^{N}p_{i}x_{i}.} The standard deviation of

3721-542: The word-final sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter. The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter [REDACTED] ( shin ). Sigma's original name may have been san , but due to the complicated early history of the Greek epichoric alphabets , san came to be identified as

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