50-402: BMAG may refer to: British Mensa Annual Gathering Birmingham Museum & Art Gallery Berliner Maschinenbau AG - German manufacturer of locomotives Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title BMAG . If an internal link led you here, you may wish to change
100-433: A mean score of 100 with a standard deviation of 15; the 98th-percentile score under these conditions is 130.8, assuming a normal distribution . Most national groups test using well-established IQ test batteries, but American Mensa and German Mensa each developed their own application tests, which they administer and monitor themselves. However, American Mensa does not provide a score comparable to scores on other tests;
150-603: A normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable . The general form of its probability density function is f ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.} The parameter μ {\textstyle \mu }
200-405: A fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed. A normal distribution is sometimes informally called a bell curve . However, many other distributions are bell-shaped (such as
250-758: A generic normal distribution with density f {\textstyle f} , mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the cumulative distribution function is F ( x ) = Φ ( x − μ σ ) = 1 2 [ 1 + erf ( x − μ σ 2 ) ] . {\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]\,.} The complement of
300-488: A high IQ. The society was ostensibly to be non-political in its aims and free from all other social distinctions, such as race and religion. However, Berrill and Ware were both disappointed with the resulting society. Berrill had intended Mensa as "an aristocracy of the intellect" and was unhappy that the majority of members came from working or lower-class homes, while Ware said: "I do get disappointed that so many members spend so much time solving puzzles." American Mensa
350-546: A known approximate solution, x 0 {\textstyle x_{0}} , to the desired Φ ( x ) {\textstyle \Phi (x)} . x 0 {\textstyle x_{0}} may be a value from a distribution table, or an intelligent estimate followed by a computation of Φ ( x 0 ) {\textstyle \Phi (x_{0})} using any desired means to compute. Use this value of x 0 {\textstyle x_{0}} and
400-624: A separate charitable U.S. corporation, edits and publishes its own Mensa Research Journal , in which both Mensans and non-Mensans are published on various topics surrounding the concept and measure of intelligence. Mensa has many events for members, from the local to the international level. Several countries hold a large event called the Annual Gathering (AG). It is held in a different city every year, with speakers, dances, leadership workshops, children's events, games, and other activities. The American and Canadian AGs are usually held during
450-542: A variance of 1 2 {\displaystyle {\frac {1}{2}}} , and Stephen Stigler once defined the standard normal as φ ( z ) = e − π z 2 , {\displaystyle \varphi (z)=e^{-\pi z^{2}},} which has a simple functional form and a variance of σ 2 = 1 2 π . {\textstyle \sigma ^{2}={\frac {1}{2\pi }}.} Every normal distribution
500-478: Is Latin for ' table ', as is symbolised in the organisation's logo, and was chosen to demonstrate the round-table nature of the organisation: the coming together of equals. Australian Roland Berrill , and Lancelot Ware , a British scientist and lawyer, founded Mensa at Lincoln College , in Oxford , England in 1946, with the intention of forming a society for the most intelligent, with the only qualification being
550-424: Is a normal deviate with parameters μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^{2}} , then this X {\textstyle X} distribution can be re-scaled and shifted via the formula Z = ( X − μ ) / σ {\textstyle Z=(X-\mu )/\sigma } to convert it to
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#1732801827722600-730: Is a version of the standard normal distribution, whose domain has been stretched by a factor σ {\textstyle \sigma } (the standard deviation) and then translated by μ {\textstyle \mu } (the mean value): f ( x ∣ μ , σ 2 ) = 1 σ φ ( x − μ σ ) . {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)\,.} The probability density must be scaled by 1 / σ {\textstyle 1/\sigma } so that
650-778: Is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution. Normal distributions form an exponential family with natural parameters θ 1 = μ σ 2 {\textstyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}} and θ 2 = − 1 2 σ 2 {\textstyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}} , and natural statistics x and x . The dual expectation parameters for normal distribution are η 1 = μ and η 2 = μ + σ . The cumulative distribution function (CDF) of
700-394: Is also used quite often. The normal distribution is often referred to as N ( μ , σ 2 ) {\textstyle N(\mu ,\sigma ^{2})} or N ( μ , σ 2 ) {\textstyle {\mathcal {N}}(\mu ,\sigma ^{2})} . Thus when a random variable X {\textstyle X}
750-417: Is called a normal deviate . Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem . It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance
800-838: Is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z)={\frac {e^{\frac {-z^{2}}{2}}}{\sqrt {2\pi }}}\,.} The variable z {\textstyle z} has a mean of 0 and a variance and standard deviation of 1. The density φ ( z ) {\textstyle \varphi (z)} has its peak 1 2 π {\textstyle {\frac {1}{\sqrt {2\pi }}}} at z = 0 {\textstyle z=0} and inflection points at z = + 1 {\textstyle z=+1} and z = − 1 {\textstyle z=-1} . Although
850-412: Is equivalent to saying that the standard normal distribution Z {\textstyle Z} can be scaled/stretched by a factor of σ {\textstyle \sigma } and shifted by μ {\textstyle \mu } to yield a different normal distribution, called X {\textstyle X} . Conversely, if X {\textstyle X}
900-437: Is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors , often have distributions that are nearly normal. Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of
950-457: Is normally distributed with mean μ {\textstyle \mu } and standard deviation σ {\textstyle \sigma } , one may write X ∼ N ( μ , σ 2 ) . {\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).} Some authors advocate using the precision τ {\textstyle \tau } as
1000-402: Is the mean or expectation of the distribution (and also its median and mode ), while the parameter σ 2 {\textstyle \sigma ^{2}} is the variance . The standard deviation of the distribution is σ {\textstyle \sigma } (sigma). A random variable with a Gaussian distribution is said to be normally distributed , and
1050-868: Is very close to zero, and simplifies formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution . Alternatively, the reciprocal of the standard deviation τ ′ = 1 / σ {\textstyle \tau '=1/\sigma } might be defined as the precision , in which case the expression of the normal distribution becomes f ( x ) = τ ′ 2 π e − ( τ ′ ) 2 ( x − μ ) 2 / 2 . {\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.} According to Stigler, this formulation
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#17328018277221100-1910: The e a x 2 {\textstyle e^{ax^{2}}} family of derivatives may be used to easily construct a rapidly converging Taylor series expansion using recursive entries about any point of known value of the distribution, Φ ( x 0 ) {\textstyle \Phi (x_{0})} : Φ ( x ) = ∑ n = 0 ∞ Φ ( n ) ( x 0 ) n ! ( x − x 0 ) n , {\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,} where: Φ ( 0 ) ( x 0 ) = 1 2 π ∫ − ∞ x 0 e − t 2 / 2 d t Φ ( 1 ) ( x 0 ) = 1 2 π e − x 0 2 / 2 Φ ( n ) ( x 0 ) = − ( x 0 Φ ( n − 1 ) ( x 0 ) + ( n − 2 ) Φ ( n − 2 ) ( x 0 ) ) , n ≥ 2 . {\displaystyle {\begin{aligned}\Phi ^{(0)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt\\\Phi ^{(1)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}\\\Phi ^{(n)}(x_{0})&=-\left(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0})\right),&n\geq 2\,.\end{aligned}}} An application for
1150-861: The Q {\textstyle Q} -function, all of which are simple transformations of Φ {\textstyle \Phi } , are also used occasionally. The graph of the standard normal cumulative distribution function Φ {\textstyle \Phi } has 2-fold rotational symmetry around the point (0,1/2); that is, Φ ( − x ) = 1 − Φ ( x ) {\textstyle \Phi (-x)=1-\Phi (x)} . Its antiderivative (indefinite integral) can be expressed as follows: ∫ Φ ( x ) d x = x Φ ( x ) + φ ( x ) + C . {\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.} The cumulative distribution function of
1200-910: The American Independence Day (4 July) or Canada Day (1 July) weekends respectively. Since 1990, American Mensa has sponsored the annual Mensa Mind Games competition, at which the Mensa Select award is given to five board games that are "original, challenging, and well designed". In Europe, since 2008 international meetings have been held under the name EMAG (European Mensa Annual Gathering), starting in Cologne that year. The next meetings were in Utrecht (2009), Prague (2010), Paris (2011), Stockholm (2012), Bratislava (2013), Zürich (2014), Berlin (2015), Kraków (2016), Barcelona (2017), Belgrade (2018) and Ghent (2019). The 2020 event
1250-632: The Cauchy , Student's t , and logistic distributions). (For other names, see Naming .) The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution . The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution . This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it
1300-850: The double factorial . An asymptotic expansion of the cumulative distribution function for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion . A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation: Φ ( x ) ≈ 1 2 + 1 2 π ∑ k = 0 n ( − 1 ) k x ( 2 k + 1 ) 2 k k ! ( 2 k + 1 ) . {\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}\,.} The recursive nature of
1350-406: The integral is still 1. If Z {\textstyle Z} is a standard normal deviate , then X = σ Z + μ {\textstyle X=\sigma Z+\mu } will have a normal distribution with expected value μ {\textstyle \mu } and standard deviation σ {\textstyle \sigma } . This
1400-749: The American Mensa Bulletin , the British Mensa Magazine , Serbian MozaIQ , the Australian TableAus , the Mexican El Mensajero , and the French, formerly Contacts , now MensaMag . Aside from national publications, some local or regional groups have their own newsletters. Mensa International publishes a Mensa World Journal , which "contains views and information about Mensa around
1450-466: The Taylor series expansion above to minimize computations. Repeat the following process until the difference between the computed Φ ( x n ) {\textstyle \Phi (x_{n})} and the desired Φ {\textstyle \Phi } , which we will call Φ ( desired ) {\textstyle \Phi ({\text{desired}})} ,
1500-459: The Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of Φ ( x ) {\textstyle \Phi (x)} , which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution. To solve, select
1550-401: The above Taylor series expansion is to use Newton's method to reverse the computation. That is, if we have a value for the cumulative distribution function , Φ ( x ) {\textstyle \Phi (x)} , but do not know the x needed to obtain the Φ ( x ) {\textstyle \Phi (x)} , we can use Newton's method to find x, and use
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1600-691: The age of two years and four months. In 2018, Mehul Garg became the youngest person in a decade to score the maximum of 162 on the Mensa IQ test. American Mensa's oldest member is 102, and British Mensa had a member aged 103. According to American Mensa's generational classifications and published demographics (as of 2023), its membership is 8 percent from the Silent generation (born 1924–1942), 37 percent Baby Boomers (born 1943–1960), 30 percent Gen-X (born 1961–1981), 10 percent Millennial (born 1982–2000), 12 percent Generation Z (born 2001–2020) and
1650-590: The children themselves; many offer activities, resources, and newsletters specifically geared toward gifted children and their parents. Kashe Quest , the youngest member of American Mensa; Adam Kirby, the youngest member of British Mensa; and several Australian Mensa members joined at age two. Elise Tan-Roberts of the UK and Miranda Elise Margolis of the US are the youngest people ever to join Mensa, having gained full membership at
1700-431: The density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss , for example, once defined the standard normal as φ ( z ) = e − z 2 π , {\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},} which has
1750-439: The distribution then becomes f ( x ) = τ 2 π e − τ ( x − μ ) 2 / 2 . {\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.} This choice is claimed to have advantages in numerical computations when σ {\textstyle \sigma }
1800-563: The largest having over 2,000 members and the smallest having fewer than 100. Members may form special interest groups (SIGs) at international, national, and local levels; these SIGs represent a wide variety of interests, ranging from motorcycle clubs to entrepreneurial co-operations. Some SIGs are associated with various geographic groups, whereas others act independently of official hierarchy. There are also electronic SIGs (eSIGs), which operate primarily as email lists, where members may or may not meet each other in person. The Mensa Foundation,
1850-419: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=BMAG&oldid=932728693 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Mensa International Mensa is the largest and oldest high-IQ society in
1900-520: The monthly publication of American Mensa, and Mensa Magazine , the monthly publication of British Mensa. Individuals who live in a country with a national group join the national group, while those living in countries without a recognized chapter may join Mensa International directly. The largest national groups are: Larger national groups are further subdivided into local groups. For example, American Mensa has 134 local groups, with
1950-412: The parameter defining the width of the distribution, instead of the standard deviation σ {\textstyle \sigma } or the variance σ 2 {\textstyle \sigma ^{2}} . The precision is normally defined as the reciprocal of the variance, 1 / σ 2 {\textstyle 1/\sigma ^{2}} . The formula for
2000-1207: The probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range [ − x , x ] {\textstyle [-x,x]} . That is: erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt\,.} These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions . However, many numerical approximations are known; see below for more. The two functions are closely related, namely Φ ( x ) = 1 2 [ 1 + erf ( x 2 ) ] . {\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]\,.} For
2050-603: The remaining 3 percent other. The American Mensa general membership identifies as 64 percent male, 32 percent female, 3 percent unknown, and less than 1 percent gender non-conforming or other. Normal distribution I ( μ , σ ) = ( 1 / σ 2 0 0 2 / σ 2 ) {\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} In probability theory and statistics ,
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2100-581: The standard normal cumulative distribution function, Q ( x ) = 1 − Φ ( x ) {\textstyle Q(x)=1-\Phi (x)} , is often called the Q-function , especially in engineering texts. It gives the probability that the value of a standard normal random variable X {\textstyle X} will exceed x {\textstyle x} : P ( X > x ) {\textstyle P(X>x)} . Other definitions of
2150-783: The standard normal distribution can be expanded by Integration by parts into a series: Φ ( x ) = 1 2 + 1 2 π ⋅ e − x 2 / 2 [ x + x 3 3 + x 5 3 ⋅ 5 + ⋯ + x 2 n + 1 ( 2 n + 1 ) ! ! + ⋯ ] . {\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]\,.} where ! ! {\textstyle !!} denotes
2200-600: The standard normal distribution, usually denoted with the capital Greek letter Φ {\textstyle \Phi } , is the integral Φ ( x ) = 1 2 π ∫ − ∞ x e − t 2 / 2 d t . {\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\,.} The related error function erf ( x ) {\textstyle \operatorname {erf} (x)} gives
2250-520: The standard normal distribution. This variate is also called the standardized form of X {\textstyle X} . The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter ϕ {\textstyle \phi } ( phi ). The alternative form of the Greek letter phi, φ {\textstyle \varphi } ,
2300-576: The test serves only to qualify a person for membership. In some national groups, a person may take a Mensa-offered test only once, although one may later submit an application with results from a different qualifying test. The Mensa test is also available in some developing countries such as Brazil, India, Indonesia and Pakistan, and societies in developing countries have been growing at a rapid pace. Mensa International consists of around 134,000 members in 100 countries and in 54 national groups. The national groups issue periodicals, such as Mensa Bulletin ,
2350-483: The world". This journal is generally included in each national magazine. The Mensa Foundation publishes the Mensa Research Journal , which "highlights scholarly articles and recent research related to intelligence". Unlike most Mensa publications, this journal is available to non-members. All national Mensa subsidiaries accept children under the age of 18. However, some national Mensas do not test
2400-670: The world. It is a non-profit organisation open to people who score at the 98th percentile or higher on a standardised, supervised IQ or other approved intelligence test. Mensa formally comprises national groups and the umbrella organisation Mensa International , with a registered office in Caythorpe, Lincolnshire , England, which is separate from the British Mensa office in Wolverhampton . The word mensa ( / ˈ m ɛ n s ə / , Latin: [ˈmẽːs̠ä] )
2450-856: Was postponed and took place in 2021 in Brno. The 2022 event was held in Strasbourg, and the 2023 event was held in Rotterdam. In the Asia-Pacific region, there is an Asia-Pacific Mensa Annual Gathering (AMAG), with rotating countries hosting the event. This has included Gold Coast, Australia (2017), Cebu, Philippines (2018), New Zealand (2019), and South Korea (2020). The governing body of Mensa International consists of: All national Mensa groups publish members-only newsletters or magazines, which include articles and columns written by members, and information about upcoming Mensa events. Examples include
2500-701: Was the second major branch of Mensa thanks to the efforts of Margot Seitelman . Mensa's requirement for membership is a score at or above the 98th percentile on certain standardized IQ or other approved intelligence tests, such as the Stanford–Binet Intelligence Scales . The minimum accepted score on the Stanford–Binet is 132, while for the Cattell it is 148, and 130 in the Wechsler tests ( WAIS , WISC ). Most IQ tests are designed to yield
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