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62-487: Intertel is open to those who have scored at or above the 99th percentile , or the top one percent, on a standardized test of intelligence. It has been identified as one of the notable high-IQ societies established since the late 1960s with admissions requirements that are stricter and more exclusive than Mensa. Intertel was founded in 1966 by Ralph Haines, following the example of Mensa founders Roland Berrill and Lancelot Ware , who wanted to create an association adapted to

124-603: 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 }

186-404: A constant that is a function of the sample size N : There is the additional requirement that the midpoint of the range ( 1 , N ) {\displaystyle (1,N)} , corresponding to the median , occur at p = 0.5 {\displaystyle p=0.5} : and our revised function now has just one degree of freedom, looking like this: The second way in which

248-405: 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

310-758: 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

372-546: 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

434-410: A limit is too high or low. In finance, value at risk is a standard measure to assess (in a model-dependent way) the quantity under which the value of the portfolio is not expected to sink within a given period of time and given a confidence value. There are many formulas or algorithms for a percentile score. Hyndman and Fan identified nine and most statistical and spreadsheet software use one of

496-508: A one-to-one correspondence in the wider region. One author has suggested a choice of C = 1 2 ( 1 + ξ ) {\displaystyle C={\tfrac {1}{2}}(1+\xi )} where ξ is the shape of the Generalized extreme value distribution which is the extreme value limit of the sampled distribution. (Sources: Matlab "prctile" function, ) where Furthermore, let The inverse relationship

558-400: A score from the distribution, although compared to interpolation methods, results can be a bit crude. The Nearest-Rank Methods table shows the computational steps for exclusive and inclusive methods. Interpolation methods, as the name implies, can return a score that is between scores in the distribution. Algorithms used by statistical programs typically use interpolation methods, for example,

620-420: A subdivision into 100 groups. The 25th percentile is also known as the first quartile ( Q 1 ), the 50th percentile as the median or second quartile ( Q 2 ), and the 75th percentile as the third quartile ( Q 3 ). For example, the 50th percentile (median) is the score below (or at or below , depending on the definition) which 50% of the scores in the distribution are found. A related quantity

682-542: 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

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744-424: 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

806-730: 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

868-778: 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

930-394: 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}

992-417: 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

1054-838: 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

1116-412: 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}

1178-437: 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

1240-404: Is no standard definition of percentile; however, all definitions yield similar results when the number of observations is very large and the probability distribution is continuous. In the limit, as the sample size approaches infinity, the 100 p percentile (0< p <1) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p , as p approximates

1302-457: 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

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1364-436: Is plotted along an axis scaled to standard deviations , or sigma ( σ {\displaystyle \sigma } ) units. Mathematically, the normal distribution extends to negative infinity on the left and positive infinity on the right. Note, however, that only a very small proportion of individuals in a population will fall outside the −3 σ to +3 σ range. For example, with human heights very few people are above

1426-506: Is restricted to a narrower region: [Source: Some software packages, including NumPy and Microsoft Excel (up to and including version 2013 by means of the PERCENTILE.INC function). Noted as an alternative by NIST . ] Note that the x ↔ p {\displaystyle x\leftrightarrow p} relationship is one-to-one for p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} ,

1488-436: Is the percentile rank of a score, expressed in percent , which represents the fraction of scores in its distribution that are less than it, an exclusive definition. Percentile scores and percentile ranks are often used in the reporting of test scores from norm-referenced tests , but, as just noted, they are not the same. For percentile ranks, a score is given and a percentage is computed. Percentile ranks are exclusive: if

1550-402: 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

1612-440: Is to use linear interpolation between adjacent ranks. All of the following variants have the following in common. Given the order statistics we seek a linear interpolation function that passes through the points ( v i , i ) {\displaystyle (v_{i},i)} . This is simply accomplished by where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } uses

1674-490: Is undefined, it does not need to be because it is multiplied by x mod 1 = 0 {\displaystyle x{\bmod {1}}=0} .) As we can see, x is the continuous version of the subscript i , linearly interpolating v between adjacent nodes. There are two ways in which the variant approaches differ. The first is in the linear relationship between the rank x , the percent rank P = 100 p {\displaystyle P=100p} , and

1736-868: 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

1798-1910: 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

1860-861: 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

1922-632: 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

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1984-850: 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

2046-450: The floor function to represent the integral part of positive x , whereas x mod 1 {\displaystyle x{\bmod {1}}} uses the mod function to represent its fractional part (the remainder after division by 1). (Note that, though at the endpoint x = N {\displaystyle x=N} , v ⌊ x ⌋ + 1 {\displaystyle v_{\lfloor x\rfloor +1}}

2108-513: The gifted needs without any specific restriction of admission (with the exception of a minimum IQ ). Intertel thus became the second oldest organization of this kind, Mensa being the first. Aligned with one of the goals stated in its constitution, Intertel's members participate in research on high intelligence. In 1978, Intertel established the international "Hollingworth Award" in memory of renowned psychologist Leta Stetter Hollingworth , who specialized in research on gifted children. This award

2170-406: 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

2232-407: The "INC" version, the second variant, does not; in fact, any number smaller than 1 N + 1 {\displaystyle {\frac {1}{N+1}}} is also excluded and would cause an error.) The inverse is restricted to a narrower region: In addition to the percentile function, there is also a weighted percentile , where the percentage in the total weight is counted instead of

2294-403: The +3 σ height level. Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3 σ is the 0.13th percentile, −2 σ the 2.28th percentile, −1 σ the 15.87th percentile, 0 σ the 50th percentile (both the mean and median of the distribution), +1 σ the 84.13th percentile, +2 σ

2356-405: The 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then bills at the nearest rate. In this way, infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time,

2418-400: The 97.72nd percentile, and +3 σ the 99.87th percentile. This is related to the 68–95–99.7 rule or the three-sigma rule. Note that in theory the 0th percentile falls at negative infinity and the 100th percentile at positive infinity, although in many practical applications, such as test results, natural lower and/or upper limits are enforced. When ISPs bill "burstable" internet bandwidth ,

2480-546: The CDF. This can be seen as a consequence of the Glivenko–Cantelli theorem . Some methods for calculating the percentiles are given below. The methods given in the calculation methods section (below) are approximations for use in small-sample statistics. In general terms, for very large populations following a normal distribution , percentiles may often be represented by reference to a normal curve plot. The normal distribution

2542-466: 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}})} ,

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2604-459: 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

2666-401: 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

2728-431: 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

2790-439: 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 }

2852-408: The list such that no more than P percent of the data is strictly less than the value and at least P percent of the data is less than or equal to that value. This is obtained by first calculating the ordinal rank and then taking the value from the ordered list that corresponds to that rank. The ordinal rank n is calculated using this formula An alternative to rounding used in many applications

2914-446: The methods they describe. Algorithms either return the value of a score that exists in the set of scores (nearest-rank methods) or interpolate between existing scores and are either exclusive or inclusive. The figure shows a 10-score distribution, illustrates the percentile scores that result from these different algorithms, and serves as an introduction to the examples given subsequently. The simplest are nearest-rank methods that return

2976-504: The only one of the three variants with this property; hence the "INC" suffix, for inclusive , on the Excel function. (The primary variant recommended by NIST . Adopted by Microsoft Excel since 2010 by means of PERCENTIL.EXC function. However, as the "EXC" suffix indicates, the Excel version excludes both endpoints of the range of p , i.e., p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} , whereas

3038-412: 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

3100-415: The percentile rank for a specified score is 90%, then 90% of the scores were lower. In contrast, for percentiles a percentage is given and a corresponding score is determined, which can be either exclusive or inclusive. The score for a specified percentage (e.g., 90th) indicates a score below which (exclusive definition) or at or below which (inclusive definition) other scores in the distribution fall. There

3162-471: The percentile.exc and percentile.inc functions in Microsoft Excel. The Interpolated Methods table shows the computational steps. One definition of percentile, often given in texts, is that the P -th percentile ( 0 < P ≤ 100 ) {\displaystyle (0<P\leq 100)} of a list of N ordered values (sorted from least to greatest) is the smallest value in

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3224-1207: 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

3286-413: The same unit of measurement as the input scores, not in percent ; for example, if the scores refer to human weight , the corresponding percentiles will be expressed in kilograms or pounds. In the limit of an infinite sample size , the percentile approximates the percentile function , the inverse of the cumulative distribution function . Percentiles are a type of quantiles , obtained adopting

3348-581: 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

3410-783: 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

3472-600: 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

3534-520: 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 } ,

3596-562: The sum of the weights. Then the formulas above are generalized by taking or and The 50% weighted percentile is known as the weighted median . 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 ,

3658-406: The total number. There is no standard function for a weighted percentile. One method extends the above approach in a natural way. Suppose we have positive weights w 1 , w 2 , w 3 , … , w N {\displaystyle w_{1},w_{2},w_{3},\dots ,w_{N}} associated, respectively, with our N sorted sample values. Let

3720-406: The usage is below this amount: so, the remaining 5% of the time, the usage is above that amount. Physicians will often use infant and children's weight and height to assess their growth in comparison to national averages and percentiles which are found in growth charts . The 85th percentile speed of traffic on a road is often used as a guideline in setting speed limits and assessing whether such

3782-404: The variants differ is in the definition of the function near the margins of the [ 0 , 1 ] {\displaystyle [0,1]} range of p : f ( p , N ) {\displaystyle f(p,N)} should produce, or be forced to produce, a result in the range [ 1 , N ] {\displaystyle [1,N]} , which may mean the absence of

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3844-533: Was annually presented until at least 1993, first sponsored by Intertel and then the Intertel Foundation. Percentile In statistics , a k -th percentile , also known as percentile score or centile , is a score below which a given percentage k of scores in its frequency distribution falls (" exclusive " definition) or a score at or below which a given percentage falls (" inclusive " definition). Percentiles are expressed in

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