Misplaced Pages

#55944

52-473: In vehicle design, the H-point is also measured relative to other features, for example the h-point to vehicle floor ( H30 ) or h-point to pavement ( H5 ). In other words, a vehicle said to have a "high H-point" may have an H-point that is "high" relative to the vehicle floor, the road surface, or both. Technically, the H-point measurement uses the hip joint of a 50th percentile male occupant, viewed laterally, and

104-402: A i {\displaystyle a_{i}} , i = 1 , … , m {\displaystyle i=1,\ldots ,m} are the model parameters. The parameters a i {\displaystyle a_{i}} must be selected so that Q ( p ) {\displaystyle Q(p)} is a quantile function. Two four-parametric quantile mixtures,

156-402: A normal distribution , percentiles may often be represented by reference to a normal curve plot. The normal distribution 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

208-432: A random variable X , the quantile function Q : [ 0 , 1 ] → R {\displaystyle Q\colon [0,1]\to \mathbb {R} } maps its input p to a threshold value x so that the probability of X being less or equal than x is p . In terms of the distribution function F , the quantile function Q returns the value x such that which can be written as inverse of

260-404: 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

312-414: A description of the H-point machine. Occupant posture-prediction models are used in computer simulations and form the basis for crash test dummy positioning. Regulatory definition: For the purpose of U.S. regulation and GTRs (Global Technical Regulations)—and for clear communication in safety and seating design—the H-point is defined as the actual hip point of the seated crash test dummy itself, whereas

364-416: 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 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

416-505: A given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The demands of simulation methods, for example in modern computational finance , are focusing increasing attention on methods based on quantile functions, as they work well with multivariate techniques based on either copula or quasi-Monte-Carlo methods and Monte Carlo methods in finance . The evaluation of quantile functions often involves numerical methods , such as

468-410: 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

520-502: 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

572-625: A range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function (after the percentile ), percent-point function , inverse cumulative distribution function (after the cumulative distribution function or c.d.f.) or inverse distribution function . With reference to a continuous and strictly monotonic cumulative distribution function (c.d.f.) F X : R → [ 0 , 1 ] {\displaystyle F_{X}\colon \mathbb {R} \to [0,1]} of

SECTION 10

#1732869229056

624-422: A result, approximate representations are usually used. Thorough composite rational and polynomial approximations have been given by Wichura and Acklam. Non-composite rational approximations have been developed by Shaw. A non-linear ordinary differential equation for the normal quantile, w ( p ), may be given. It is with the centre (initial) conditions This equation may be solved by several methods, including

676-400: 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,

728-421: 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 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,

780-398: 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 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

832-489: Is highly relevant to national and international vehicle design standards such as global technical regulations (GTR) . For example, a vehicle design standard known as the Society of Automotive Engineers (SAE) J1100 Interior Measurement Index sets parameters for such measurements as H30 (H-point to vehicle floor); H5 (H-point to pavement surface), H61 (H-point to interior ceiling) and H25 (H-point to windowsill). As with

884-402: Is one way of prescribing a probability distribution, and it is an alternative to the probability density function (pdf) or probability mass function , the cumulative distribution function (cdf) and the characteristic function . The quantile function, Q , of a probability distribution is the inverse of its cumulative distribution function F . The derivative of the quantile function, namely

936-500: 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]} ,

988-451: Is the unique function satisfying the Galois inequalities If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have In general, even though the distribution function F may fail to possess a left or right inverse , the quantile function Q behaves as an "almost sure left inverse" for the distribution function, in

1040-440: 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

1092-490: 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

SECTION 20

#1732869229056

1144-857: The Ford Five Hundred and the Fiat 500L . Sports cars and vehicles with higher aerodynamic considerations, by contrast, may employ lower H-points relative to the road surface. When an automobile features progressively higher H-points at each successive seating row, the seating is called stadium seating , as in the Dodge Journey , and Ford Flex . Vehicle interior ergonomics are integral to an automotive design education. The Society of Automotive Engineers (SAE) has adopted tools for vehicle design, including statistical models for predicting driver eye location and seat position as well as an H-point mannequin for measuring seats and interior package geometry. See SAE J826 for

1196-468: The R-point (or SgRP, seating reference point) is the theoretical hip point used by manufacturers when designing a vehicle—and more specifically describes the relative location of the seated dummy's hip point when the seat is set in the rearmost and lowermost seating position. Percentile In statistics , a k -th percentile , also known as percentile score or centile , is a score below which

1248-450: 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}}

1300-411: 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 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

1352-436: The quantile density function , is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. Consider a statistical application where a user needs to know key percentage points of a given distribution. For example, they require the median and 25% and 75% quartiles as in the example above or 5%, 95%, 2.5%, 97.5% levels for other applications such as assessing

1404-512: The statistical significance of an observation whose distribution is known; see the quantile entry. Before the popularization of computers, it was not uncommon for books to have appendices with statistical tables sampling the quantile function. Statistical applications of quantile functions are discussed extensively by Gilchrist. Monte-Carlo simulations employ quantile functions to produce non-uniform random or pseudorandom numbers for use in diverse types of simulation calculations. A sample from

1456-407: 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

1508-583: The 100 p percentile (0< p <1) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p , as p approximates 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

1560-562: 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 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 ,

1612-405: 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,

H-point - Misplaced Pages Continue

1664-400: The c.d.f. In the general case of distribution functions that are not strictly monotonic and therefore do not permit an inverse c.d.f. , the quantile is a (potentially) set valued functional of a distribution function F , given by the interval It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of F ) Here we capture the fact that

1716-511: The cases of the normal , Student , beta and gamma distributions have been given and solved. The normal distribution is perhaps the most important case. Because the normal distribution is a location-scale family , its quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal distribution, known as the probit function. Unfortunately, this function has no closed-form representation using basic algebraic functions; as

1768-419: The classical power series approach. From this solutions of arbitrarily high accuracy may be developed (see Steinbrecher and Shaw, 2008). This has historically been one of the more intractable cases, as the presence of a parameter, ν, the degrees of freedom, makes the use of rational and other approximations awkward. Simple formulas exist when the ν = 1, 2, 4 and the problem may be reduced to

1820-535: The exponential distribution above, which is one of the few distributions where a closed-form expression can be found (others include the uniform , the Weibull , the Tukey lambda (which includes the logistic ) and the log-logistic ). When the cdf itself has a closed-form expression, one can always use a numerical root-finding algorithm such as the bisection method to invert the cdf. Other methods rely on an approximation of

1872-586: The front and back seats. By the early 2000s there had been a global trend toward higher H-points relative to both the road surface and the vehicle's interior floor. Referring to the trend in a 2004 article, The Wall Street Journal noted an advantage: "the higher the H-Point, the higher you ride in the car, and in some cases, the more comfortable you feel behind the wheel". Buses, minivans , SUVs and CUVs generally have higher H-points than sedans , though certain sedans feature higher H-points than most, for example

1924-599: The inverse via interpolation techniques. Further algorithms to evaluate quantile functions are given in the Numerical Recipes series of books. Algorithms for common distributions are built into many statistical software packages. General methods to numerically compute the quantile functions for general classes of distributions can be found in the following libraries: Quantile functions may also be characterized as solutions of non-linear ordinary and partial differential equations . The ordinary differential equations for

1976-408: 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

2028-479: The location of other automotive design "hard points," the H-point has major ramifications in the overall vehicle design, including roof height, aerodynamics, handling (especially at highway speeds), visibility (both within the vehicle and from the vehicle into traffic), seating comfort, driver fatigue, ease of entry and exit, interior packaging, safety , restraint and airbag design and collision performance. As an example, higher H-points can provide more legroom, both in

2080-446: 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

2132-480: The normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture, are presented by Karvanen. The non-linear ordinary differential equation given for normal distribution is a special case of that available for any quantile function whose second derivative exists. In general the equation for a quantile, Q ( p ), may be given. It is augmented by suitable boundary conditions, where and ƒ ( x )

H-point - Misplaced Pages Continue

2184-500: 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

2236-464: 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

2288-408: The quantile function returns the minimum value of x from amongst all those values whose c.d.f value exceeds p , which is equivalent to the previous probability statement in the special case that the distribution is continuous. Note that the infimum function can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing. The quantile

2340-444: 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 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

2392-571: The sense that For example, the cumulative distribution function of Exponential( λ ) (i.e. intensity λ and expected value ( mean ) 1/ λ ) is The quantile function for Exponential( λ ) is derived by finding the value of Q for which 1 − e − λ Q = p {\displaystyle 1-e^{-\lambda Q}=p} : for 0 ≤  p  < 1. The quartiles are therefore: Quantile functions are used in both statistical applications and Monte Carlo methods . The quantile function

2444-646: The solution of a polynomial when ν is even. In other cases the quantile functions may be developed as power series. The simple cases are as follows: where and In the above the "sign" function is +1 for positive arguments, −1 for negative arguments and zero at zero. It should not be confused with the trigonometric sine function. Analogously to the mixtures of densities , distributions can be defined as quantile mixtures where Q i ( p ) {\displaystyle Q_{i}(p)} , i = 1 , … , m {\displaystyle i=1,\ldots ,m} are quantile functions and

2496-415: 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 . Percentile function In probability and statistics , the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value. Intuitively, the quantile function associates with

2548-449: 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 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

2600-406: 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

2652-406: 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

SECTION 50

#1732869229056

2704-404: 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

#55944