1.
Censoring (statistics)
–
In statistics, engineering, economics, and medical research, censoring is a condition in which the value of a measurement or observation is only partially known. For example, suppose a study is conducted to measure the impact of a drug on mortality rate, in such a study, it may be known that an individuals age at death is at least 75 years. Such a situation could occur if the individual withdrew from the study at age 75, censoring also occurs when a value occurs outside the range of a measuring instrument. For example, a bathroom scale might only measure up to 300 pounds, if a 350 lb individual is weighed using the scale, the observer would only know that the individuals weight is at least 300 pounds. The problem of censored data, in which the value of some variable is partially known, is related to the problem of missing data. Censoring should not be confused with the related idea truncation, with censoring, observations result either in knowing the exact value that applies, or in knowing that the value lies within an interval. With truncation, observations never result in values outside a given range, note that in statistics, truncation is not the same as rounding. Left censoring – a data point is below a certain value, interval censoring – a data point is somewhere on an interval between two values. Right censoring – a data point is above a certain value, type I censoring occurs if an experiment has a set number of subjects or items and stops the experiment at a predetermined time, at which point any subjects remaining are right-censored. Random censoring is when each subject has a time that is statistically independent of their failure time. The observed value is the minimum of the censoring and failure times, interval censoring can occur when observing a value requires follow-ups or inspections. Left and right censoring are special cases of interval censoring, with the beginning of the interval at zero or the end at infinity, respectively. Estimation methods for using left-censored data vary, and not all methods of estimation may be applicable to, or the most reliable, a common misconception with time interval data is to class as left censored intervals where the start time is unknown. In these cases we have a bound on the time interval. Special techniques may be used to handle censored data, Tests with specific failure times are coded as actual failures, censored data are coded for the type of censoring and the known interval or limit. Special software programs can conduct a maximum likelihood estimation for summary statistics, confidence intervals, Reliability testing often consists of conducting a test on an item to determine the time it takes for a failure to occur. Sometimes a failure is planned and expected but does not occur, operator error, equipment malfunction, test anomaly, the test result was not the desired time-to-failure but can be used as a time-to-termination. The use of censored data is unintentional but necessary, sometimes engineers plan a test program so that, after a certain time limit or number of failures, all other tests will be terminated

2.
Reliability engineering
–
Reliability engineering is engineering that emphasizes dependability in the lifecycle management of a product. Dependability, or reliability, describes the ability of a system or component to function under stated conditions for a period of time. Reliability may also describe the ability to function at a moment or interval of time. Reliability engineering represents a sub-discipline within systems engineering, Testability, Maintainability and maintenance are often defined as a part of reliability engineering in Reliability Programs. Reliability plays a key role in the cost-effectiveness of systems, Reliability engineering deals with the estimation, prevention and management of high levels of lifetime engineering uncertainty and risks of failure. Although stochastic parameters define and affect reliability, according to some authors on reliability engineering. You cannot really find a cause by only looking at statistics. Reliability engineering relates closely to safety engineering and to safety, in that they use common methods for their analysis. Reliability engineering focuses on costs of failure caused by system downtime, cost of spares, repair equipment, personnel, Safety engineering normally emphasizes not cost, but preserving life and nature, and therefore deals only with particular dangerous system-failure modes. High reliability levels also result from good engineering and from attention to detail, the word reliability can be traced back to 1816, by poet Coleridge. Before World War II the name has been linked mostly to repeatability, a test was considered reliable if the same results would be obtained repeatedly. The development of reliability engineering was here on a path with quality. The modern use of the reliability was defined by the U. S. military in the 1940s, characterizing a product that would operate when expected. In World War II, many reliability issues were due to inherent unreliability of electronics, in 1945, M. A. Miner published the seminal paper titled Cumulative Damage in Fatigue in an ASME journal. The IEEE formed the Reliability Society in 1948, in 1950, on the military side, a group called the Advisory Group on the Reliability of Electronic Equipment, AGREE, was born. The famous military standard 781 was created at that time, around this period also the much-used military handbook 217 was published by RCA and was used for the prediction of failure rates of components. The emphasis on component reliability and empirical research alone slowly decreases, More pragmatic approaches, as used in the consumer industries, are being used. In the 1980s, televisions were made up of solid-state semiconductors

3.
Life insurance
–
Depending on the contract, other events such as terminal illness or critical illness can also trigger payment. The policy holder typically pays a premium, either regularly or as one lump sum, other expenses can also be included in the benefits. Life policies are legal contracts and the terms of the contract describe the limitations of the insured events. Specific exclusions are often written into the contract to limit the liability of the insurer, common examples are claims relating to suicide, fraud, war, riot, and civil commotion. Life-based contracts tend to fall into two categories, Protection policies – designed to provide a benefit, typically a lump sum payment. A common form of a protection policy design is term insurance, investment policies – where the main objective is to facilitate the growth of capital by regular or single premiums. Common forms are whole life, universal life, and variable life policies, an early form of life insurance dates to Ancient Rome, burial clubs covered the cost of members funeral expenses and assisted survivors financially. The first company to offer insurance in modern times was the Amicable Society for a Perpetual Assurance Office, founded in London in 1706 by William Talbot. Each member made a payment per share on one to three shares with consideration to age of the members being twelve to fifty-five. At the end of the year a portion of the contribution was divided among the wives and children of deceased members. The Amicable Society started with 2000 members and he was unsuccessful in his attempts at procuring a charter from the government. His disciple, Edward Rowe Mores, was able to establish the Society for Equitable Assurances on Lives, Mores also gave the name actuary to the chief official - the earliest known reference to the position as a business concern. The first modern actuary was William Morgan, who served from 1775 to 1830, in 1776 the Society carried out the first actuarial valuation of liabilities and subsequently distributed the first reversionary bonus and interim bonus among its members. It also used regular valuations to balance competing interests, the Society sought to treat its members equitably and the Directors tried to ensure that policyholders received a fair return on their investments. Premiums were regulated according to age, and anybody could be admitted regardless of their state of health, the sale of life insurance in the U. S. began in the 1760s. Between 1787 and 1837 more than two dozen life insurance companies were started, but fewer than half a dozen survived. S. The person responsible for making payments for a policy is the policy owner, the owner and insured may or may not be the same person. For example, if Joe buys a policy on his own life, but if Jane, his wife, buys a policy on Joes life, she is the owner and he is the insured

4.
Bathtub curve
–
The bathtub curve is widely used in reliability engineering. It describes a form of the hazard function which comprises three parts, The first part is a decreasing failure rate, known as early failures. The second part is a constant failure rate, known as random failures, the third part is an increasing failure rate, known as wear-out failures. The name is derived from the shape of a bathtub, steep sides. In the mid-life of a product—generally, once it reaches consumers—the failure rate is low, in the late life of the product, the failure rate increases, as age and wear take their toll on the product. Many consumer product life cycles strongly exhibit the bathtub curve, the term Military Specification is often used to describe systems in which the infant mortality section of the bathtub curve has been burned out or removed. This is done mainly for life-critical or system-critical applications as it reduces the possibility of the system failing early in its life. Manufacturers will do this at some cost generally by means similar to environmental stress screening, in reliability engineering, the cumulative distribution function corresponding to a bathtub curve may be analysed using a Weibull chart. Gompertz–Makeham law of mortality Klutke, G. Kiessler, P. C, a critical look at the bathtub curve

5.
Poisson point process
–
In probability, statistics and related fields, a Poisson point process or Poisson process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is defined on the real line. In this setting, it is used, for example, in queueing theory to model events, such as the arrival of customers at a store or phone calls at an exchange. In this setting, the process is used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics. On more abstract spaces, the Poisson point process serves as an object of study in its own right. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, the process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. The process was discovered independently and repeatedly in different settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. The point process depends on a mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings. In the first case, the constant, known as the rate or intensity, is the density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process, depending on the setting, the process has several equivalent definitions as well as definitions of varying generality owing to its many applications and characterizations. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process, despite its different forms and varying generality, the Poisson point process has two key properties. The Poisson point process is related to the Poisson distribution, which implies that the probability of a Poisson random variable N being equal to n is given by, P = Λ n n. E − Λ where n. denotes n factorial and Λ is the single Poisson parameter that is used to define the Poisson distribution. If a Poisson point process is defined on some underlying space and this property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being called a purely or completely random process. For all the instances of the Poisson point process, the two key properties of the Poisson distribution and complete independence play an important role, if a Poisson point process has a constant parameter, say, λ, then it is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the number of Poisson points existing in some bounded region. The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a process, a type of stochastic process

6.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

7.
Statistics
–
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes

8.
Outline of statistics
–
The following outline is provided as an overview and topical guide to statistics, Statistics – collection, analysis, interpretation, and presentation of data. Statistics can be described as all of the following, Academic discipline, one with academic departments, curricula and degrees, national and international societies, scientific field – widely recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature. Such a field will usually be represented by one or more scientific journals, formal science – branch of knowledge concerned with formal systems. Mathematical science – field of science that is mathematical in nature

9.
Probability distribution
–
For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. In more technical terms, the probability distribution is a description of a phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey, a probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of numbers or a higher-dimensional vector space, or it may be a list of non-numerical values, for example. Probability distributions are divided into two classes. A discrete probability distribution can be encoded by a discrete list of the probabilities of the outcomes, on the other hand, a continuous probability distribution is typically described by probability density functions. The normal distribution represents a commonly encountered continuous probability distribution, more complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A probability distribution whose sample space is the set of numbers is called univariate. Important and commonly encountered univariate probability distributions include the distribution, the hypergeometric distribution. The multivariate normal distribution is a commonly encountered multivariate distribution, to define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. For example, the probability that an object weighs exactly 500 g is zero. Continuous probability distributions can be described in several ways, the cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. As probability theory is used in diverse applications, terminology is not uniform. The following terms are used for probability distribution functions, Distribution. Probability distribution, is a table that displays the probabilities of outcomes in a sample. Could be called a frequency distribution table, where all occurrences of outcomes sum to 1. Distribution function, is a form of frequency distribution table. Probability distribution function, is a form of probability distribution table

10.
Mean
–
In mathematics, mean has several different definitions depending on the context. An analogous formula applies to the case of a probability distribution. Not every probability distribution has a mean, see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite, for example, the arithmetic mean of a set of numbers x1, x2. Xn is typically denoted by x ¯, pronounced x bar, if the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean to distinguish it from the population mean. For a finite population, the mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, outside of probability and statistics, a wide range of other notions of mean are often used in geometry and analysis, examples are given below. The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product. X ¯ =1 n For example, the mean of five values,4,36,45,50,75 is,1 /5 =243000005 =30. The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed. AM, GM, and HM satisfy these inequalities, A M ≥ G M ≥ H M Equality holds if, in descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an average. The mean of a set of observations is the average of the values, however, for skewed distributions. For example, mean income is typically skewed upwards by a number of people with very large incomes. By contrast, the income is the level at which half the population is below. The mode income is the most likely income, and favors the larger number of people with lower incomes, the mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. In this context, it is known as the expected value

11.
Arithmetic mean
–
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, the term arithmetic mean is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the mean is used frequently in fields such as economics, sociology, and history. For example, per capita income is the average income of a nations population. While the arithmetic mean is used to report central tendencies, it is not a robust statistic. In a more obscure usage, any sequence of values that form a sequence between two numbers x and y can be called arithmetic means between x and y. The arithmetic mean is the most commonly used and readily understood measure of central tendency, in statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the values of each. For example, let us consider the monthly salary of 10 employees of a firm,2500,2700,2400,2300,2550,2650,2750,2450,2600,2400. The arithmetic mean is 2500 +2700 +2400 +2300 +2550 +2650 +2750 +2450 +2600 +240010 =2530, If the data set is a statistical population, then the mean of that population is called the population mean. If the data set is a sample, we call the statistic resulting from this calculation a sample mean. The arithmetic mean of a variable is denoted by a bar, for example as in x ¯. The arithmetic mean has several properties that make it useful, especially as a measure of central tendency and these include, If numbers x 1, …, x n have mean x ¯, then + ⋯ + =0. The mean is the single number for which the residuals sum to zero. If the arithmetic mean of a population of numbers is desired, the arithmetic mean may be contrasted with the median. The median is defined such that half the values are larger than, and half are smaller than, If elements in the sample data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample 1,2,3,4, the average is 2.5, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as 1,2,4,8,16, in this case, the arithmetic average is 6.2 and the median is 4

12.
Geometric mean
–
In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i. e. for a set of numbers x1, x2. As another example, the mean of the three numbers 4,1, and 1/32 is the cube root of their product, which is 1/2. A geometric mean is used when comparing different items—finding a single figure of merit for these items—when each item has multiple properties that have different numeric ranges. So, a 20% change in environmental sustainability from 4 to 4.8 has the effect on the geometric mean as a 20% change in financial viability from 60 to 72. The geometric mean can be understood in terms of geometry, the geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. The geometric mean applies only to numbers of the same sign, the geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. The above figure uses capital pi notation to show a series of multiplications. For example, in a set of four numbers, the product of 1 ×2 ×3 ×4 is 24, note that the exponent 1 / n on the left side is equivalent to the taking nth root. For example,241 /4 =244, the geometric mean of a data set is less than the data sets arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the mean, a mixture of the two which always lies in between. The geometric mean can also be expressed as the exponential of the mean of logarithms. This is sometimes called the log-average and this is less likely to occur with the sum of the logarithms for each number. Instead, the mean is simply 1 n, where n is the number of steps from the initial to final state. If the values are a 0, …, a n and this is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources. In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of time of computer programs. However, by presenting appropriately normalized values and using the arithmetic mean, however, this reasoning has been questioned

13.
Harmonic mean
–
In mathematics, the harmonic mean is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired, the harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the mean of 1,2. The third formula in the equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. From the following formula, H = n ⋅ ∏ j =1 n x j ∑ i =1 n. it is apparent that the harmonic mean is related to the arithmetic and geometric means. Thus, the harmonic mean cannot be arbitrarily large by changing some values to bigger ones. The harmonic mean is one of the three Pythagorean means, the arithmetic mean is often mistakenly used in places calling for the harmonic mean. In the speed example below for instance, the mean of 50 is incorrect. The harmonic mean is related to the other Pythagorean means, as seen in the formula in the above equation. This can be seen by interpreting the denominator to be the mean of the product of numbers n times. That is, for the first term, we multiply all n numbers except the first, for the second, we multiply all n numbers except the second, and so on. The numerator, excluding the n, which goes with the mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means, the general formula is H = n A = n A. For the special case of just two numbers, x 1 and x 2, the mean can be written H =2 x 1 x 2 x 1 + x 2. In this special case, the mean is related to the arithmetic mean A = x 1 + x 22. Since G A ≤1 by the inequality of arithmetic and geometric means and it also follows that G = A H, meaning the two numbers geometric mean equals the geometric mean of their arithmetic and harmonic means. Three positive numbers H, G, and A are respectively the harmonic, geometric, W n is associated to the dataset x 1. X n, the harmonic mean is defined by H = ∑ i =1 n w i ∑ i =1 n w i x i = −1

14.
Median
–
The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. In simple terms, it may be thought of as the value of a data set. For example, in the set, the median is 6. The median is a commonly used measure of the properties of a set in statistics. The basic advantage of the median in describing data compared to the mean is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a typical value. For example, in understanding statistics like household income or assets which vary greatly, Median income, for example, may be a better way to suggest what a typical income is. The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest, if there is an odd number of numbers, the middle one is picked. For example, consider the set of numbers,1,3,3,6,7,8,9 This set contains seven numbers, the median is the fourth of them, which is 6. If there are a number of observations, then there is no single middle value. For example, in the set,1,2,3,4,5,6,8,9 The median is the mean of the middle two numbers, this is ÷2, which is 4.5. The formula used to find the number of a data set of n numbers is ÷2. This either gives the number or the halfway point between the two middle values. For example, with 14 values, the formula will give 7.5, and you will also be able to find the median using the Stem-and-Leaf Plot. There is no accepted standard notation for the median. In any of these cases, the use of these or other symbols for the needs to be explicitly defined when they are introduced. The median is used primarily for skewed distributions, which it summarizes differently from the arithmetic mean, the median is 2 in this case, and it might be seen as a better indication of central tendency than the arithmetic mean of 4. The widely cited empirical relationship between the locations of the mean and the median for skewed distributions is, however. There are, however, various relationships for the difference between them, see below

15.
Mode (statistics)
–
The mode is the value that appears most often in a set of data. The mode of a probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled, the mode of a continuous probability distribution is the value x at which its probability density function has its maximum value, so the mode is at the peak. Like the statistical mean and median, the mode is a way of expressing, in a single number, the numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions. The mode is not necessarily unique to a distribution, since the probability mass function or probability density function may take the same maximum value at several points x1, x2. The most extreme case occurs in uniform distributions, where all values occur equally frequently, when a probability density function has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal, in symmetric unimodal distributions, such as the normal distribution, the mean, median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample is 6, given the list of data the mode is not unique - the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal. For a sample from a distribution, such as, the concept is unusable in its raw form. The mode is then the value where the histogram reaches its peak, the following MATLAB code example computes the mode of a sample, The algorithm requires as a first step to sort the sample in ascending order. It then computes the derivative of the sorted list. Unlike mean and median, the concept of mode makes sense for nominal data. For example, taking a sample of Korean family names, one might find that Kim occurs more often than any other name, then Kim would be the mode of the sample. In any voting system where a plurality victory, a single modal value determines the victor. Unlike median, the concept of mode makes sense for any random variable assuming values from a space, including the real numbers. For example, a distribution of points in the plane will typically have a mean and a mode, the median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median, for the remainder, the assumption is that we have a real-valued random variable

16.
Variance
–
The variance has a central role in statistics. It is used in statistics, statistical inference, hypothesis testing, goodness of fit. This makes it a central quantity in numerous such as physics, biology, chemistry, cryptography, economics. The variance of a random variable X is the value of the squared deviation from the mean of X, μ = E . This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, the variance can also be thought of as the covariance of a random variable with itself, Var = Cov . The variance is also equivalent to the second cumulant of a probability distribution that generates X, the variance is typically designated as Var , σ X2, or simply σ2. On computational floating point arithmetic, this equation should not be used, if a continuous distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either. Many other distributions for which the value does exist also do not have a finite variance because the integral in the variance definition diverges. An example is a Pareto distribution whose index k satisfies 1 < k ≤2. e, the normal distribution with parameters μ and σ is a continuous distribution whose probability density function is given by f =12 π σ2 e −22 σ2. In this distribution, E = μ and the variance Var is related with σ via Var = ∫ − ∞ ∞22 π σ2 e −22 σ2 d x = σ2. The role of the distribution in the central limit theorem is in part responsible for the prevalence of the variance in probability. The exponential distribution with parameter λ is a distribution whose support is the semi-infinite interval. Its probability density function is given by f = λ e − λ x, the variance is equal to Var = ∫0 ∞2 λ e − λ x d x = λ −2. So for an exponentially distributed random variable, σ2 = μ2, the Poisson distribution with parameter λ is a discrete distribution for k =0,1,2, …. Its probability mass function is given by p = λ k k, E − λ, and it has expected value μ = λ. The variance is equal to Var = ∑ k =0 ∞ λ k k, E − λ2 = λ, So for a Poisson-distributed random variable, σ2 = μ. The binomial distribution with n and p is a discrete distribution for k =0,1,2, …, n. Its probability mass function is given by p = p k n − k, the variance is equal to Var = ∑ k =0 n p k n − k 2 = n p

17.
Standard deviation
–
In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. The standard deviation of a variable, statistical population, data set. It is algebraically simpler, though in practice less robust, than the absolute deviation. A useful property of the deviation is that, unlike the variance. There are also other measures of deviation from the norm, including mean absolute deviation, in addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a deviation is often called the standard error of the estimate or standard error of the mean when referring to a mean. It is computed as the deviation of all the means that would be computed from that population if an infinite number of samples were drawn. It is very important to note that the deviation of a population. The reported margin of error of a poll is computed from the error of the mean and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment. For a finite set of numbers, the deviation is found by taking the square root of the average of the squared deviations of the values from their average value. For example, the marks of a class of eight students are the eight values,2,4,4,4,5,5,7,9. These eight data points have the mean of 5,2 +4 +4 +4 +5 +5 +7 +98 =5 and this formula is valid only if the eight values with which we began form the complete population. If the values instead were a sample drawn from some large parent population. In that case the result would be called the standard deviation. Dividing by n −1 rather than by n gives an estimate of the variance of the larger parent population. This is known as Bessels correction, as a slightly more complicated real-life example, the average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches

18.
Percentile
–
A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value below which 20% of the observations may be found, the term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. For example, if a score is at the 86th percentile, the 25th percentile is also known as the first quartile, the 50th percentile as the median or second quartile, and the 75th percentile as the third quartile. In general, percentiles and quartiles are specific types of quantiles, when ISPs bill burstable internet bandwidth, 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, the usage is below this amount, just the same, the remaining 5% of the time, the usage is above that amount. Physicians will often use infant and childrens 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 used as a guideline in setting speed limits. The methods given in the Definitions section 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 is plotted along an axis scaled to standard deviations, mathematically, the normal distribution extends to negative infinity on the left and positive infinity on the right. Note, however, that only a small proportion of individuals in a population will fall outside the −3 to +3 range. For example, with human heights very few people are above the +3 sigma height level, Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile and this is related to the 68–95–99.7 rule or the three-sigma rule. There is no definition of percentile, however all definitions yield similar results when the number of observations is very large. Some methods for calculating the percentiles are given below and this is obtained by first calculating the ordinal rank and then taking the value from the ordered list that corresponds to that rank. A percentile calculated using the Nearest Rank method will always be a member of the ordered list. The 100th percentile is defined to be the largest value in the ordered list, Example 1, Consider the ordered list, which contains five data values. What are the 5th, 30th, 40th, 50th and 100th percentiles of this list using the Nearest Rank method

19.
Interquartile range
–
In other words, the IQR is the 1st quartile subtracted from the 3rd quartile, these quartiles can be clearly seen on a box plot on the data. It is an estimator, defined as the 25% trimmed range. The interquartile range is a measure of variability, based on dividing a set into quartiles. Quartiles divide a rank-ordered data set into four equal parts, the values that separate parts are called the first, second, and third quartiles, and they are denoted by Q1, Q2, and Q3, respectively. Unlike total range, the range has a breakdown point of 25%. The IQR is used to box plots, simple graphical representations of a probability distribution. For a symmetric distribution, half the IQR equals the median absolute deviation, the median is the corresponding measure of central tendency. The IQR can be used to identify outliers, the quartile deviation or semi-interquartile range is defined as half the IQR. If P is normally distributed, then the score of the first quartile, z1, is -0.67. However, a distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and -0.67. A better test of normality, such as Q-Q plot would be indicated here, the interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 −1.5 IQR or above Q3 +1.5 IQR, in a boxplot, the highest and lowest occurring value within this limit are indicated by whiskers of the box and any outliers as individual points. Midhinge Interdecile range Robust measures of scale

20.
Shape of a probability distribution
–
The shape of a distribution may be considered either descriptively, using terms such as J-shaped, or numerically, using quantitative measures such as skewness and kurtosis. A bimodal distribution would have two points rather than one. The shape of a distribution is sometimes characterised by the behaviours of the tails, for example, a flat distribution can be said either to have no tails, or to have short tails. A normal distribution is usually regarded as having short tails, while a distribution has exponential tails. Shape parameter List of probability distributions Yule, G. U, an Introduction to the Theory of Statistics, 14th Edition, Griffin, London. Den Dekker A. J. Sijbers J, data distributions in magnetic resonance images, a review, Physica Medica

21.
Central limit theorem
–
If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution. The central limit theorem has a number of variants, in its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, in more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory. When the variance of the i. i. d, Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i. i. d, Random variables with power law tail distributions decreasing as | x |−α −1 where 0 < α <2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Suppose we are interested in the sample average S n, = X1 + ⋯ + X n n of these random variables, by the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n → ∞. The classical central limit theorem describes the size and the form of the stochastic fluctuations around the deterministic number µ during this convergence. For large enough n, the distribution of Sn is close to the distribution with mean µ. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi, formally, the theorem can be stated as follows, Lindeberg–Lévy CLT. Suppose is a sequence of i. i. d, Random variables with E = µ and Var = σ2 < ∞. Then as n approaches infinity, the random variables √n converge in distribution to a normal N, n → d N. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | =0, the theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the limit theorem the random variables Xi have to be independent. The theorem also requires that random variables | Xi | have moments of order. Suppose is a sequence of independent random variables, each with finite expected value μi, in practice it is usually easiest to check Lyapunov’s condition for δ =1. If a sequence of random variables satisfies Lyapunov’s condition, then it also satisfies Lindeberg’s condition, the converse implication, however, does not hold. In the same setting and with the notation as above. Suppose that for every ε >0 lim n → ∞1 s n 2 ∑ i =1 n E =0 where 1 is the indicator function

22.
Moment (mathematics)
–
In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points. If the points represent mass, then the moment is the total mass, the first moment divided by the total mass is the center of mass. The mathematical concept is related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the same is not true on unbounded intervals. The n-th moment of a continuous function f of a real variable about a value c is μ n = ∫ − ∞ ∞ n f d x. It is possible to define moments for random variables in a general fashion than moments for real values—see moments in metric spaces. The moment of a function, without explanation, usually refers to the above expression with c =0. For the second and higher moments, the moment are usually used rather than the moments about zero. Other moments may also be defined, for example, the n-th inverse moment about zero is E and the n-th logarithmic moment about zero is E . The n-th moment about zero of a probability density function f is the value of Xn and is called a raw moment or crude moment. The moments about its mean μ are called moments, these describe the shape of the function. If f is a probability density function, then the value of the integral above is called the moment of the probability distribution. When E = ∫ − ∞ ∞ | x n | d F = ∞, if the n-th moment about any point exists, so does the -th moment about every point. The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one, the first raw moment is the mean, usually denoted μ ≡ μ1 ≡ E . The second central moment is the variance and its positive square root is the standard deviation σ ≡1 /2. The normalised n-th central moment or standardised moment is the central moment divided by σn. These normalised central moments are dimensionless quantities, which represent the distribution independently of any change of scale. For an electric signal, the first moment is its DC level, the third central moment is the measure of the lopsidedness of the distribution, any symmetric distribution will have a third central moment, if defined, of zero

23.
Skewness
–
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined, the qualitative interpretation of the skew is complicated and unintuitive. Skew must not be thought to refer to the direction the curve appears to be leaning, in fact, conversely, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the tail is fat. Further, in multimodal distributions and discrete distributions, skewness is also difficult to interpret, importantly, the skewness does not determine the relationship of mean and median. In cases where it is necessary, data might be transformed to have a normal distribution, consider the two distributions in the figure just below. Within each graph, the values on the side of the distribution taper differently from the values on the left side. A left-skewed distribution usually appears as a right-leaning curve, positive skew, The right tail is longer, the mass of the distribution is concentrated on the left of the figure. A right-skewed distribution usually appears as a left-leaning curve, Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the sequence, whose values are evenly distributed around a central value of 50. If the distribution is symmetric, then the mean is equal to the median, if, in addition, the distribution is unimodal, then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4, note, however, that the converse is not true in general, i. e. zero skewness does not imply that the mean is equal to the median. Paul T. von Hippel points out, Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew and this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long, most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they contradict the textbook interpretation of the median. It is sometimes referred to as Pearsons moment coefficient of skewness, or simply the moment coefficient of skewness, the last equality expresses skewness in terms of the ratio of the third cumulant κ3 to the 1. 5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant, the skewness is also sometimes denoted Skew. Starting from a standard cumulant expansion around a distribution, one can show that skewness =6 /standard deviation + O

24.
Kurtosis
–
In probability theory and statistics, kurtosis is a measure of the tailedness of the probability distribution of a real-valued random variable. Depending on the measure of kurtosis that is used, there are various interpretations of kurtosis. The standard measure of kurtosis, originating with Karl Pearson, is based on a version of the fourth moment of the data or population. This number is related to the tails of the distribution, not its peak, hence, for this measure, higher kurtosis is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations. The kurtosis of any normal distribution is 3. It is common to compare the kurtosis of a distribution to this value, distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is flat-topped as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution, an example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be leptokurtic and it is also common practice to use an adjusted version of Pearsons kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the normal distribution. Some authors use kurtosis by itself to refer to the excess kurtosis, for the reason of clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant. Alternative measures of kurtosis are, the L-kurtosis, which is a version of the fourth L-moment. These are analogous to the measures of skewness that are not based on ordinary moments. The kurtosis is the fourth standardized moment, defined as Kurt = μ4 σ4 = E 2, several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant, other choices include γ2, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. The kurtosis is bounded below by the squared skewness plus 1, μ4 σ4 ≥2 +1, the lower bound is realized by the Bernoulli distribution. There is no limit to the excess kurtosis of a general probability distribution. A reason why some authors favor the excess kurtosis is that cumulants are extensive, formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. Xn be independent random variables for which the fourth moment exists, the excess kurtosis of Y is Kurt −3 =12 ∑ i =1 n σ i 4 ⋅, where σ i is the standard deviation of X i. In particular if all of the Xi have the same variance, the reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed

25.
L-moment
–
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. Standardised L-moments are called L-moment ratios and are analogous to standardized moments, just as for conventional moments, a theoretical distribution has a set of population L-moments. Sample L-moments can be defined for a sample from the population, in particular, the first four population L-moments are λ1 = E X λ2 = /2 λ3 = /3 λ4 = /4. Note that the coefficients of the k-th L-moment are the same as in the term of the binomial transform. The first two of these L-moments have conventional names, λ1 = mean, L-mean or L-location, the L-scale is equal to half the mean difference. Grouping these by order statistic counts the number of ways an element of an n-element sample can be the jth element of an r-element subset, sample L-moments can also be defined indirectly in terms of probability weighted moments, which leads to a more efficient algorithm for their computation. A set of L-moment ratios, or scaled L-moments, is defined by τ r = λ r / λ2, r =3,4, …, the most useful of these are τ3, called the L-skewness, and τ4, the L-kurtosis. L-moment ratios lie within the interval, tighter bounds can be found for some specific L-moment ratios, in particular, the L-kurtosis τ4 lies in [-¼, 1), and 14 ≤ τ4 <1. A quantity analogous to the coefficient of variation, but based on L-moments, can also be defined, τ = λ2 / λ1, for a non-negative random variable, this lies in the interval and is identical to the Gini coefficient. L-moments are statistical quantities that are derived from probability weighted moments which were defined earlier, PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel, the Tukey, and the Wakeby distributions. There are two ways that L-moments are used, in both cases analogously to the conventional moments, As summary statistics for data. To derive estimators for the parameters of probability distributions, applying the method of moments to the L-moments rather than conventional moments, in addition to doing these with standard moments, the latter is more commonly done using maximum likelihood methods, however using L-moments provides a number of advantages. Specifically, L-moments are more robust than conventional moments, and existence of higher L-moments only requires that the random variable have finite mean, one disadvantage of L-moment ratios for estimation is their typically smaller sensitivity. As an example consider a dataset with a few data points, if the ordinary standard deviation of this data set is taken it will be highly influenced by this one point, however, if the L-scale is taken it will be far less sensitive to this data value. Consequently L-moments are far more meaningful when dealing with outliers in data than conventional moments, however, there are also other better suited methods to achieve an even higher robustness than just replacing moments by L-moments. One example of this is using L-moments as summary statistics in extreme value theory, a finite variance is required in addition in order for the standard errors of estimates of the L-moments to be finite. Some appearances of L-moments in the literature include the book by David & Nagaraja. A number of comparisons of L-moments with ordinary moments have been reported

26.
Correlation and dependence
–
In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price, correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a day based on the correlation between electricity demand and weather. In this example there is a relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship, formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence, however, when used in a technical sense, correlation refers to any of several specific types of relationship between mean values. There are several correlation coefficients, often denoted ρ or r, the most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables. Other correlation coefficients have been developed to be more robust than the Pearson correlation – that is, mutual information can also be applied to measure dependence between two variables. It is obtained by dividing the covariance of the two variables by the product of their standard deviations, karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton. The Pearson correlation is defined only if both of the deviations are finite and nonzero. It is a corollary of the Cauchy–Schwarz inequality that the correlation cannot exceed 1 in absolute value, the correlation coefficient is symmetric, corr = corr. As it approaches zero there is less of a relationship, the closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are independent, Pearsons correlation coefficient is 0, for example, suppose the random variable X is symmetrically distributed about zero, and Y = X2. Then Y is completely determined by X, so that X and Y are perfectly dependent, however, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence. If we have a series of n measurements of X and Y written as xi, N, then the sample correlation coefficient can be used to estimate the population Pearson correlation r between X and Y. If x and y are results of measurements that contain measurement error, for the case of a linear model with a single independent variable, the coefficient of determination is the square of r, Pearsons product-moment coefficient. If, as the one variable increases, the other decreases, to illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers. As we go from each pair to the pair x increases

27.
Pearson correlation coefficient
–
In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price, correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a day based on the correlation between electricity demand and weather. In this example there is a relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship, formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence, however, when used in a technical sense, correlation refers to any of several specific types of relationship between mean values. There are several correlation coefficients, often denoted ρ or r, the most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables. Other correlation coefficients have been developed to be more robust than the Pearson correlation – that is, mutual information can also be applied to measure dependence between two variables. It is obtained by dividing the covariance of the two variables by the product of their standard deviations, karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton. The Pearson correlation is defined only if both of the deviations are finite and nonzero. It is a corollary of the Cauchy–Schwarz inequality that the correlation cannot exceed 1 in absolute value, the correlation coefficient is symmetric, corr = corr. As it approaches zero there is less of a relationship, the closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are independent, Pearsons correlation coefficient is 0, for example, suppose the random variable X is symmetrically distributed about zero, and Y = X2. Then Y is completely determined by X, so that X and Y are perfectly dependent, however, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence. If we have a series of n measurements of X and Y written as xi, N, then the sample correlation coefficient can be used to estimate the population Pearson correlation r between X and Y. If x and y are results of measurements that contain measurement error, for the case of a linear model with a single independent variable, the coefficient of determination is the square of r, Pearsons product-moment coefficient. If, as the one variable increases, the other decreases, to illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers. As we go from each pair to the pair x increases

28.
Spearman's rank correlation coefficient
–
In statistics, Spearmans rank correlation coefficient or Spearmans rho, named after Charles Spearman and often denoted by the Greek letter ρ or as r s, is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function, if there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other. Spearmans coefficient is appropriate for both continuous and discrete variables, including ordinal variables, both Spearmans ρ and Kendalls τ can be formulated as special cases of a more general correlation coefficient. The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the ranked variables, cov is the covariance of the rank variables. σ rg X and σ rg Y are the deviations of the rank variables. The first equation—normalizing by the standard deviation—may even be used even when ranks are normalized to because it is both to translation and linear scaling. The standard error of the coefficient was determined by Pearson in 1907 and it is σ r s =0.6325 n −1 There are several other numerical measures that quantify the extent of statistical dependence between pairs of observations. An alternative name for the Spearman rank correlation is the “grade correlation”, in this, in continuous distributions, the grade of an observation is, by convention, always one half less than the rank, and hence the grade and rank correlations are the same in this case. More generally, the “grade” of an observation is proportional to an estimate of the fraction of a less than a given value. Thus this corresponds to one possible treatment of tied ranks, while unusual, the term “grade correlation” is still in use. The sign of the Spearman correlation indicates the direction of association between X and Y, if Y tends to increase when X increases, the Spearman correlation coefficient is positive. If Y tends to decrease when X increases, the Spearman correlation coefficient is negative, a Spearman correlation of zero indicates that there is no tendency for Y to either increase or decrease when X increases. The Spearman correlation increases in magnitude as X and Y become closer to being perfect monotone functions of each other, when X and Y are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfect monotone increasing relationship implies that for any two pairs of data values Xi, Yi and Xj, Yj, that Xi − Xj, a perfect monotone decreasing relationship implies that these differences always have opposite signs. The Spearman correlation coefficient is often described as being nonparametric and this can have two meanings, First, a perfect Spearman correlation results when X and Y are related by any monotonic function. Contrast this with the Pearson correlation, which gives a perfect value when X and Y are related by a linear function. In this example, the raw data in the table below is used to calculate the correlation between the IQ of a person with the number of hours spent in front of TV per week, to do so use the following steps, reflected in the table below. Sort the data by the first column, create a new column x i and assign it the ranked values 1,2,3. n

29.
Partial correlation
–
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. The coefficient of alienation, and its relation with joint variance through correlation are available in Guilford, a simple way to compute the sample partial correlation for some data is to solve the two associated linear regression problems, get the residuals, and calculate the correlation between the residuals. Let X and Y be, as above, random variables taking real values, if we write xi, yi and zi to denote the ith of N i. i. d. Note that in some formulations the regression includes a constant term and it can be computationally expensive to solve the linear regression problems. Actually, the partial correlation can be easily computed from three th-order partial correlations. The zeroth-order partial correlation ρXY·Ø is defined to be the regular correlation coefficient ρXY, naïvely implementing this computation as a recursive algorithm yields an exponential time complexity. However, this computation has the overlapping subproblems property, such that using dynamic programming or simply caching the results of the recursive calls yields a complexity of O. Note in the case where Z is a variable, this reduces to. If we define P = Ω−1, we have, ρ X i X j ⋅ V ∖ = − p i j p i i p j j, let three variables X, Y, Z be chosen from a joint probability distribution over n variables V. Further let vi,1 ≤ i ≤ N, be N n-dimensional i. i. d, samples taken from the joint probability distribution over V. We then consider the N-dimensional vectors x, y and z and it can be shown that the residuals RX coming from the linear regression of X on Z, if also considered as an N-dimensional vector rX, have a zero scalar product with the vector z generated by Z. This means that the vector lies on an -dimensional hyperplane Sz that is perpendicular to z. The same also applies to the residuals RY generating a vector rY, the desired partial correlation is then the cosine of the angle φ between the projections rX and rY of x and y, respectively, onto the hyperplane perpendicular to z. With the assumption that all involved variables are multivariate Gaussian, the partial correlation ρXY·Z is zero if and this property does not hold in the general case. To test if a partial correlation ρ ^ X Y ⋅ Z vanishes, Fishers z-transform of the partial correlation can be used. The null hypothesis is H0, ρ ^ X Y ⋅ Z =0, note that this z-transform is approximate and that the actual distribution of the sample correlation coefficient is not straightforward. However, an exact t-test based on a combination of the regression coefficient, the partial correlation coefficient. The distribution of the partial correlation was described by Fisher

30.
Scatter plot
–
A scatter plot is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. If the points are color-coded, one variable can be displayed. A scatter plot can be used either one continuous variable that is under the control of the experimenter. The measured or dependent variable is plotted along the vertical axis. If no dependent variable exists, either type of variable can be plotted on either axis, a scatter plot can suggest various kinds of correlations between variables with a certain confidence interval. For example, weight and height, weight would be on y axis, correlations may be positive, negative, or null. If the pattern of dots slopes from lower left to upper right, if the pattern of dots slopes from upper left to lower right, it indicates a negative correlation. A line of best fit can be drawn in order to study the relationship between the variables, an equation for the correlation between the variables can be determined by established best-fit procedures. For a linear correlation, the procedure is known as linear regression and is guaranteed to generate a correct solution in a finite time. No universal best-fit procedure is guaranteed to generate a solution for arbitrary relationships. A scatter plot is very useful when we wish to see how two comparable data sets agree with each other. In this case, an identity line, i. e. a y=x line, one of the most powerful aspects of a scatter plot, however, is its ability to show nonlinear relationships between variables. The ability to do this can be enhanced by adding a line such as LOESS. Furthermore, if the data are represented by a model of simple relationships. The scatter diagram is one of the seven basic tools of quality control, scatter charts can be built in the form of bubble, marker, or/and line charts. The researcher would then plot the data in a plot, assigning lung capacity to the horizontal axis. A person with a capacity of 400 cl who held his/her breath for 21.7 seconds would be represented by a single dot on the scatter plot at the point in the Cartesian coordinates. For a set of data variables X1, X2, xk, the scatter plot matrix shows all the pairwise scatter plots of the variables on a single view with multiple scatterplots in a matrix format

31.
Statistical graphics
–
Statistical graphics, also known as graphical techniques, are graphics in the field of statistics used to visualize quantitative data. Whereas statistics and data analysis procedures generally yield their output in numeric or tabular form and they include plots such as scatter plots, histograms, probability plots, spaghetti plots, residual plots, box plots, block plots and biplots. Exploratory data analysis relies heavily on such techniques, in addition, the choice of appropriate statistical graphics can provide a convincing means of communicating the underlying message that is present in the data to others. If one is not using statistical graphics, then one is forfeiting insight into one or more aspects of the structure of the data. Statistical graphics have been central to the development of science and date to the earliest attempts to analyse data, many familiar forms, including bivariate plots, statistical maps, bar charts, and coordinate paper were used in the 18th century. Since the 1970s statistical graphics have been re-emerging as an important analytic tool with the revitalisation of computer graphics, famous graphics were designed by, William Playfair who produced what could be called the first line, bar, pie, and area charts. See the plots page for more examples of statistical graphics. nist. gov. The Visual Display of Quantitative Information, trend Compass Alphabetic gallery of graphical techniques DataScope a website devoted to data visualization and statistical graphics

32.
Bar chart
–
A bar chart or bar graph is a chart or graph that presents grouped data with rectangular bars with lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally, a vertical bar chart is sometimes called a Line graph. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories, one axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one, diagrams of the velocity of a constantly accelerating object against time published in The Latitude of Forms about 300 years before can be interpreted as proto bar charts. Bar charts have a discrete range, bar charts are usually scaled so that all the data can fit on the chart. Bars on the chart may be arranged in any order, bar charts arranged from highest to lowest incidence are called Pareto charts. Normally, bars showing frequency will be arranged in chronological sequence, bar graphs/charts provide a visual presentation of categorical data. Categorical data is a grouping of data into groups, such as months of the year, age group, shoe sizes. In a column bar chart, the appear along the horizontal axis. Bar graphs can also be used for more complex comparisons of data with grouped bar charts, in a grouped bar chart, for each categorical group there are two or more bars. These bars are color-coded to represent a particular grouping, alternatively, a stacked bar chart could be used. The stacked bar chart stacks bars that represent different groups on top of each other, the height of the resulting bar shows the combined result of the groups. However, stacked bar charts are not suited to datasets where some groups have negative values, in such cases, grouped bar chart are preferable. Grouped bar graphs present the information in the same order in each grouping. Stacked bar graphs present the information in the sequence on each bar. See Extension, EasyTimeline to include bar charts in Wikipedia, enhanced Metafile Format to use in office suits, as MS PowerPoint. Histogram, similar appearance - for continuous data http, //www. wikihow. com/Make-Bar-Graphs must see how to do a bargraph be happy, ) Directory of graph software, free online graph creation tool at the website for the National Center for Education Statistics