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121-515: In statistics , an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set . An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in

242-422: A n ( X ) ) / s | {\displaystyle \scriptstyle \delta =|(X-mean(X))/s|} . If δ > Rejection Region, the data point is an outlier. If δ ≤ Rejection Region, the data point is not an outlier. The modified Thompson Tau test is used to find one outlier at a time (largest value of δ is removed if it is an outlier). Meaning, if a data point is found to be an outlier, it

363-405: A biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is The cumulative distribution function can be expressed as: where ⌊ k ⌋ {\displaystyle \lfloor k\rfloor } is the "floor" under k , i.e. the greatest integer less than or equal to k . It can also be represented in terms of

484-469: A population , for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics . Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. Consider independent identically distributed (IID) random variables with

605-407: A ) and Bernoulli( p ) distribution): Asymptotically, this bound is reasonably tight; see for details. One can also obtain lower bounds on the tail F ( k ; n , p ) , known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that which implies the simpler but looser bound For p = 1/2 and k ≥ 3 n /8 for even n , it

726-418: A continuous value for determining if an instance is an outlier instance. The choice of how to deal with an outlier should depend on the cause. Some estimators are highly sensitive to outliers, notably estimation of covariance matrices . Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that

847-400: A data point (or points) is excluded from the data analysis , this should be clearly stated on any subsequent report. The possibility should be considered that the underlying distribution of the data is not approximately normal, having " fat tails ". For instance, when sampling from a Cauchy distribution , the sample variance increases with the sample size, the sample mean fails to converge as

968-418: A decade earlier in 1795. The modern field of statistics emerged in the late 19th and early 20th century in three stages. The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson , who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing

1089-427: A different population than the rest of the sample set. Estimators capable of coping with outliers are said to be robust: the median is a robust statistic of central tendency , while the mean is not. In the case of normally distributed data, the three sigma rule means that roughly 1 in 22 observations will differ by twice the standard deviation or more from the mean, and 1 in 370 will deviate by three times

1210-453: A diverse subset L ⊂ H {\displaystyle L\subset H} : where g j ( t , α ) {\displaystyle g_{j}(t,\alpha )} is the hypothesis induced by learning algorithm g j {\displaystyle g_{j}} trained on training set t with hyperparameters α {\displaystyle \alpha } . Instance hardness provides

1331-458: A given probability distribution : standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables. The population being examined is described by a probability distribution that may have unknown parameters. A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters . The probability distribution of

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1452-484: A given probability of containing the true value is to use a credible interval from Bayesian statistics : this approach depends on a different way of interpreting what is meant by "probability" , that is as a Bayesian probability . In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because

1573-471: A given situation and carry the computation, several methods have been proposed: the method of moments , the maximum likelihood method, the least squares method and the more recent method of estimating equations . Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time. The best illustration for

1694-555: A mathematical discipline only took shape at the very end of the 17th century, particularly in Jacob Bernoulli 's posthumous work Ars Conjectandi . This was the first book where the realm of games of chance and the realm of the probable (which concerned opinion, evidence, and argument) were combined and submitted to mathematical analysis. The method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it

1815-1033: A meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit ), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation. Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables , whereas ratio and interval measurements are grouped together as quantitative variables , which can be either discrete or continuous , due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with

1936-407: A normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. The two common approaches to exclude outliers are truncation (or trimming) and Winsorising . Trimming discards the outliers whereas Winsorising replaces the outliers with

2057-499: A novice is the predicament encountered by a criminal trial. The null hypothesis, H 0 , asserts that the defendant is innocent, whereas the alternative hypothesis, H 1 , asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H 0 (status quo) stands in opposition to H 1 and is maintained unless H 1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H 0 " in this case does not imply innocence, but merely that

2178-404: A population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if

2299-421: A probability density function). If no outliers occur, x should belong to the intersection of all X i 's. When outliers occur, this intersection could be empty, and we should relax a small number of the sets X i (as small as possible) in order to avoid any inconsistency. This can be done using the notion of q - relaxed intersection . As illustrated by the figure, the q -relaxed intersection corresponds to

2420-412: A problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics, such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population (an operation called a census ). This may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize

2541-542: A property which is used in various ways, such as in Wald's confidence intervals . A closed form Bayes estimator for p also exists when using the Beta distribution as a conjugate prior distribution . When using a general Beta ⁡ ( α , β ) {\displaystyle \operatorname {Beta} (\alpha ,\beta )} as a prior, the posterior mean estimator is: The Bayes estimator

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2662-497: A sample using indexes such as the mean or standard deviation , and inferential statistics , which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location ) seeks to characterize the distribution's central or typical value, while dispersion (or variability ) characterizes

2783-407: A sequence of outcomes is called a Bernoulli process ; for a single trial, i.e., n = 1 , the binomial distribution is a Bernoulli distribution . The binomial distribution is the basis for the binomial test of statistical significance . The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N . If

2904-465: A statistician would use a modified, more structured estimation method (e.g., difference in differences estimation and instrumental variables , among many others) that produce consistent estimators . The basic steps of a statistical experiment are: Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of

3025-544: A successful result, then the expected value of X is: This follows from the linearity of the expected value along with the fact that X is the sum of n identical Bernoulli random variables, each with expected value p . In other words, if X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} are identical (and independent) Bernoulli random variables with parameter p , then X = X 1 + ... + X n and The variance is: This similarly follows from

3146-637: A test and confidence intervals . Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling. Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. The use of modern computers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on

3267-399: A transformation is sensible to contemplate depends on the question one is trying to answer." A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information , while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics

3388-401: A transient malfunction. There may have been an error in data transmission or transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in

3509-419: A value accurately rejecting the null hypothesis (sometimes referred to as the p-value ). The standard approach is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis

3630-786: Is asymptotically efficient and as the sample size approaches infinity ( n → ∞ ), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability. Using the Bayesian estimator with the Beta distribution can be used with Thompson sampling . For the special case of using the standard uniform distribution as a non-informative prior , Beta ⁡ ( α = 1 , β = 1 ) = U ( 0 , 1 ) {\displaystyle \operatorname {Beta} (\alpha =1,\beta =1)=U(0,1)} ,

3751-475: Is "far out". In various domains such as, but not limited to, statistics , signal processing , finance , econometrics , manufacturing , networking and data mining , the task of anomaly detection may take other approaches. Some of these may be distance-based and density-based such as Local Outlier Factor (LOF). Some approaches may use the distance to the k-nearest neighbors to label observations as outliers or non-outliers. The modified Thompson Tau test

Outlier - Misplaced Pages Continue

3872-426: Is a method used to determine if an outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set's standard deviation, average and provides a statistically determined rejection zone; thus providing an objective method to determine if a data point is an outlier. How it works: First, a data set's average is determined. Next the absolute deviation between each data point and

3993-492: Is a mixture of two distributions, which may be two distinct sub-populations, or may indicate 'correct trial' versus 'measurement error'; this is modeled by a mixture model . In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable. This can be due to incidental systematic error or flaws in the theory that generated an assumed family of probability distributions , or it may be that some observations are far from

4114-442: Is a mode. In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However, several special results have been established: For k ≤ np , upper bounds can be derived for the lower tail of the cumulative distribution function F ( k ; n , p ) = Pr ( X ≤ k ) {\displaystyle F(k;n,p)=\Pr(X\leq k)} ,

4235-470: Is also consistent both in probability and in MSE . This statistic is asymptotically normal thanks to the central limit theorem , because it is the same as taking the mean over Bernoulli samples. It has a variance of v a r ( p ^ ) = p ( 1 − p ) n {\displaystyle var({\widehat {p}})={\frac {p(1-p)}{n}}} ,

4356-597: Is an integer, then ( n + 1 ) p − 1 {\displaystyle (n+1)p-1} and ( n + 1 ) p {\displaystyle (n+1)p} is a mode. In the case that ( n + 1 ) p − 1 ∉ Z {\displaystyle (n+1)p-1\notin \mathbb {Z} } , then only ⌊ ( n + 1 ) p − 1 ⌋ + 1 = ⌊ ( n + 1 ) p ⌋ {\displaystyle \lfloor (n+1)p-1\rfloor +1=\lfloor (n+1)p\rfloor }

4477-484: Is an outlier is ultimately a subjective exercise. There are various methods of outlier detection, some of which are treated as synonymous with novelty detection. Some are graphical such as normal probability plots . Others are model-based. Box plots are a hybrid. Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on mean and standard deviation: It

4598-575: Is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected. Various attempts have been made to produce a taxonomy of levels of measurement . The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have

4719-465: Is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not

4840-834: Is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares , which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve. Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic ( bias ), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Most studies only sample part of

4961-428: Is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample , rather than use the data to learn about the population that the sample of data is thought to represent. Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution . Inferential statistical analysis infers properties of

Outlier - Misplaced Pages Continue

5082-675: Is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows: Proof: Let For p = 0 {\displaystyle p=0} only f ( 0 ) {\displaystyle f(0)} has a nonzero value with f ( 0 ) = 1 {\displaystyle f(0)=1} . For p = 1 {\displaystyle p=1} we find f ( n ) = 1 {\displaystyle f(n)=1} and f ( k ) = 0 {\displaystyle f(k)=0} for k ≠ n {\displaystyle k\neq n} . This proves that

5203-522: Is however not very tight. In particular, for p = 1 , we have that F ( k ; n , p ) = 0 (for fixed k , n with k < n ), but Hoeffding's bound evaluates to a positive constant. A sharper bound can be obtained from the Chernoff bound : where D ( a ∥ p ) is the relative entropy (or Kullback-Leibler divergence) between an a -coin and a p -coin (i.e. between the Bernoulli(

5324-485: Is more desirable to correct the erroneous value, if possible. Removing a data point solely because it is an outlier, on the other hand, is a controversial practice, often frowned upon by many scientists and science instructors, as it typically invalidates statistical results. While mathematical criteria provide an objective and quantitative method for data rejection, they do not make the practice more scientifically or methodologically sound, especially in small sets or where

5445-418: Is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study , and then look for the number of cases of lung cancer in each group. A case-control study

5566-433: Is possible to make the denominator constant: When n is known, the parameter p can be estimated using the proportion of successes: This estimator is found using maximum likelihood estimator and also the method of moments . This estimator is unbiased and uniformly with minimum variance , proven using Lehmann–Scheffé theorem , since it is based on a minimal sufficient and complete statistic (i.e.: x ). It

5687-451: Is proposed for the statistical relationship between the two data sets, an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis

5808-409: Is proposed to determine in a series of m {\displaystyle m} observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as n {\displaystyle n} such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when

5929-408: Is rejected when it is in fact true, giving a "false positive") and Type II errors (null hypothesis fails to be rejected when it is in fact false, giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Statistical measurement processes are also prone to error in regards to

6050-498: Is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set. Some work has also examined outliers for nominal (or categorical) data. In the context of a set of examples (or instances) in a data set, instance hardness measures the probability that an instance will be misclassified ( 1 − p ( y | x ) {\displaystyle 1-p(y|x)} where y

6171-471: Is the assigned class label and x represent the input attribute value for an instance in the training set t ). Ideally, instance hardness would be calculated by summing over the set of all possible hypotheses H : Practically, this formulation is unfeasible as H is potentially infinite and calculating p ( h | t ) {\displaystyle p(h|t)} is unknown for many algorithms. Thus, instance hardness can be approximated using

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6292-436: Is the case. Instead, one should use a method that is robust to outliers to model or analyze data with naturally occurring outliers. When deciding whether to remove an outlier, the cause has to be considered. As mentioned earlier, if the outlier's origin can be attributed to an experimental error, or if it can be otherwise determined that the outlying data point is erroneous, it is generally recommended to remove it. However, it

6413-402: Is true ( statistical significance ) and the probability of type II error is the probability that the estimator does not belong to the critical region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false. Referring to statistical significance does not necessarily mean that

6534-449: Is widely employed in government, business, and natural and social sciences. The mathematical foundations of statistics developed from discussions concerning games of chance among mathematicians such as Gerolamo Cardano , Blaise Pascal , Pierre de Fermat , and Christiaan Huygens . Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel ), probability theory as

6655-765: The Boolean data type , polytomous categorical variables with arbitrarily assigned integers in the integral data type , and continuous variables with the real data type involving floating-point arithmetic . But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented. Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998), van den Berg (1991). ) The issue of whether or not it

6776-455: The Poisson distribution with λ = pn . Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, p is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3. Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered

6897-470: The Stirling numbers of the second kind , and n k _ = n ( n − 1 ) ⋯ ( n − k + 1 ) {\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)} is the k {\displaystyle k} th falling power of n {\displaystyle n} . A simple bound follows by bounding

7018-487: The Western Electric Company . The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under

7139-440: The binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments , each asking a yes–no question , and each with its own Boolean -valued outcome : success (with probability p ) or failure (with probability q = 1 − p ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and

7260-546: The forecasting , prediction , and estimation of unobserved values either in or associated with the population being studied. It can include extrapolation and interpolation of time series or spatial data , as well as data mining . Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis , linear algebra , stochastic analysis , differential equations , and measure-theoretic probability theory . Formal discussions on inference date back to

7381-432: The limit to the true value of such parameter. Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and consistent estimators which converges in probability to the true value of such parameter. This still leaves the question of how to obtain estimators in

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7502-719: The mathematicians and cryptographers of the Islamic Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages , which contains one of the first uses of permutations and combinations , to list all possible Arabic words with and without vowels. Al-Kindi 's Manuscript on Deciphering Cryptographic Messages gave a detailed description of how to use frequency analysis to decipher encrypted messages, providing an early example of statistical inference for decoding . Ibn Adlan (1187–1268) later made an important contribution on

7623-475: The mode of a binomial B ( n ,  p ) distribution is equal to ⌊ ( n + 1 ) p ⌋ {\displaystyle \lfloor (n+1)p\rfloor } , where ⌊ ⋅ ⌋ {\displaystyle \lfloor \cdot \rfloor } is the floor function . However, when ( n + 1) p is an integer and p is neither 0 nor 1, then the distribution has two modes: ( n + 1) p and ( n + 1) p − 1 . When p

7744-611: The n trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them ( p q ) must be added ( n k ) {\textstyle {\binom {n}{k}}} times, hence Pr ( X = k ) = ( n k ) p k ( 1 − p ) n − k {\textstyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}} . In creating reference tables for binomial distribution probability, usually,

7865-417: The regularized incomplete beta function , as follows: which is equivalent to the cumulative distribution function of the F -distribution : Some closed-form bounds for the cumulative distribution function are given below . If X ~ B ( n , p ) , that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding

7986-509: The Binomial moments via the higher Poisson moments : This shows that if c = O ( n p ) {\displaystyle c=O({\sqrt {np}})} , then E ⁡ [ X c ] {\displaystyle \operatorname {E} [X^{c}]} is at most a constant factor away from E ⁡ [ X ] c {\displaystyle \operatorname {E} [X]^{c}} Usually

8107-399: The assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end ( King effect ). There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation

8228-538: The average are determined. Thirdly, a rejection region is determined using the formula: where t α / 2 {\displaystyle \scriptstyle {t_{\alpha /2}}} is the critical value from the Student t distribution with n -2 degrees of freedom, n is the sample size, and s is the sample standard deviation. To determine if a value is an outlier: Calculate δ = | ( X − m e

8349-436: The center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected (and not due to any anomalous condition). Outliers, being the most extreme observations, may include the sample maximum or sample minimum , or both, depending on whether they are extremely high or low. However,

8470-439: The collection, analysis, interpretation or explanation, and presentation of data , or as a branch of mathematics . Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in the context of uncertainty and decision-making in the face of uncertainty. In applying statistics to

8591-540: The concepts of standard deviation , correlation , regression analysis and the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among others. Pearson developed the Pearson product-moment correlation coefficient , defined as a product-moment, the method of moments for the fitting of distributions to samples and the Pearson distribution , among many other things. Galton and Pearson founded Biometrika as

8712-542: The concepts of sufficiency , ancillary statistics , Fisher's linear discriminator and Fisher information . He also coined the term null hypothesis during the Lady tasting tea experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation". In his 1930 book The Genetical Theory of Natural Selection , he applied statistics to various biological concepts such as Fisher's principle (which A. W. F. Edwards called "probably

8833-425: The data that they generate. Many of these errors are classified as random (noise) or systematic ( bias ), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics is a mathematical body of science that pertains to

8954-414: The data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object (but not the temperature in the room) than the mean; naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may indicate data points that belong to

9075-439: The data-set, measurement error , or that the population has a heavy-tailed distribution . In the case of measurement error, one wishes to discard them or use statistics that are robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution . A frequent cause of outliers

9196-406: The effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements with different levels using

9317-571: The estimator: When estimating p with very rare events and a small n (e.g.: if x = 0 ), then using the standard estimator leads to p ^ = 0 , {\displaystyle {\widehat {p}}=0,} which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator p ^ b {\displaystyle {\widehat {p}}_{b}} , leading to: Another method

9438-495: The evidence was insufficient to convict. So the jury does not necessarily accept H 0 but fails to reject H 0 . While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test , which tests for type II errors . What statisticians call an alternative hypothesis is simply a hypothesis that contradicts the null hypothesis. Working from a null hypothesis , two broad categories of error are recognized: Standard deviation refers to

9559-503: The exception of the case where ( n + 1) p is an integer. In this case, there are two values for which f is maximal: ( n + 1) p and ( n + 1) p − 1 . M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode . Equivalently, M − p < np ≤ M + 1 − p . Taking the floor function , we obtain M = floor( np ) . Suppose

9680-426: The expected number. In general, if the nature of the population distribution is known a priori , it is possible to test if the number of outliers deviate significantly from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability p ) of a given distribution, the number of outliers will follow a binomial distribution with parameter p , which can generally be well-approximated by

9801-478: The expected value assumes on a given sample (also called prediction). Mean squared error is used for obtaining efficient estimators , a widely used class of estimators. Root mean square error is simply the square root of mean squared error. Many statistical methods seek to minimize the residual sum of squares , and these are called " methods of least squares " in contrast to Least absolute deviations . The latter gives equal weight to small and big errors, while

9922-474: The experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness . The Hawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed. An example of an observational study

10043-402: The extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while Standard error refers to an estimate of difference between sample mean and population mean. A statistical error is the amount by which an observation differs from its expected value . A residual is the amount an observation differs from the value the estimator of

10164-450: The extent to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory , which deals with the analysis of random phenomena. A standard statistical procedure involves the collection of data leading to a test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis

10285-581: The fact that the variance of a sum of independent random variables is the sum of the variances. The first 6 central moments , defined as μ c = E ⁡ [ ( X − E ⁡ [ X ] ) c ] {\displaystyle \mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right]} , are given by The non-central moments satisfy and in general where { c k } {\displaystyle \textstyle \left\{{c \atop k}\right\}} are

10406-432: The first journal of mathematical statistics and biostatistics (then called biometry ), and the latter founded the world's first university statistics department at University College London . The second wave of the 1910s and 20s was initiated by William Sealy Gosset , and reached its culmination in the insights of Ronald Fisher , who wrote the textbooks that were to define the academic discipline in universities around

10527-402: The former gives more weight to large errors. Residual sum of squares is also differentiable , which provides a handy property for doing regression . Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares . Also in a linear regression model the non deterministic part of the model

10648-605: The given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction— inductively inferring from samples to the parameters of a larger or total population. A common goal for a statistical research project is to investigate causality , and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables . There are two major types of causal statistical studies: experimental studies and observational studies . In both types of studies,

10769-405: The lower and upper quartiles respectively, then one could define an outlier to be any observation outside the range: for some nonnegative constant k {\displaystyle k} . John Tukey proposed this test, where k = 1.5 {\displaystyle k=1.5} indicates an "outlier", and k = 3 {\displaystyle k=3} indicates data that

10890-403: The mode is 0 for p = 0 {\displaystyle p=0} and n {\displaystyle n} for p = 1 {\displaystyle p=1} . Let 0 < p < 1 {\displaystyle 0<p<1} . We find From this follows So when ( n + 1 ) p − 1 {\displaystyle (n+1)p-1}

11011-424: The most celebrated argument in evolutionary biology ") and Fisherian runaway , a concept in sexual selection about a positive feedback runaway effect found in evolution . The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of " Type II " error, power of

11132-412: The nearest "nonsuspect" data. Exclusion can also be a consequence of the measurement process, such as when an experiment is not entirely capable of measuring such extreme values, resulting in censored data. In regression problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the estimated coefficients, using a measure such as Cook's distance . If

11253-412: The overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably. Although in principle the acceptable level of statistical significance may be subject to debate, the significance level is the largest p-value that allows

11374-415: The population data. Numerical descriptors include mean and standard deviation for continuous data (like income), while frequency and percentage are more useful in terms of describing categorical data (like education). When a census is not feasible, a chosen subset of the population called a sample is studied. Once a sample that is representative of the population is determined, data is collected for

11495-544: The population. Sampling theory is part of the mathematical discipline of probability theory . Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures . The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from

11616-501: The posterior mean estimator becomes: (A posterior mode should just lead to the standard estimator.) This method is called the rule of succession , which was introduced in the 18th century by Pierre-Simon Laplace . When relying on Jeffreys prior , the prior is Beta ⁡ ( α = 1 2 , β = 1 2 ) {\displaystyle \operatorname {Beta} (\alpha ={\frac {1}{2}},\beta ={\frac {1}{2}})} , which leads to

11737-575: The probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from A Manual of Astronomy 2:558 by Chauvenet.) Other methods flag observations based on measures such as the interquartile range . For example, if Q 1 {\displaystyle Q_{1}} and Q 3 {\displaystyle Q_{3}} are

11858-423: The probability that there are at most k successes. Since Pr ( X ≥ k ) = F ( n − k ; n , 1 − p ) {\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)} , these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for k ≥ np . Hoeffding's inequality yields the simple bound which

11979-494: The problem of how to analyze big data . When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples . Statistics itself also provides tools for prediction and forecasting through statistical models . To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling assures that inferences and conclusions can safely extend from

12100-470: The publication of Natural and Political Observations upon the Bills of Mortality by John Graunt . Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology . The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics

12221-416: The same probability of being achieved (regardless of positions of successes within the sequence). There are ( n k ) {\textstyle {\binom {n}{k}}} such sequences, since the binomial coefficient ( n k ) {\textstyle {\binom {n}{k}}} counts the number of ways to choose the positions of the k successes among

12342-461: The same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation . Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies , they are also applied to other kinds of data—like natural experiments and observational studies —for which

12463-526: The same rate p ) is given by the probability mass function : for k = 0, 1, 2, ..., n , where is the binomial coefficient . The formula can be understood as follows: p q is the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are "successes" and the remaining n − k trials result in "failure". Since the trials are independent with probabilities remaining constant between them, any sequence of n trials with k successes (and n − k failures) has

12584-439: The sample data to draw inferences about the population represented while accounting for randomness. These inferences may take the form of answering yes/no questions about the data ( hypothesis testing ), estimating numerical characteristics of the data ( estimation ), describing associations within the data ( correlation ), and modeling relationships within the data (for example, using regression analysis ). Inference can extend to

12705-409: The sample maximum and minimum are not always outliers because they may not be unusually far from other observations. Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the average temperature of 10 objects in a room, and nine of them are between 20 and 25 degrees Celsius , but an oven is at 175 °C, the median of

12826-399: The sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, drawing the sample contains an element of randomness; hence, the numerical descriptors from the sample are also prone to uncertainty. To draw meaningful conclusions about the entire population, inferential statistics are needed. It uses patterns in

12947-404: The sample size increases, and outliers are expected at far larger rates than for a normal distribution. Even a slight difference in the fatness of the tails can make a large difference in the expected number of extreme values. A set membership approach considers that the uncertainty corresponding to the i th measurement of an unknown random vector x is represented by a set X i (instead of

13068-405: The sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures. There are also methods of experimental design that can lessen these issues at the outset of a study, strengthening its capability to discern truths about

13189-482: The sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis : descriptive statistics , which summarize data from

13310-412: The sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable . Either

13431-589: The sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution , not a binomial one. However, for N much larger than n , the binomial distribution remains a good approximation, and is widely used. If the random variable X follows the binomial distribution with parameters n ∈ N {\displaystyle \mathbb {N} } and p ∈ [0, 1] , we write X ~ B ( n , p ) . The probability of getting exactly k successes in n independent Bernoulli trials (with

13552-609: The set of all x which belong to all sets except q of them. Sets X i that do not intersect the q -relaxed intersection could be suspected to be outliers. In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a hierarchical Bayes model , or a mixture model . Statistics When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples . Representative sampling assures that inferences and conclusions can reasonably extend from

13673-500: The standard deviation. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see Poisson distribution – and not indicate an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times

13794-408: The statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean , unbiased sample variance and sample covariance . A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on

13915-477: The table is filled in up to n /2 values. This is because for k > n /2 , the probability can be calculated by its complement as Looking at the expression f ( k , n , p ) as a function of k , there is a k value that maximizes it. This k value can be found by calculating and comparing it to 1. There is always an integer M that satisfies f ( k , n , p ) is monotone increasing for k < M and monotone decreasing for k > M , with

14036-401: The test to reject the null hypothesis. This test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic . Therefore, the smaller the significance level, the lower the probability of committing type I error. Binomial distribution In probability theory and statistics ,

14157-420: The true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed random variables . One approach that does yield an interval that can be interpreted as having

14278-416: The two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds. Statistics rarely give a simple Yes/No type answer to the question under analysis. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of

14399-485: The unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score , the chi square statistic and Student's t-value . Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient . Furthermore, an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at

14520-640: The use of sample size in frequency analysis. Although the term statistic was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the German Gottfried Achenwall in 1749 who started using the term as a collection of quantitative information, in the modern use for this science. The earliest writing containing statistics in Europe dates back to 1663, with

14641-468: The world. Fisher's most important publications were his 1918 seminal paper The Correlation between Relatives on the Supposition of Mendelian Inheritance (which was the first to use the statistical term, variance ), his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments , where he developed rigorous design of experiments models. He originated

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