Misplaced Pages

#239760

111-577: One of several widely publicized statistical systems of forecasts of player performance, PECOTA player forecasts are marketed by BP as a fantasy baseball product. Since 2003, annual PECOTA forecasts have been published both in the Baseball Prospectus annual books and, in more detailed form, on the BaseballProspectus.com subscription-based website. PECOTA also inspired some analogous projection systems for other professional sports: KUBIAK for

222-469: 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

333-432: A state , a country" ) is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data . In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing

444-762: A baseline a weighted average of a player's performance in his previous three years. Like PECOTA, many of those systems also adjust the projections for aging, park effects and regression toward the mean . Like PECOTA, they may also adjust for the competitive difficulty of each of the two major leagues. The systems differ from one another, however, in the types and intensities of age adjustments, regression-effect estimates, park adjustments, and league-difficulty adjustments that they may make as well as in whether they use similarity scores. PECOTA also makes projections for many more players than do other systems, because PECOTA relies on adjusted minor league statistics as well as major league statistics and tries to make projections for all of

555-411: A crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments . 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 the sample to the population as

666-418: 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

777-433: A distinct way that leads to a very different set of "comparables" than James' method. Furthermore, Silver describes the following distinct feature: The PECOTA similarity scores are based primarily on looking at a three-year window of a pitcher’s performance. Thus, we might look at what a pitcher did from ages 35–37, and compare that against the most similar age 35–37 performances, after adjusting for parks, league effects, and

888-458: 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

999-538: A given player's past performance statistics to the performance of "comparable" Major League ballplayers by means of similarity scores . As is described in the Baseball Prospectus website's glossary: PECOTA compares each player against a database of roughly 20,000 major league batter seasons since World War II. In addition, it also draws upon a database of roughly 15,000 translated minor league seasons (1997–2006) for players that spent most of their previous season in

1110-484: 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

1221-471: 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

SECTION 10

#1732851195240

1332-548: 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

1443-1028: 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

1554-427: A mistake,' Silver said. Silver found that the most predictive statistics, by a considerable margin, are a pitcher's strikeout rate and walk rate. Home runs allowed, lefty-righty breakdowns and other data tell less about a pitcher's future". Instead of focusing on making point estimates of a player's future performance (such as batting average, home runs, and strike-outs), PECOTA relies on the historical performance of

1665-499: 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

1776-410: A number of different career paths. I think Gary used something like thirteen or fifteen separate career paths, and all that PECOTA is really doing is carrying that to the logical extreme, where there is essentially a separate career path for every player in major league history. The comparability scores are the mechanism by which it picks and chooses from among those career paths. PECOTA relies on fitting

1887-471: A pitcher has a high xFIP, but also induces a lot of ground balls and popups, his SIERA will be lower than his xFIP. The calculations for it are as follows: where SO is strikeouts , PA is plate appearances , BB is bases on balls , GB is ground ball , FB is fly ball , and PU is pop-up In 1999, Voros McCracken became the first to detail and publicize these effects to the baseball research community when he wrote on rec.sport.baseball, "I've been working on

1998-476: A pitcher's 'Batting Average on Balls In Play' ( BABIP ). His research found the opposite to be true: that while a pitcher's ability to cause strikeouts or prevent home runs remained somewhat constant from season to season, his ability to prevent hits on balls in play did not. To better evaluate pitchers in light of his theory, McCracken developed " Defense-Independent ERA " (dERA), the most well-known defense-independent pitching statistic. McCracken's formula for dERA

2109-468: A pitcher's actual home run total with an expected home run total (xHR). where xHR is calculated using the league average home run per fly ball rate (lgHR/FB) multiplied by the number of fly balls the pitcher has allowed. Typically, the lgHR/FB is around 10.5%, meaning 10.5% of fly balls go for home runs. In 2015, it was 11.4%. Baseball Prospectus invented this statistic, which takes into account balls in play and adjusts for balls in play. For example, if

2220-505: A pitcher's future performance in a given area by using information about his past performance in other areas. As baseball analyst and journalist Alan Schwarz writes, "Silver ... designed a sophisticated variance algorithm that has examined every big-league pitcher's statistics since 1946 to determine which numbers best forecast effectiveness, specifically earned run average . His findings are counterintuitive to most fans. 'When you try to predict future E.R.A.'s with past E.R.A.'s, you're making

2331-540: A pitching evaluation tool and thought I'd post it here to get some feedback. I call it 'Defensive Independent Pitching' and what it does is evaluate a pitcher base[d] strictly on the statistics his defense has no ability to affect..." Until the publication of a more widely read article in 2001, however, on Baseball Prospectus , most of the baseball research community believed that individual pitchers had an inherent ability to prevent hits on balls in play. McCracken reasoned that if this ability existed, it would be noticeable in

SECTION 20

#1732851195240

2442-404: 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

2553-412: 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

2664-497: 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

2775-411: A slightly different FIP equation, instead using 3*(BB+HBP-IBB) rather than simply 3*(BB) where "HBP" stands for batters hit by pitch and "IBB" stands for intentional base on balls. Dave Studeman of The Hardball Times derived Expected Fielding Independent Pitching (xFIP), a regressed version of FIP. Calculated like FIP, it differs in that it normalizes the number of home runs the pitcher allows, replacing

2886-460: 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

2997-637: 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

3108-399: 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

3219-419: 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

3330-421: A whole host of other things. This is different from the similarity scores you might see at baseball-reference.com or in other places, which attempt to evaluate the totality of a player’s career up to a given age. Once a set of "comparables" is determined for each player, his future performance forecast is based on the historical performance of his "comparables". For example, a 26-year-old's forecast performance in

3441-450: 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

PECOTA - Misplaced Pages Continue

3552-434: Is a constant that renders league FIP for the time period in question equal to league ERA for the same period. It is calculated as: where lgERA is the league average ERA, lgHR is the number of home runs in the league, lgBB is the number of walks in the league, lgK is the number of strikeouts in the league, and lgIP is the number of innings pitched in the league. The Hardball Times , a popular baseball statistics website, uses

3663-488: Is a wide variation in career BABIP among pitchers, and this seems to correlate with career success. For instance, no pitcher in the Hall of Fame has a below-average career BABIP. Each of the following formulae uses innings pitched (IP), a measure of the number of outs a team made while a pitcher was in the game. Since most outs rely on fielding, the results from calculations using IP are not truly independent of team defense. While

3774-575: 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

3885-465: 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

3996-834: 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

4107-428: 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

4218-431: Is intended to measure a pitcher 's effectiveness based only on statistics that do not involve fielders (except the catcher ). These include home runs allowed, strikeouts , hit batters , walks , and, more recently, fly ball percentage, ground ball percentage, and (to a much lesser extent) line drive percentage. By focusing on these statistics and ignoring what happens once a ball is put in play, which – on most plays –

4329-474: Is more complicated than the standard method of applying an age adjustment based on the 'average' course of development of all players throughout history. However, it is also leaps and bounds more representative of reality, and more accurate to boot. Although Silver was the creator of PECOTA, producing PECOTA forecasts was a team effort: "I might be 'the PECOTA guy,' but it very much is a team effort," Silver has said of

4440-418: 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

4551-451: 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

PECOTA - Misplaced Pages Continue

4662-456: Is regarded by many in the sabermetric community as the most important piece of baseball research in many years. As Jonah Keri wrote in 2012, "When Voros McCracken wrote his seminal piece on pitching and defense 11 years ago, he helped change the way people—fans, writers, even general managers—think about run prevention in baseball. Where once we used to throw most of the blame for a hit on the pitcher who gave it up, McCracken helped us realize that

4773-408: 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

4884-402: 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

4995-589: Is very complicated, with a number of steps. DIP ERA is not as useful for knuckleballers and other "trick" pitchers, a factor that McCracken mentioned a few days after his original announcement of his research findings in 1999, in a posting on the rec.sport.baseball.analysis Usenet site on November 23, 1999, when he wrote: "Also to [note] is that, anecdotally, I believe pitchers with trick deliveries (e.g. Knuckleballers) might post consistently lower $ H numbers than other pitchers. I looked at Tim Wakefield 's career and that seems to bear out slightly". In later postings on

5106-449: 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

5217-760: 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

5328-707: The National Football League , SCHOENE and CARMELO for the National Basketball Association , and VUKOTA for the National Hockey League . PECOTA forecasts a player's performance in all of the major categories used in typical fantasy baseball games; it also forecasts production in advanced sabermetric categories developed by Baseball Prospectus (e.g., VORP and EqA ). In addition, PECOTA forecasts several summary diagnostics such as breakout rates, improve rates, and attrition rates, as well as

5439-575: The Pittsburgh Pirates were in process of developing MITT ("Managing, Information, Tools and Talent"), a proprietary database that integrates scouting reports, medical and contract information, and performance statistics and projections. First introduced in 2003, PECOTA projections are produced each year and published both in the Baseball Prospectus annual monographs and on the BaseballProspectus.com website. PECOTA has undergone several improvements since 2003. The 2006 version introduced metrics for

5550-477: 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

5661-546: 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

SECTION 50

#1732851195240

5772-432: 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

5883-707: 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

5994-475: The 2003 through 2007 seasons shows that PECOTA's average error between the predicted and actual team wins declined: 2003 5.91 wins; 2004 7.71 wins; 2005 5.14 wins; 2006 4.94 wins; 2007 4.31 wins. Silver conjectures that the improvement has come in part from taking defense into account in the forecasts beginning in 2005. In 2008 the average error was 8.5 wins. Statistical Statistics (from German : Statistik , orig. "description of

6105-401: The 2006 season in predicting OPS . It performed nearly as well as the best of the other systems in predicting ERA . Although PECOTA projections are made for well over 1000 hitters each season, the evaluation of the system included only slightly over 100 players who had a minimum of 500 major league AB and had also been included in projections by the other systems. Nate Silver's own comparison of

6216-553: The BP staff. "We all do it. It's my baby, but it takes a village to run a PECOTA". For example, PECOTA draws on Clay Davenport 's translations (the so-called Davenport Translations or DT's) of minor league and international baseball statistics to estimate the major league equivalent performance of each player. In this way, PECOTA is able to make projections for more than 1,600 players each year, including many players with little or no prior major league experience. The 2009 preseason forecasts were

6327-454: The Baseball Prospectus team, initially with Clay Davenport in charge of the effort, and later, through the 2013 season, with Colin Wyers heading up both production and improvements in PECOTA. Most of the other popular forecasting systems do not use a "comparable players" approach. Instead most rely on direct projections from a player's past performance to his future performance, typically by using as

6438-439: 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

6549-427: The coming season will be based on how the most comparable Major League 26-year-olds performed in their subsequent season. Separate sets of predictions are developed for hitters and pitchers. PECOTA also relies a lot on the use of peripheral statistics to forecast a given player's future performance. For example, drawing on the insights coming out of the use of defense-independent pitching statistics , PECOTA forecasts

6660-535: 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

6771-538: 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

SECTION 60

#1732851195240

6882-409: The creators of DICE, FIP and similar statistics all suggest they are "defense independent", others have pointed out that their formulas involve (IP). IP is a statistical measure of how many outs were made while a pitcher was pitching. This includes those made by fielders who are typically involved in more than two thirds of the outs. These critics claim this makes pitchers' DICE or FIP highly dependent on

6993-489: The current year. Tom Tango independently derived a similar formula, known as Fielding Independent Pitching, which is very close to the results of dERA and DICE. In that equation, "HR" is home runs, "BB" is walks, "K" is strikeouts, and "IP" is innings pitched. That equation usually gives a number that is nothing close to a normal ERA (this is the FIP core), so the equation used is more often (but not always) this one: where C

7104-425: 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

7215-478: The database is large enough to provide a meaningfully large set of appropriate comparables. When it isn't, the program is designed to 'cheat' by expanding its tolerance for dissimilar players until a reasonable sample size is reached. PECOTA uses nearest neighbor analysis to match the individual player with a set of other players who are most similar to him. Although drawing on the underlying concept of Bill James ' similarity scores, PECOTA calculates these scores in

7326-404: The defensive play of their fielders. A simple formula, known as Defense-Independent Component ERA (DICE), was created by Clay Dreslough in 1998: In that equation, "HR" is home runs, "BB" is walks, "HBP" is hit batters, "K" is strikeouts, and "IP" is innings pitched. That equation gives a number that is better at predicting a pitcher's ERA in the following year than the pitcher's actual ERA in

7437-406: 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

7548-450: The estimated number of runs scored and allowed by the roster of players under the given playing-time assumptions. PECOTA has been used in preseason forecasts of how many wins teams will attain and in mid-season simulations of the number of wins each team will attain and its odds of reaching the playoffs. In 2006, PECOTA's preseason forecasts compared favorably to other forecasting systems (including Las Vegas betting line odds) in predicting

7659-495: 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

7770-478: 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

7881-474: 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

7992-435: The expert advice of the Baseball Prospectus staff. The number of runs a team will score and allow during the coming season is estimated based on the playing times and PECOTA's predicted individual performance of each player, using a "Marginal Lineup Value" algorithm created by David Tate and further developed by Keith Woolner . A team's expected wins is based on applying an improved version of Bill James' Pythagorean Formula to

8103-402: 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

8214-450: 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

8325-410: The field of play (BHFP). Controversy over DIP was heightened when Tom Tippett at Diamond Mind published his own findings in 2003. Tippett concluded that the differences between pitchers in preventing hits on balls in play were at least partially the result of the pitcher's skill. Tippett analyzed certain groups of pitchers that appear to be able to reduce the number of hits allowed on balls hit into

8436-409: The field of play (BHFP). Like McCracken, Tippett found that pitchers' BABIP was more volatile on an annual basis than the rates at which they gave up home runs or walks. It was this greater volatility that had led McCracken to conclude pitchers had "little or no control" over hits on balls in play. But Tippett also found large and significant differences between pitchers' career BABIP. In many cases, it

8547-432: 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

8658-541: The forecasting of hurricane paths: players can go in many directions, so preparing for just one is foolish". Silver has written, This procedure requires us to become comfortable with probabilistic thinking. While a majority of players of a certain type may progress a certain way – say, peak early – there will always be exceptions. Moreover, the comparable players may not always perform in accordance with their true level of ability. They will sometimes appear to exceed it in any given season, and other times fall short, because of

8769-402: 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

8880-605: 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,

8991-571: The given player's "comparables" to produce a probability distribution of the given player's predicted performance during the next five years. Alan Schwarz has emphasized this feature of PECOTA: "What separates Pecota from the gaggle of projection systems that outsiders have developed over many decades is how it recognizes, even flaunts, the uncertainty of predicting a player's skills. Rather than generate one line of expected statistics, Pecota presents seven – some optimistic, some pessimistic – each with its own confidence level. The system greatly resembles

9102-439: The last ones for which Silver took primary responsibility. In March 2009, Silver announced that PECOTA's extremely complex and laborious set of database manipulations and calculations would be moving to a different platform . Although Baseball Prospectus had been the owner of PECOTA since Silver sold it to them in 2003 – and Silver stewarded and took responsibility for the forecasts – henceforth PECOTA forecasts would be generated by

9213-613: The market valuation of players based on the predicted performance levels. The 2007 version introduced adjustments for league effects, to account for differences in the competitive environment of the two major leagues. The 2008 update took into account differences in players' performance during the first and second halves of the previous season as well as platoon splits (how well a player performed against hitters or pitchers who were left- or right-handed). It also took account of baserunning. In 2009, Baseball Prospectus introduced in-season PECOTA projections, to update and supplement its beginning of

9324-529: The market values of the players. The logic and methodology underlying PECOTA have been described in several publications, but the detailed formulas are proprietary and have not been shared with the baseball research community. Silver described the inspiration for his approach as follows: The basic idea behind PECOTA is really a fusion of two different things – [Bill] James's work on similarity scores and Gary Huckabay's work on Vlad, [Baseball Prospectus's] previous projection system, which tried to assign players to

9435-596: The minor leagues. ... PECOTA considers four broad categories of attributes in determining a player's comparability: 1. Production metrics – such as batting average, isolated power, and unintentional walk rate for hitters, or strikeout rate and groundball rate for pitchers. 2. Usage metrics, including career length and plate appearances or innings pitched. 3. Phenotypic attributes, including handedness, height, weight, career length (for major leaguers), and minor league level (for prospects). 4. Fielding Position (for hitters) or starting/relief role (for pitchers). ... In most cases,

9546-424: 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

9657-402: The number of wins teams would earn during the season. An independent evaluation by the website Vegas Watch showed that PECOTA had the lowest error in predicting Major League team wins in 2008 of all the best known forecasts, both those that were sabermetrically based and those that relied on individual expertise. In 2009, however, PECOTA lagged behind all the well-known forecasters. A summary for

9768-412: 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

9879-414: The performance of alternative projection systems for hitters in 2007 also showed that PECOTA led the field, though a couple of others were close. Although designed primarily for predicting individual player performance, PECOTA has been applied also to predicting team performance. For this purpose, projected team depth charts are established with projected playing times for each team member, drawing on

9990-408: The pitcher has little control over, DIP claims to offer a clearer picture of the pitcher's true ability. The most controversial part of DIP is the idea that pitchers have little influence over what happens to balls that are put into play. Some people believe this has been well-established (see below), primarily by showing the large variability of most pitchers' BABIP from year to year. However, there

10101-514: The players on major league expanded rosters (40 players per team) as well as other prospects . Beginning in 2000, the Cleveland Indians developed a proprietary analytical database called DiamondView to evaluate scouting information gathered by the team; this system later incorporated player performance indicators and financial indicators, for purposes of evaluating and projecting the performance of all major league players. During 2008–2009,

10212-415: 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

10323-544: 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

10434-494: 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

10545-466: 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

10656-412: The rec.sport.baseball site during 1999 and 2000 (prior to the publication of his widely read article on BaseballProspectus.com in 2001), McCracken also discussed other pitcher characteristics that might influence BABIP. In 2002 McCracken created and published version 2.0 of dERA, which incorporates the ability of knuckleballers and other types of pitchers to affect the number of hits allowed on balls hit in

10767-461: 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

10878-439: 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

10989-399: 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

11100-503: The sample size problems that we described earlier. PECOTA accounts for these sorts of factors by creating not a single forecast point, as other systems do, but rather a range of possible outcomes that the player could expect to achieve at different levels of probability. Instead of telling you that it's going to rain, we tell you that there's an 80% chance of rain, because 80% of the time that these atmospheric conditions have emerged on Tuesday, it has rained on Wednesday. Surely, this approach

11211-405: 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

11322-412: 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

11433-515: The season projections. In 2012, PECOTA substantially changed the way it weighed past years' performance in establishing the baseline for projections. In addition, 10-year forecasts and percentile forecasts were added to the individual player PECOTA cards that are published on-line. Although Baseball Prospectus promotes PECOTA commercially as "deadly accurate," all projection systems are subject to considerable uncertainty. A comparison found that PECOTA had outperformed several other forecasting systems for

11544-408: 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

11655-491: 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. Defense independent pitching statistics In baseball , fielding independent pitching ( FIP ) (also referred to as defense independent pitching (DIP))

11766-420: 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

11877-416: 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

11988-485: 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

12099-620: 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

12210-462: 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

12321-467: Was these differences that accounted for the pitchers' relative success. However, improvements to DIP that look at more nuanced defense-independent stats than strikeouts, home runs, and walks (such as groundball rate), have been able to account for many of the BABIP differences that Tippet identified without reintroducing the noise from defense variability. Despite other criticisms, the work by McCracken on DIP

#239760