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176-415: 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 a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using

352-410: A b f ( x ) d x . {\displaystyle P\left(a\leq X\leq b\right)=\int _{a}^{b}f(x)\,dx.} This is the definition of a probability density function , so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for X {\displaystyle X} to take any single value

528-401: A {\displaystyle a} (that is, a ≤ X ≤ a {\displaystyle a\leq X\leq a} ) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval [ a , b ] {\displaystyle [a,b]} is replaced by any measurable set A {\displaystyle A} ,

704-439: A , b ] ⊂ R {\displaystyle I=[a,b]\subset \mathbb {R} } the probability of X {\displaystyle X} belonging to I {\displaystyle I} is given by the integral of f {\displaystyle f} over I {\displaystyle I} : P ( a ≤ X ≤ b ) = ∫

880-638: A base-10 positional system. On May 25, 1577, King Philip II of Spain ordered by royal cédula the preparation of a general description of Spain's holdings in the Indies. Instructions and a questionnaire, issued in 1577 by the Office of the Cronista Mayor, were distributed to local officials in the Viceroyalties of New Spain and Peru to direct the gathering of information. The questionnaire, composed of fifty items,

1056-533: A census for tax purposes, which was partially responsible for the development of the Zealot movement and several failed rebellions against Rome ultimately ending in the Jewish Diaspora . The Gospel of Luke makes reference to Quirinius' census in relation to the birth of Jesus ; based on variant readings of this passage, a minority of biblical scholars, including N. T. Wright , speculate that this passage refers to

1232-474: A distribution (sample or population): central tendency (or location ) seeks to characterize the distribution's central or typical value, while dispersion (or variability ) characterizes 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

1408-457: A measurable space ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Given that probabilities of events of the form { ω ∈ Ω ∣ X ( ω ) ∈ A } {\displaystyle \{\omega \in \Omega \mid X(\omega )\in A\}} satisfy Kolmogorov's probability axioms ,

1584-421: 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 the statistic, though, may have unknown parameters. Consider now

1760-419: 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 a population , for example by testing hypotheses and deriving estimates. It is assumed that the observed data set

1936-408: A sampling frame such as an address register. Census counts are necessary to adjust samples to be representative of a population by weighting them as is common in opinion polling . Similarly, stratification requires knowledge of the relative sizes of different population strata, which can be derived from census enumerations. In some countries, the census provides the official counts used to apportion

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2112-412: A sampling frame to count the population. This is the only way to be sure that everyone has been included, as otherwise those not responding would not be followed up on and individuals could be missed. The fundamental premise of a census is that the population is not known, and a new estimate is to be made by the analysis of primary data. The use of a sampling frame is counterintuitive as it suggests that

2288-480: A "permanent" address, which might be a family home for students or long-term migrants. A precise definition of residence is needed, to decide whether visitors to a country should be included in the population count. This is becoming more important as students travel abroad for education for a period of several years. Other groups causing problems with enumeration are newborn babies, refugees, people away on holiday, people moving home around census day, and people without

2464-495: A Bernoulli distribution with parameter p {\displaystyle p} . This is a transformation of discrete random variable. For a distribution function F {\displaystyle F} of an absolutely continuous random variable, an absolutely continuous random variable must be constructed. F i n v {\displaystyle F^{\mathit {inv}}} , an inverse function of F {\displaystyle F} , relates to

2640-477: A clan or tribe, or by a juridical person such as a corporation, cooperative, or government agency. The holding's land may consist of one or more parcels, located in one or more separate areas or one or more territorial or administrative divisions, providing the parcels share the same production means, such as labor, farm buildings, machinery or draught animals. Historical censuses used crude enumeration assuming absolute accuracy. Modern approaches take into account

2816-414: A coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function . On the other hand, absolutely continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as

2992-416: 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

3168-484: A defined territory, simultaneity and defined periodicity", and recommends that population censuses be taken at least every ten years. UN recommendations also cover census topics to be collected, official definitions, classifications, and other useful information to coordinate international practices. The UN 's Food and Agriculture Organization (FAO), in turn, defines the census of agriculture as "a statistical operation for collecting, processing and disseminating data on

3344-1094: A discrete probability distribution, there is a countable set A {\displaystyle A} with P ( X ∈ A ) = 1 {\displaystyle P(X\in A)=1} and a probability mass function p {\displaystyle p} . If E {\displaystyle E} is any event, then P ( X ∈ E ) = ∑ ω ∈ A p ( ω ) δ ω ( E ) , {\displaystyle P(X\in E)=\sum _{\omega \in A}p(\omega )\delta _{\omega }(E),} or in short, P X = ∑ ω ∈ A p ( ω ) δ ω . {\displaystyle P_{X}=\sum _{\omega \in A}p(\omega )\delta _{\omega }.} Similarly, discrete distributions can be represented with

3520-1024: A discrete random variable X {\displaystyle X} , let u 0 , u 1 , … {\displaystyle u_{0},u_{1},\dots } be the values it can take with non-zero probability. Denote Ω i = X − 1 ( u i ) = { ω : X ( ω ) = u i } , i = 0 , 1 , 2 , … {\displaystyle \Omega _{i}=X^{-1}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,1,2,\dots } These are disjoint sets , and for such sets P ( ⋃ i Ω i ) = ∑ i P ( Ω i ) = ∑ i P ( X = u i ) = 1. {\displaystyle P\left(\bigcup _{i}\Omega _{i}\right)=\sum _{i}P(\Omega _{i})=\sum _{i}P(X=u_{i})=1.} It follows that

3696-415: A family home during vacations, or children whose parents have separated who effectively have two family homes. Census enumeration has always been based on finding people where they live, as there is no systematic alternative: any list used to find people is likely to be derived from census activities in the first place. Recent UN guidelines provide recommendations on enumerating such complex households. In

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3872-499: A fishing analogy can be found in "Trout, Catfish and Roach..." which won an award from the Royal Statistical Society for excellence in official statistics in 2011. Triple system enumeration has been proposed as an improvement as it would allow evaluation of the statistical dependence of pairs of sources. However, as the matching process is the most difficult aspect of census estimation this has never been implemented for

4048-414: A fixed address. People with second homes, because they are working in another part of the country or have a holiday cottage, are difficult to fix at a particular address; this sometimes causes double counting or houses being mistakenly identified as vacant. Another problem is where people use a different address at different times e.g. students living at their place of education in term time but returning to

4224-434: 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 the unknown parameter is called a pivotal quantity or pivot. Widely used pivots include

4400-484: 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

4576-469: 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

4752-500: A large city, it might be appropriate to give the average income for black males aged between 50 and 60. However, doing this for a town that only has two black males in this age group would be a breach of privacy because either of those persons, knowing his own income and the reported average, could determine the other man's income. Typically, census data are processed to obscure such individual information. Some agencies do this by intentionally introducing small statistical errors to prevent

4928-546: 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

5104-1028: 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

5280-467: A more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure . In the measure-theoretic formalization of probability theory , a random variable is defined as a measurable function X {\displaystyle X} from a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} to

5456-418: A multivariate distribution (a joint probability distribution ) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution , the hypergeometric distribution , and the normal distribution . A commonly encountered multivariate distribution

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5632-534: A myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as hypercubes or balls . However, this is not always the case, and there exist phenomena with supports that are actually complicated curves γ : [ a , b ] → R n {\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{n}} within some space R n {\displaystyle \mathbb {R} ^{n}} or similar. In these cases,

5808-469: A national enumeration. It would also be difficult to identify three different sources that were sufficiently different to make the triple system effort worthwhile. The DSE approach has another weakness in that it assumes there is no person counted twice (over count). In de facto residence definitions this would not be a problem but in de jure definitions individuals risk being recorded on more than one form leading to double counting. A particular problem here

5984-499: 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

6160-433: A number in [ 0 , 1 ] ⊆ R {\displaystyle [0,1]\subseteq \mathbb {R} } . The probability function P {\displaystyle P} can take as argument subsets of the sample space itself, as in the coin toss example, where the function P {\displaystyle P} was defined so that P (heads) = 0.5 and P (tails) = 0.5 . However, because of

6336-433: A population and housing census – numbers of people, their distribution, their living conditions and other key data – is critical for development." This is because this type of data is essential for policymakers so that they know where to invest. Many countries have outdated or inaccurate data about their populations and thus have difficulty in addressing the needs of the population. The UNFPA said: "The unique advantage of

6512-560: A population, not just the number of individuals. Censuses typically began as the only method of collecting national demographic data and are now part of a larger system of different surveys. Although population estimates remain an important function of a census, including exactly the geographic distribution of the population or the agricultural population, statistics can be produced about combinations of attributes, e.g., education by age and sex in different regions. Current administrative data systems allow for other approaches to enumeration with

6688-404: 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

6864-421: A random variable takes values from a continuum then by convention, any individual outcome is assigned probability zero. For such continuous random variables , only events that include infinitely many outcomes such as intervals have probability greater than 0. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then,

7040-460: A realist approach to measurement, acknowledging that under any definition of residence there is a true value of the population but this can never be measured with complete accuracy. An important aspect of the census process is to evaluate the quality of the data. Many countries use a post-enumeration survey to adjust the raw census counts. This works similarly to capture-recapture estimation for animal populations. Among census experts, this method

7216-678: A separate registration conducted during the reign of Herod the Great , several years before Quirinius' census. The 15-year indiction cycle established by Diocletian in AD   297 was based on quindecennial censuses and formed the basis for dating in late antiquity and under the Byzantine Empire . In the Middle Ages , the Caliphate began conducting regular censuses soon after its formation, beginning with

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7392-425: A set of probability zero, where 1 A {\displaystyle 1_{A}} is the indicator function of A {\displaystyle A} . This may serve as an alternative definition of discrete random variables. A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution . Expressed formally,

7568-400: A sine, sin ⁡ ( t ) {\displaystyle \sin(t)} , whose limit when t → ∞ {\displaystyle t\rightarrow \infty } does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies

7744-460: 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

7920-516: A sufficient sample size to specifying an adequate null hypothesis. Statistical measurement processes are also prone to error in regards to 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

8096-636: 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

8272-519: 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 is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize

8448-778: A uniform distribution between 0 and 1. To construct a random Bernoulli variable for some 0 < p < 1 {\displaystyle 0<p<1} , we define X = { 1 , if  U < p 0 , if  U ≥ p {\displaystyle X={\begin{cases}1,&{\text{if }}U<p\\0,&{\text{if }}U\geq p\end{cases}}} so that Pr ( X = 1 ) = Pr ( U < p ) = p , Pr ( X = 0 ) = Pr ( U ≥ p ) = 1 − p . {\displaystyle \Pr(X=1)=\Pr(U<p)=p,\quad \Pr(X=0)=\Pr(U\geq p)=1-p.} This random variable X has

8624-418: 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

8800-453: 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 a given probability distribution : standard statistical inference and estimation theory defines

8976-403: Is a countable set with P ( X ∈ A ) = 1 {\displaystyle P(X\in A)=1} . Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function p ( x ) = P ( X = x ) {\displaystyle p(x)=P(X=x)} . In the case where

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9152-433: Is a mathematical body of science that pertains to 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

9328-467: Is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events ( subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads , and 0.5 for X = tails (assuming that the coin is fair ). More commonly, probability distributions are used to compare

9504-553: Is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable X {\displaystyle X} has an absolutely continuous probability distribution if there is a function f : R → [ 0 , ∞ ] {\displaystyle f:\mathbb {R} \to [0,\infty ]} such that for each interval I = [

9680-575: 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

9856-465: 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

10032-453: Is called dual system enumeration (DSE). A sample of households is visited by interviewers who record the details of the household as of census day. These data are then matched to census records, and the number of people missed can be estimated by considering the number of people who are included in one count but not the other. This allows adjustments to the count for non-response, varying between different demographic groups. An explanation using

10208-833: 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

10384-551: Is common to denote as P ( X ∈ E ) {\displaystyle P(X\in E)} the probability that a certain value of the variable X {\displaystyle X} belongs to a certain event E {\displaystyle E} . The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms , that is: The concept of probability function

10560-401: Is defined as F ( x ) = P ( X ≤ x ) . {\displaystyle F(x)=P(X\leq x).} The cumulative distribution function of any real-valued random variable has the properties: Conversely, any function F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } that satisfies the first four of

10736-401: Is important in considering individuals who have multiple or temporary addresses. Every person should be identified uniquely as a resident in one place; but the place where they happen to be on Census Day, their de facto residence , may not be the best place to count them. Where an individual uses services may be more useful, and this is at their usual residence. An individual may be recorded at

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10912-469: Is interested; researchers in particular have an interest in the role of Census Field Officers (CFO) and their assistants. Data can be represented visually or analysed in complex statistical models, to show the difference between certain areas, or to understand the association between different personal characteristics. Census data offer a unique insight into small areas and small demographic groups which sample data would be unable to capture with precision. In

11088-508: Is made more rigorous by defining it as the element of a probability space ( X , A , P ) {\displaystyle (X,{\mathcal {A}},P)} , where X {\displaystyle X} is the set of possible outcomes, A {\displaystyle {\mathcal {A}}} is the set of all subsets E ⊂ X {\displaystyle E\subset X} whose probability can be measured, and P {\displaystyle P}

11264-690: Is named after the counting of the Israelite population according to the house of the Fathers after the exodus from Egypt. A second census was taken while the Israelites were camped in the " plains of Moab ". King David performed a census that produced disastrous results. His son, King Solomon , had all of the foreigners in Israel counted. One of the world's earliest preserved censuses was held in China in AD   2 during

11440-538: Is not known if there are any residents or how many people there are in each household. Depending on the mode of enumeration, a form is sent to the householder, an enumerator calls, or administrative records for the dwelling are accessed. As a preliminary to the dispatch of forms, census workers will check for any address problems on the ground. While it may seem straightforward to use the postal service file for this purpose, this can be out of date and some dwellings may contain several independent households. A particular problem

11616-417: 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

11792-462: Is possible because this measurement does not require as much precision from the underlying equipment. Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. An alternative description of

11968-423: Is still conducted in a similar way to the post-enumeration survey employed in a traditional census. Other countries that have a population register use this as a basis for all the census statistics needed by users. This is most common among Nordic countries but requires many distinct registers to be combined, including population, housing, employment, and education. These registers are then combined and brought up to

12144-415: Is students who often have a term time and family address. Several countries have used a system known as short form/long form. This is a sampling strategy that randomly chooses a proportion of people to send a more detailed questionnaire to (the long form). Everyone receives the short-form questions. This means more data are collected, but without imposing a burden on the whole population. This also reduces

12320-420: Is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system. This kind of complicated support appears quite frequently in dynamical systems . It is not simple to establish that the system has a probability measure, and

12496-424: Is the multivariate normal distribution . Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function. Some key concepts and terms, widely used in

12672-412: Is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set: a set of real numbers , a set of descriptive labels, a set of vectors , a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be  Ω = { "heads", "tails" } . To define probability distributions for the specific case of random variables (so

12848-406: Is the area under the probability density function from   − ∞   {\displaystyle \ -\infty \ } to   x   , {\displaystyle \ x\ ,} as shown in figure 1. A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of

13024-624: Is the probability distribution of a random variable that can take on only a countable number of values ( almost surely ) which means that the probability of any event E {\displaystyle E} can be expressed as a (finite or countably infinite ) sum: P ( X ∈ E ) = ∑ ω ∈ A ∩ E P ( X = ω ) , {\displaystyle P(X\in E)=\sum _{\omega \in A\cap E}P(X=\omega ),} where A {\displaystyle A}

13200-399: Is the probability function, or probability measure , that assigns a probability to each of these measurable subsets E ∈ A {\displaystyle E\in {\mathcal {A}}} . Probability distributions usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g.

13376-402: 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

13552-491: Is typically collected about the household structure and the housing. For this reason, international documents refer to censuses of population and housing. Normally the census response is made by a household, indicating details of individuals resident there. An important aspect of census enumerations is determining which individuals can be counted and which cannot be counted. Broadly, three definitions can be used: de facto residence; de jure residence; and permanent residence. This

13728-416: Is what is termed " communal establishments ", a category that includes student residences, religious orders, homes for the elderly, people in prisons, etc. As these are not easily enumerated by a single householder, they are often treated differently and visited by special teams of census workers to ensure they are classified appropriately. Individuals are normally counted within households , and information

13904-449: 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

14080-756: 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

14256-1158: The Dirac delta function as a generalized probability density function f {\displaystyle f} , where f ( x ) = ∑ ω ∈ A p ( ω ) δ ( x − ω ) , {\displaystyle f(x)=\sum _{\omega \in A}p(\omega )\delta (x-\omega ),} which means P ( X ∈ E ) = ∫ E f ( x ) d x = ∑ ω ∈ A p ( ω ) ∫ E δ ( x − ω ) = ∑ ω ∈ A ∩ E p ( ω ) {\displaystyle P(X\in E)=\int _{E}f(x)\,dx=\sum _{\omega \in A}p(\omega )\int _{E}\delta (x-\omega )=\sum _{\omega \in A\cap E}p(\omega )} for any event E . {\displaystyle E.} For

14432-546: The Han dynasty , and is still considered by scholars to be quite accurate. The population was registered as having 57,671,400 individuals in 12,366,470 households but on this occasion only taxable families had been taken into account, indicating the income and the number of soldiers who could be mobilized. Another census was held in AD   144. The oldest recorded census in India is thought to have occurred around 330   BC during

14608-510: The Poisson distribution , the Bernoulli distribution , the binomial distribution , the geometric distribution , the negative binomial distribution and categorical distribution . When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally,

14784-477: 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

14960-473: The discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices. A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities —that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely

15136-544: 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

15312-400: The half-open interval [0, 1) . These random variates X {\displaystyle X} are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated. For example, suppose U {\displaystyle U} has

15488-706: 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

15664-668: The nomarch , "whence he gained his living". Under the Ptolemies and the Romans several censuses were conducted in Egypt by government officials. There are several accounts of ancient Greek city states carrying out censuses. Censuses are mentioned several times in the Biblical narrative. God commands a per capita tax to be paid with the census for the upkeep of the Tabernacle . The Book of Numbers

15840-437: The population size is already known. However, a census is also used to collect attribute data on the individuals in the nation, not only to assess population size. This process of sampling marks the difference between a historical census, which was a house-to-house process or the product of an imperial decree, and the modern statistical project. The sampling frame used by a census is almost always an address register. Thus, it

16016-820: The probability distribution of X {\displaystyle X} is the image measure X ∗ P {\displaystyle X_{*}\mathbb {P} } of X {\displaystyle X} , which is a probability measure on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} satisfying X ∗ P = P X − 1 {\displaystyle X_{*}\mathbb {P} =\mathbb {P} X^{-1}} . Absolutely continuous and discrete distributions with support on R k {\displaystyle \mathbb {R} ^{k}} or N k {\displaystyle \mathbb {N} ^{k}} are extremely useful to model

16192-409: 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 the limit to

16368-412: The 19th and 20th centuries collected paper documents which had to be collated by hand, so the statistical information obtained was quite basic. The government that owned the data could publish statistics on the state of the nation. The results were used to measure changes in the population and apportion representation. Population estimates could be compared to those of other countries. By the beginning of

16544-437: The 20th century, censuses were recording households and some indications of their employment. In some countries, census archives are released for public examination after many decades, allowing genealogists to track the ancestry of interested people. Archives provide a substantial historical record which may challenge established views. Information such as job titles and arrangements for the destitute and sick may also shed light on

16720-460: The UK, all census formats are scanned and stored electronically before being destroyed, replacing the need for physical archives. The record linking to perform an administrative census would not be possible without large databases being stored on computer systems. There are sometimes problems in introducing new technology. The US census had been intended to use handheld computers, but cost escalated, and this

16896-676: The according equality still holds: P ( X ∈ A ) = ∫ A f ( x ) d x . {\displaystyle P(X\in A)=\int _{A}f(x)\,dx.} An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous. There are many examples of absolutely continuous probability distributions: normal , uniform , chi-squared , and others . Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case,

17072-417: The burden on the statistical office. Indeed, in the UK until 2001 all residents were required to fill in the whole form but only a 10% sample was coded and analysed in detail. New technology means that all data are now scanned and processed. During the 2011 Canadian census there was controversy about the cessation of the mandatory long-form census; the head of Statistics Canada , Munir Sheikh , resigned upon

17248-438: The census is also an important tool for identifying forms of social, demographic or economic exclusions, such as inequalities relating to race, ethics, and religion as well as disadvantaged groups such as those with disabilities and the poor. An accurate census can empower local communities by providing them with the necessary information to participate in local decision-making and ensuring they are represented. The importance of

17424-536: The census is that it represents the entire statistical universe, down to the smallest geographical units, of a country or region. Planners need this information for all kinds of development work, including: assessing demographic trends; analysing socio-economic conditions; designing evidence-based poverty-reduction strategies; monitoring and evaluating the effectiveness of policies; and tracking progress toward national and internationally agreed development goals." In addition to making policymakers aware of population issues,

17600-443: The census of agriculture , data is collected at the agricultural holding unit. An agricultural holding is an economic unit of agricultural production under single management comprising all livestock kept and all land used wholly or partly for agricultural production purposes, without regard to title, legal form, or size. Single management may be exercised by an individual or household, jointly by two or more individuals or households, by

17776-474: The census of agriculture for development is that it gives a snapshot of the structure of the agricultural sector in a country and, when compared with previous censuses, provides an opportunity to identify trends and structural transformations of the sector, and points towards areas for policy intervention. Census data are used as a benchmark for current statistics and their value is increased when they are employed together with other data sources. Early censuses in

17952-563: The census of agriculture, users need census data to: Although the census provides useful statistical information about a population, the availability of this information could sometimes lead to abuses, political or otherwise, by the linking of individuals' identities to anonymous census data. This is particularly important when individuals' census responses are made available in microdata form, but even aggregate-level data can result in privacy breaches when dealing with small areas and/or rare subpopulations. For instance, when reporting data from

18128-564: The central government. Differing release strategies of governments have led to an international project ( IPUMS ) to co-ordinate access to microdata and corresponding metadata. Such projects such as SDMX also promote standardising metadata, so that best use can be made of the minimal data available. Censuses in Egypt first appeared in the late Middle Kingdom and developed in the New Kingdom Pharaoh Amasis , according to Herodotus , required every Egyptian to declare annually to

18304-429: 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 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

18480-515: The combination of data from registers, surveys and other sources. Censuses have evolved in their use of technology: censuses in 2010 used many new types of computing. In Brazil, handheld devices were used by enumerators to locate residences on the ground. In many countries, census returns could be made via the Internet as well as in paper form. DSE is facilitated by computer matching techniques that can be automated, such as propensity score matching . In

18656-532: 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

18832-534: 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

19008-402: The cumulative distribution function F {\displaystyle F} has the form F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t {\displaystyle F(x)=P(X\leq x)=\int _{-\infty }^{x}f(t)\,dt} where f {\displaystyle f} is a density of

19184-427: The distribution is by means of the cumulative distribution function , which describes the probability that the random variable is no larger than a given value (i.e.,   P ( X < x )   {\displaystyle \ {\boldsymbol {\mathcal {P}}}(X<x)\ } for some   x   {\displaystyle \ x\ } ). The cumulative distribution function

19360-406: 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

19536-495: 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

19712-439: The existence of a probability measure is ergodic theory . Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively. Most algorithms are based on a pseudorandom number generator that produces numbers X {\displaystyle X} that are uniformly distributed in

19888-478: 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

20064-473: 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

20240-402: 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

20416-465: The face of uncertainty. In applying statistics to 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

20592-548: The federal government's decision to do so. The use of alternative enumeration strategies is increasing but these are not as simple as many people assume and are only used in developed countries. The Netherlands has been most advanced in adopting a census using administrative data . This allows a simulated census to be conducted by linking several different administrative databases at an agreed time. Data can be matched, and an overall enumeration established allowing for discrepancies between different data sources. A validation survey

20768-431: 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

20944-423: The form of conditional distributions ( histograms ) can be derived interactively from the estimated mixture model without any further access to the original database. As the final product does not contain any protected microdata, the model-based interactive software can be distributed without any confidentiality concerns. Another method is simply to release no data at all, except very large scale data directly to

21120-402: 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

21296-605: 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,

21472-785: The historical structure of society. Political considerations influence the census in many countries. In Canada in 2010 for example, the government under the leadership of Stephen Harper abolished the mandatory long-form census. This abolition was a response to protests from some Canadians who resented the personal questions. The long-form census was reinstated by the Justin Trudeau government in 2016. As governments assumed responsibility for schooling and welfare, large government research departments made extensive use of census data. Population projections could be made, to help plan for provision in local government and regions. Central government could also use census data to allocate funding. Even in

21648-466: The identification of individuals in marginal populations; others swap variables for similar respondents. Whatever is done to reduce the privacy risk, new improved electronic analysis of data can threaten to reveal sensitive individual information. This is known as statistical disclosure control . Another possibility is to present survey results by means of statistical models in the form of a multivariate distribution mixture. The statistical information in

21824-499: The internet. It is also possible that the hidden nature of an administrative census means that users are not engaged with the importance of contributing their data to official statistics. Alternatively, population estimations may be carried out remotely with geographic information system (GIS) and remote sensing technologies. According to the United Nations Population Fund (UNFPA), "The information generated by

22000-506: The inverse is not true, there exist singular distributions , which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution . Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous , i.e. refer to absolutely continuous distributions as continuous distributions. For

22176-449: The literature on the topic of probability distributions, are listed below. In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable X {\displaystyle X} with regard to a probability distribution p {\displaystyle p}

22352-741: The main problem is the following. Let t 1 ≪ t 2 ≪ t 3 {\displaystyle t_{1}\ll t_{2}\ll t_{3}} be instants in time and O {\displaystyle O} a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set O {\displaystyle O} would be equal in interval [ t 1 , t 2 ] {\displaystyle [t_{1},t_{2}]} and [ t 2 , t 3 ] {\displaystyle [t_{2},t_{3}]} , which might not happen; for example, it could oscillate similar to

22528-429: The members of a given population , usually displayed in the form of statistics. This term is used mostly in connection with national population and housing censuses ; other common censuses include censuses of agriculture , traditional culture, business, supplies, and traffic censuses. The United Nations (UN) defines the essential features of population and housing censuses as "individual enumeration, universality within

22704-524: The mid 20th century, census data was only directly accessible to large government departments. However, computers meant that tabulations could be used directly by university researchers, large businesses and local government offices. They could use the detail of the data to answer new questions and add to local and specialist knowledge. Nowadays, census data are published in a wide variety of formats to be accessible to business, all levels of government, media, students and teachers, charities, and any citizen who

22880-418: 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

23056-437: The most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function P : A → R {\displaystyle P\colon {\mathcal {A}}\to \mathbb {R} } whose input space A {\displaystyle {\mathcal {A}}} is a σ-algebra , and gives a real number probability as its output, particularly,

23232-859: The number of dots on the die, has the probability   1 6   ) . {\displaystyle \ {\tfrac {1}{6}}~).} The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is   p ( “ 2 ” ) + p ( “ 4 ” ) + p ( “ 6 ” ) = 1 6 + 1 6 + 1 6 = 1 2   . {\displaystyle \ p({\text{“}}2{\text{”}})+p({\text{“}}4{\text{”}})+p({\text{“}}6{\text{”}})={\tfrac {1}{6}}+{\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{2}}~.} In contrast, when

23408-403: The number of elected representatives to regions (sometimes controversially – e.g., Utah v. Evans ). In many cases, a carefully chosen random sample can provide more accurate information than attempts to get a population census. A census is often construed as the opposite of a sample as it intends to count everyone in a population, rather than a fraction. However, population censuses do rely on

23584-497: The one ordered by the second Rashidun caliph , Umar . The Domesday Book was undertaken in AD   1086 by William I of England so that he could properly tax the land he had recently conquered. In 1183, a census was taken of the crusader Kingdom of Jerusalem , to ascertain the number of men and amount of money that could possibly be raised against an invasion by Saladin , sultan of Egypt and Syria . The first national census of France ( L'État des paroisses et des feux )

23760-412: 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

23936-415: 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

24112-544: 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

24288-493: The probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it. One example is shown in the figure to the right, which displays the evolution of a system of differential equations (commonly known as the Rabinovich–Fabrikant equations ) that can be used to model the behaviour of Langmuir waves in plasma . When this phenomenon

24464-533: The probability that X {\displaystyle X} takes any value except for u 0 , u 1 , … {\displaystyle u_{0},u_{1},\dots } is zero, and thus one can write X {\displaystyle X} as X ( ω ) = ∑ i u i 1 Ω i ( ω ) {\displaystyle X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )} except on

24640-458: The probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero decimal digits in the remaining omitted digits ignored by the precision level. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability. This

24816-493: 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

24992-540: The problems of overcount and undercount and the coherence of census enumerations with other official sources of data. For instance, during the 2020 U.S. Census, the Census Bureau counted people primarily by collecting answers sent by mail, on the internet, over the phone, or using shared information through proxies. These methods accounted for 95.5 percent of all occupied housing units in the United States. This reflects

25168-454: The properties above is the cumulative distribution function of some probability distribution on the real numbers. Any probability distribution can be decomposed as the mixture of a discrete , an absolutely continuous and a singular continuous distribution , and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions. A discrete probability distribution

25344-465: 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

25520-450: The random variable X {\displaystyle X} has a one-point distribution if it has a possible outcome x {\displaystyle x} such that P ( X = x ) = 1. {\displaystyle P(X{=}x)=1.} All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1. An absolutely continuous probability distribution

25696-408: The random variable X {\displaystyle X} with regard to the distribution P {\displaystyle P} . Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions , which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but

25872-643: The range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if p ( n ) = 1 2 n {\displaystyle p(n)={\tfrac {1}{2^{n}}}} for n = 1 , 2 , . . . {\displaystyle n=1,2,...} , the sum of probabilities would be 1 / 2 + 1 / 4 + 1 / 8 + ⋯ = 1 {\displaystyle 1/2+1/4+1/8+\dots =1} . Well-known discrete probability distributions used in statistical modeling include

26048-490: The real numbers. A discrete probability distribution is often represented with Dirac measures , the probability distributions of deterministic random variables . For any outcome ω {\displaystyle \omega } , let δ ω {\displaystyle \delta _{\omega }} be the Dirac measure concentrated at ω {\displaystyle \omega } . Given

26224-498: The reign of Emperor Chandragupta Maurya under the leadership of Chanakya and Ashoka . The English term is taken directly from the Latin census , from censere ("to estimate"). The census played a crucial role in the administration of the Roman government, as it was used to determine the class a citizen belonged to for both military and tax purposes. Beginning in the middle republic, it

26400-523: The relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space . The sample space, often represented in notation by   Ω   , {\displaystyle \ \Omega \ ,}

26576-443: The same level of detail but raise concerns about privacy and the possibility of biasing estimates. A census can be contrasted with sampling in which information is obtained only from a subset of a population; typically, main population estimates are updated by such intercensal estimates . Modern census data are commonly used for research, business marketing , and planning, and as a baseline for designing sample surveys by providing

26752-588: 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 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

26928-460: 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

27104-439: 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

27280-399: 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

27456-431: The sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables . In the discrete case, it is sufficient to specify a probability mass function   p   {\displaystyle \ p\ } assigning a probability to each possible outcome (e.g. when throwing a fair die , each of the six digits “1” to “6” , corresponding to

27632-405: 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

27808-412: 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

27984-462: 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 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

28160-530: The standard of a statistical register by comparing the data from different sources and ensuring the quality is sufficient for official statistics to be produced. A recent innovation is the French instigation of a rolling census program with different regions enumerated each year so that the whole country is completely enumerated every 5 to 10 years. In Europe, in connection with the 2010 census round, many countries adopted alternative census methodologies, often based on

28336-479: The structure of agriculture, covering the whole or a significant part of a country." "In a census of agriculture, data are collected at the holding level." The word is of Latin origin: during the Roman Republic , the census was a list of all adult males fit for military service. The modern census is essential to international comparisons of any type of statistics, and censuses collect data on many attributes of

28512-448: The temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density function , and the probability distribution is by definition the integral of the probability density function. The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time , may demand

28688-494: 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. Census A census (from Latin censere , 'to assess') is the procedure of systematically acquiring, recording, and calculating population information about

28864-478: The third by Karl Friedrich Wilhelm Dieterici in 1859. In 1931, Walter Willcox published a table in his book, International Migrations: Volume II Interpretations , that estimated the 1929 world population to be roughly 1.8 billion. Continuous probability distribution In probability theory and statistics , a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment . It

29040-420: 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

29216-417: 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

29392-416: 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

29568-617: 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

29744-432: The use of more general probability measures . A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate , while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate . A univariate distribution gives the probabilities of a single random variable taking on various different values;

29920-463: The values which the random variable may take. Thus the cumulative distribution function has the form F ( x ) = P ( X ≤ x ) = ∑ ω ≤ x p ( ω ) . {\displaystyle F(x)=P(X\leq x)=\sum _{\omega \leq x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in

30096-439: The widespread use of random variables , which transform the sample space into a set of numbers (e.g., R {\displaystyle \mathbb {R} } , N {\displaystyle \mathbb {N} } ), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It

30272-459: 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

30448-443: Was abandoned, with the contract being sold to Brazil. The online response has some advantages, but one of the functions of the census is to make sure everyone is counted accurately. A system that allowed people to enter their address without verification would be open to abuse. Therefore, households have to be verified on the ground, typically by an enumerator visit or post out . Paper forms are still necessary for those without access to

30624-524: Was designed to elicit basic information about the nature of the land and the life of its peoples. The replies, known as " relaciones geográficas ", were written between 1579 and 1585 and were returned to the Cronista Mayor in Spain by the Council of the Indies. The earliest estimate of the world population was made by Giovanni Battista Riccioli in 1661; the next by Johann Peter Süssmilch in 1741, revised in 1762;

30800-518: Was undertaken in 1328, mostly for fiscal purposes. It estimated the French population at 16 to 17 million. In the 15th century, the Inca Empire had a unique way to record census information. The Incas did not have any written language but recorded information collected during censuses and other numeric information as well as non-numeric data on quipus , strings from llama or alpaca hair or cotton cords with numeric and other values encoded by knots in

30976-489: Was usually carried out every five years. It provided a register of citizens and their property from which their duties and privileges could be listed. It is said to have been instituted by the Roman king Servius Tullius in the 6th century BC, at which time the number of arms-bearing citizens was supposedly counted at around 80,000. When the Romans conquered Judea in AD   6, the legate Publius Sulpicius Quirinius organized

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