Statistics

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Statistics is a mathematical science pertaining to collection, analysis, interpretation and presentation of data. It is applicable to a wide variety of academic disciplines from the physical and social sciences to the humanities, as well as to business, government, medicine and industry.

Given a collection of data, statistics may be employed to summarize or describe the data; this use is called descriptive statistics. In addition, patterns in the data may be modeled, in a way that accounts for randomness and uncertainty in the observations, in order to draw inferences about the larger population; this use is called inferential statistics. Both of these uses may be termed applied statistics. There is also a discipline of mathematical statistics concerned with the theoretical basis of the subject.

The word statistics is also the plural of statistic (singular), which refers to the result of applying a statistical algorithm to a set of data.

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Historical overview

The word statistics ultimately derives from the modern Latin term statisticum collegium ("council of state") and the Italian word statista ("statesman" or "politician"). The German Statistik, first introduced by Gottfried Achenwall (1749), originally designated the analysis of data about the state, signifying the "science of state". It acquired the meaning of the collection and classification of data generally in the early 19th century. It was introduced into English by Sir John Sinclair.

Thus, the original principal purpose of statistics was data to be used by governmental and (often centralized) administrative bodies. The collection of data about states and localities continues, largely through national and international statistical services; in particular, censuses provide regular information about the population.

Statistics eventually merged with the more mathematically oriented field of inverse probability, referring to the estimation of a parameter from experimental data in the experimental sciences (most notably astronomy). Today the use of statistics has broadened far beyond the service of a state or government, to include such areas as business, natural and social sciences, and medicine, among others.

Because of its history and wide applicability, statistics is generally regarded not as a subfield of mathematics but as a distinct, albeit allied, field. Many large universities maintain separate mathematics and statistics departments. Statistics is also taught in departments as diverse as psychology, education, and public health.

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Important contributors to statistics

See also list of statisticians.

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Conceptual overview

In applying statistics to a scientific, industrial, or societal problem, one begins with a population to be studied. This might be a population of people in a country, of crystal grains in a rock, or of goods manufactured by a particular factory. The population may even consist of a single process observed at various times; data collected about this kind of "population" constitute what is called a time series.

For practical reasons, rather than compiling data about the entire population, one instead studies a chosen subset of the population, called a sample. Data are collected about the sample in some kind of experimental setting. The data are then subjected to statistical analysis, which serves two related purposes: description and inference.

  • Descriptive statistics deals with the description problem: Can the data be summarized in a useful way, either numerically or graphically, to yield insight about the population in question? Basic examples of numerical descriptors include the mean and standard deviation. Graphical summarizations include various kinds of charts and graphs.
  • Inferential statistics is used to model patterns in the data, accounting for randomness and drawing inferences about the larger population. These inferences may take the form of answers to yes/no questions (hypothesis testing), estimates of numerical characteristics (estimation), prediction of future observations, descriptions of association (correlation), or modeling of relationships (regression). Other modeling techniques include ANOVA, time series, and data mining.

The concept of correlation is particularly noteworthy. Statistical analysis of a data set may reveal that two variables (that is, two properties of the population under consideration) tend to vary together, as if they are connected. For example, a study of annual income and age of death among people might find that poor people tend to have shorter lives, on average, than affluent people. The two variables are said to be correlated. However, one cannot immediately infer the existence of a causal relationship between the two variables; see correlation implies causation (logical fallacy).

If the sample is representative of the population, then inferences and conclusions made from the sample can be extended to the population as a whole. A major problem lies in determining the extent to which the chosen sample is representative. Statistics offers methods to estimate and correct for randomness (uncertainty) in the sample and in the data collection procedure, as well as methods for designing robust experiments in the first place; see experimental design.

The fundamental mathematical concept employed in understanding such randomness is probability. Mathematical statistics (also called statistical theory) is the branch of applied mathematics that uses probability theory and analysis to examine the theoretical basis of statistics.

The use of any statistical method is valid only when the system or population under consideration satisfies the basic mathematical assumptions of the method. Misuse of statistics can produce subtle but serious errors in description and interpretation — subtle in that even experienced professionals sometimes make such errors, and serious in that they may affect social policy, medical practice and the reliability of structures such as bridges and nuclear power plants.

Even when statistics is correctly applied, the results can be difficult to interpret for a non-expert. For example, the statistical significance of a trend in the data — which measures the extent to which the trend may be caused by random variation in the sample — may not agree with one's intuitive sense of its significance. The set of basic statistical skills (and skepticism) needed by people to deal with information in their everyday lives is referred to as statistical literacy.

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Statistical methods

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Experimental and observational studies

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 a response or dependent variable. There are two major types of causal statistical studies, experimental studies and observational studies. In both types of studies, 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 is in how the study is actually conducted.

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 may have 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 the response are investigated.

An example of an experimental study is the famous Hawthorne studies which attempted to test changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured productivity in the plant then modified the illumination in an area of the plant to see if changes in illumination would affect productivity. Due to errors in experimental procedures, specifically the lack of a control group and blindedness, the researchers were unable to do what they planned, in what is known as the Hawthorne effect.

An example of an observational study is a study which explores the correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then perform statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers and then look at the number of cases of lung cancer in each group.

The basic steps for an experiment are to:

  1. plan the research including determining information sources, research subject selection, and ethical considerations for the proposed research and method,
  2. design the experiment concentrating on the system model and the interaction of independent and dependent variables,
  3. summarize a collection of observations to feature their commonality by suppressing details (descriptive statistics),
  4. reach consensus about what the observations tell us about the world we observe (statistical inference),
  5. document and present the results of the study.
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Levels of measurement

There are four types of measurements or measurement scales used in statistics. The four types or levels of measurement (ordinal, nominal, interval, and ratio) have different degrees of usefulness in statistical research. Ratio measurements, where both a zero value and distances between different measurements are defined, provide the greatest flexibility in statistical methods that can be used for analysing the data. Interval measurements, with meaningful distances between measurements but no meaningful zero value (such as IQ measurements or temperature measurements in degrees Celsius). Ordinal measurements have imprecise differences between consecutive values but a meaningful order to those values. Nominal measurements have no meaningful rank order among values.

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Statistical techniques

Some well known statistical tests and procedures for research observations are:

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Specialized disciplines

Some sciences use applied statistics so extensively that they have specialized terminology. These disciplines include:

Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles it is a key tool, and perhaps the only reliable tool.

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Software

The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models were almost always from the class of linear models, but powerful computers, coupled with suitable numerical algorithms, caused a resurgence of interest in nonlinear models (especially neural networks and decision trees) and the creation of new types, such as generalised linear models and multilevel models.

The computer revolution has implications for the future of statistics, with a new emphasis on "experimental" and "empirical" statistics.

Statistical packages in common use include the following:

Open source or freeware: Proprietary:
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See also

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External links

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General sites and organizations

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Link collections

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Online courses and textbooks

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Statistical software

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Other resources

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Additional references

  • Lindley, D. (1985). Making Decisions, Second Edition. London, New York: John Wiley. ISBN 0471908088 (paperback edition.)
  • Tijms, H., Understanding probability : chance rules in everyday life . Cambridge, New York: Cambridge University Press. 2004. ISBN 0521833299.
  • Desrosières, Alain. La politique des grands nombres. Histoire de la raison statistique ("The politics of great numbers. History of the statistic reason" - a very complete account of the historical formation of statistics and epistemological problems) - La Découverte, 2000. ISBN 2707133531.dv:Statistics

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