# Statistics

Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from 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, and industry.

Key concepts and terms of statistics assume probability theory; among the terms are: population, sample, sampling, sampling unit and probability. Warning: systems are known to science that violate probability theory empirically.

Once data has been collected, either through a formal sampling procedure or by recording responses to treatments in an experimental setting (cf experimental design), or by repeatedly observing a process over time (time series), graphical and numerical summaries may be obtained using descriptive statistics.

Patterns in the data are modeled to draw inferences about the larger population, using inferential statistics to account for randomness and uncertainty in the observations. These inferences may take the form of answers to essentially yes/no questions (hypothesis testing), estimates of numerical characteristics (estimation), prediction of future observations, descriptions of association (correlation), or modeling of relationships (regression).

The framework described above is sometimes referred to as applied statistics. In contrast, mathematical statistics (or simply statistical theory) is the subdiscipline of applied mathematics which uses probability theory and analysis to place statistical practice on a firm theoretical basis.

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.

## Origin

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. It acquired the meaning of the collection and classification of data generally in the early nineteenth 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. Today, however, 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.

## Statistical methods

### 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 changes 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 is 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, the researchers while unable to do what they planned were able to provide the world with 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.

### 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.

### Statistical techniques

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

## Probability

Statistics makes extensive use of the concept of probability. The probability of an event is often defined as a number between one and zero. In reality however there is virtually nothing that has a probability of 1 or 0. You could say that the sun will certainly rise in the morning, but what if an extremely unlikely event destroys the sun? What if there is a nuclear war and the sky is covered in ash and smoke?

We often round the probability of such things up or down because they are so likely or unlikely to occur, that it's easier to recognize them as a probability of one or zero.

However, this can often lead to misunderstandings and dangerous behaviour, because people are unable to distinguish between, e.g., a probability of 10−4 and a probability of 10−9, despite the very practical difference between them. If you expect to cross the road about 105 or 106 times in your life, then reducing your risk of being run over per road crossing to 10−9 will make it unlikely that you will be run over while crossing the road for your whole life, while a risk per road crossing of 10−4 will make it very likely that you will have an accident, despite the intuitive feeling that 0.01% is a very small risk.

Use of prior probabilities of 0 (or 1) causes problems in Bayesian statistics, since the posterior probability is then forced to be 0 (or 1) as well. In other words, the data is not taken into account at all! As Dennis Lindley puts it, if a coherent Bayesian attaches a prior probability of zero to the hypothesis that the Moon is made of green cheese, then even whole armies of astronauts coming back bearing green cheese cannot convince him. Lindley advocates never using prior probabilities of 0 or 1. He calls it Cromwell's rule, from a letter Oliver Cromwell wrote to the synod of the Church of Scotland on August 5th, 1650 in which he said "I beseech you, in the bowels of Christ, consider it possible that you are mistaken."

## 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.

## Software

Modern statistics is supported by computers to perform some of the very large and complex calculations required.

Whole branches of statistics have been made possible by computing, for example neural networks.

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:

 Open source or Freeware proprietary