Statistics

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Statistics (from

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).^[4] Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory, which deals with the analysis of random phenomena.

A standard statistical procedure involves the collection of data leading to a

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 is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data,^[5] or as a branch of mathematics.^[6] Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in the context of uncertainty and decision-making in the face of uncertainty.^[7][8]

In applying statistics to a problem, it is common practice to start with a

When a census is not feasible, a chosen subset of the population called a

Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include

Formal discussions on inference date back to the

The mathematical foundations of statistics developed from discussions concerning

The modern field of statistics emerged in the late 19th and early 20th century in three stages.

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 the problem of how to analyze big data.^[37]

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 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 the population.

Sampling theory is part of the

1. Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size of
   experimental variability
   . Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects.
2. experimental protocol
   that will guide the performance of the experiment and which specifies the primary analysis of the experimental data.
3. Performing the experiment following the
   experimental protocol and analyzing the data
   following the experimental protocol.
4. Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
5. Documenting and presenting the results of the study.

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 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 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)^[41] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)^[42] described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),^[43] van den Berg (1991).^[44])

The issue of whether or not it 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 a transformation is sensible to contemplate depends on the question one is trying to answer."^[45]: 82

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A descriptive statistic (in the

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.^[48] Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.^[49]

Consider

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.^[51][52]

The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H₀, asserts that the defendant is innocent, whereas the alternative hypothesis, H₁, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H₀ (status quo) stands in opposition to H₁ and is maintained unless H₁ is supported by evidence "beyond a reasonable doubt". However, "failure to reject H₀" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily accept H₀ but fails to reject H₀. While one can not "prove" a null hypothesis, one can test how close it is to being true with a

What

Working from a null hypothesis, two broad categories of error are recognized:

• Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
• Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".

A

Many statistical methods seek to minimize the

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as

Most studies only sample part of a population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value.

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 a value accurately rejecting the null hypothesis (sometimes referred to as the p-value).

• A difference that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests to account for this. One response involves going beyond reporting only the
  significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the size or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to report confidence intervals
  . Although these are produced from the same calculations as those of hypothesis tests or p-values, they describe both the size of the effect and the uncertainty surrounding it.
• Fallacy of the transposed conditional, aka
  prosecutor's fallacy: criticisms arise because the hypothesis testing approach forces one hypothesis (the null hypothesis) to be favored, since what is being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered by Bayesian inference, although it requires establishing a prior probability.^[54]
• Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
• As everything in
  fat tails p-values may be seriously mis-computed.^{[clarification needed}
  ]

Exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.

Misuse of statistics can produce subtle but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.

Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data—which measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy.

There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.^[55] A mistrust and misunderstanding of statistics is associated with the quotation, "There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional, and the book How to Lie with Statistics,^[55] by Darrell Huff, outlines a range of considerations. In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques used in particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter (2012)).^[56]

Ways to avoid misuse of statistics include using proper diagrams and avoiding

To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:[55]

• Who says so? (Does he/she have an axe to grind?)
• How does he/she know? (Does he/she have the resources to know the facts?)
• What's missing? (Does he/she give us a complete picture?)
• Did someone change the subject? (Does he/she offer us the right answer to the wrong problem?)
• Does it make sense? (Is his/her conclusion logical and consistent with what we already know?)

The concept of

Applied statistics, sometimes referred to as Statistical science,^[61] comprises descriptive statistics and the application of inferential statistics.^[62][63] Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.

Machine learning models are statistical and probabilistic models that capture patterns in the data through use of computational algorithms.

Statistics is applicable to a wide variety of

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

In business, "statistics" is a widely used

A typical "Business Statistics" course is intended for

Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This tradition has changed with the use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically.^[according to whom?] Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.

• In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
• Predictive methods of statistics in
  fractal geometry can be used to create video works.^[70]
• The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed.^[71] With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art.^{[citation needed]}
• Methods of statistics may be used predicatively in
  Markov process
  that only works some of the time, the occasion of which can be predicted using statistical methodology.
• Statistics can be used to predicatively create art, as in the statistical or
  stochastic music invented by Iannis Xenakis
  , where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tunable using statistics.

Statistical techniques are used in a wide range of types of scientific and social research, including:

In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:

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.

Foundations and major areas of statistics