# Statistics

Statistics (from

experiments.[6]

When

experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study
does not involve experimental manipulation.

Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from 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).[7] 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 on mathematical statistics are made under 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

Type II errors (null hypothesis fails to be rejected when an it is in fact false, giving a "false negative").[8] Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.[7]

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.

## Introduction

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data,[9] or as a branch of mathematics.[10] 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.[11][12]

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

categorical data
(like education).

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

correlation), and modeling relationships within the data (for example, using regression analysis). Inference can extend to 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

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

measure-theoretic probability theory.[13][14]

## History

Formal discussions on inference date back to

sample size in frequency analysis.[15]

Although the term 'statistic' was introduced by the Italian scholar

Natural and Political Observations upon the Bills of Mortality by John Graunt.[18] 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 is widely employed in government, business, and natural and social sciences.

The mathematical foundations of statistics developed from discussions concerning

method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it a decade earlier in 1795.[22]

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

Pearson product-moment correlation coefficient, defined as a product-moment,[25] the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things.[26] Galton and Pearson founded Biometrika as 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.[27]

The second wave of the 1910s and 20s was initiated by

sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information.[31] 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".[32][33] In his 1930 book The Genetical Theory of Natural Selection, he applied statistics to various biological concepts such as Fisher's principle[34] (which A. W. F. Edwards called "probably the most celebrated argument in evolutionary biology") and Fisherian runaway,[35][36][37][38][39][40] 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 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.[41]

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.[42]

## Statistical data

### Data collection

#### Sampling

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

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 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.

#### 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 dependent variables. 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 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

instrumental variables, among many others) that produce consistent estimators
.

##### Experiments

The basic steps of a statistical experiment are:

Experiments on human behavior have special concerns. The famous

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.[44]

##### Observational study

An example of an observational study 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

case-control study
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.

### Types of data

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)[46] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)[47] described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),[48] van den Berg (1991).[49])

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."[50]: 82

## Methods

### Descriptive statistics

A descriptive statistic (in the

sample, rather than use the data to learn about the population that the sample of data is thought to represent.[52]

### Inferential statistics

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.[53] 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.[54]

#### Terminology and theory of inferential statistics

##### Statistics, estimators and pivotal quantities

Consider

column vector of these IID variables.[55] 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 a function of the unknown parameter: an

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

.

Between two estimators of a given parameter, the one with lower

true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the limit
to the true value of such parameter.

Other desirable properties for estimators include:

converges in probability
to the true value of such parameter.

This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the

maximum likelihood method, the least squares method and the more recent method of estimating equations
.

##### Null hypothesis and alternative hypothesis

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.[56][57]

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

type II errors
.

What

statisticians call an alternative hypothesis
is simply a hypothesis that contradicts the null hypothesis.

##### Error

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".

Standard error
refers to an estimate of difference between sample mean and population mean.

A

residual
is the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).

Root mean square error
is simply the square root of mean squared error.

Many statistical methods seek to minimize the

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

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.[58]

##### Interval estimation

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.

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 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 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 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 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.

##### Significance

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).

The standard approach

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 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 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.

Some problems are usually associated with this framework (See

criticism of hypothesis testing
):

##### Examples

Some well-known statistical

tests
and procedures are:

### Exploratory data analysis

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

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.[60] 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,[60] 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)).[61]

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

overgeneralized and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias.[63] Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs.[62] Most people do not look for bias or errors, so they are not noticed. Thus, people may often believe that something is true even if it is not well represented.[63] To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.[64] According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."[65]

• 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?)

### Misinterpretation: correlation

The concept of

confounding variable
. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.

## Applications

### Applied statistics, theoretical statistics and mathematical statistics

Applied statistics, sometimes referred to as Statistical science,[66] comprises descriptive statistics and the application of inferential statistics.[67][68] 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.

Statistical consultants
can help organizations and companies that do not have in-house expertise relevant to their particular questions.

### Machine learning and data mining

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

### Statistics in academia

Statistics is applicable to a wide variety of

A typical statistics course covers descriptive statistics, probability, binomial and

### Statistical computing

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

neural networks) as well as the creation of new types, such as generalized linear models and multilevel models
.

Increased computing power has also led to the growing popularity of computationally intensive methods based on

.

In business, "statistics" is a widely used

economic relationships
.)

A typical "Business Statistics" course is intended for

, often include topics in statistics.

### Statistics applied to mathematics or the arts

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.

## Specialized disciplines

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

specialized terminology
. These disciplines include:

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. In these roles, it is a key tool, and perhaps the only reliable tool.[citation needed]

Foundations and major areas of statistics

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