Glossary of probability and statistics

Short description: List of definitions of terms and concepts in statistics and probability

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This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.

A

admissible decision rule

algebra of random variables

alternative hypothesis

analysis of variance

atomic event

Another name for elementary event.

B

bar chart

Bayes' theorem

Bayes estimator

Bayes factor

Bayesian inference

bias

1.  Any feature of a sample that is not representative of the larger population.

2.  The difference between the expected value of an estimator and the true value.

binary data

Data that can take only two values, usually represented by the binary digits 0 and 1.

binomial distribution

bivariate analysis

A type of quantitative statistical analysis in which exactly two variables are analyzed, for the purpose of determining the empirical relationship between them. Contrast multivariate analysis.

blocking

In experimental design, the arranging of experimental units in groups ("blocks") that are similar to one another. Blocking is often used to manage the problem of pseudoreplication.

Box–Jenkins method

box plot

C

causal study

A statistical study in which the objective is to measure the effect of some variable on the outcome of a different variable. For example, a causal study might ask the question: "How will my headache feel if I take aspirin, versus if I do not take aspirin?" Causal studies may be either experimental or observational.

central limit theorem

central moment

characteristic function

chi-squared distribution

chi-squared test

cluster analysis

cluster sampling

complementary event

completely randomized design

computational statistics

The study of statistical methods that are enabled by using computational methods, at the interface of statistics and computer science.

concomitants

In a statistical study, any variables whose values are unaffected by experimental treatments, such as a unit’s age, gender, and cholesterol level before starting an experimental diet.

conditional distribution

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X (written "Y | X") is the probability distribution of Y when X is known to be a particular value.

conditional probability

The probability of some event A, assuming the occurrence of event B. In mathematical notation, conditional probability is written P(A|B), and is read "the probability of A, given B".

conditional probability distribution

confidence interval (CI)

In inferential statistics, a range of plausible values for some unknown parameter, such as a population mean, defined as an interval with a lower bound and an upper bound. The precise values of these bounds are calculated from a pre-determined confidence level, chosen by the researcher. The confidence level represents the frequency of intervals that, over the long run, capture the true value of the unknown parameter; i.e. 95% of confidence intervals computed at the 95% confidence level contain the true value, and likewise for other confidence levels. For example, based on a study of sleep habits among a random sample of 100 people, a researcher may estimate at the 95% confidence level that the overall population sleeps somewhere between 5 and 9 hours per night. There is a 95% chance that the true population mean falls within this interval, because 95% of random samples taken from this same population will yield 95% confidence intervals that contain the true mean.

confidence level

Also confidence coefficient.

A number indicating the probability that the confidence interval (range) captures the true population mean. For example, a confidence interval with a 95% confidence level has a 95% chance of capturing the population mean. Technically, this means that, if the experiment were repeated many times, 95% of the CIs computed at this level would contain the true population mean.

confounder

A variable that influences both the dependent variable and the independent variable, causing a spurious association. The existence of hidden confounding variables is an important quantitative explanation why correlation does not imply causation: if changes in two variables appear to be correlated, it is risky to presume that one change causes the other because it is possible that one or more unidentified confounders has in fact caused the changes in both variables. A classic example is the correlation between increased consumption of ice cream and increased crime in the summer. It is irrational to assume that eating more ice cream makes people commit more crime, or vice versa; it is more likely that one or more additional variables, e.g. warmer weather, increase both ice cream consumption and crime simultaneously. In this example, warmer weather is the confounder.

conjugate prior

continuous variable

convenience sampling

correlation

Also correlation coefficient.

A numeric measure of the strength of a linear relationship between two random variables (one can use it to quantify, for example, how shoe size and height are correlated in the population). An example is the Pearson product-moment correlation coefficient, which is found by dividing the covariance of the two variables by the product of their standard deviations. Independent variables, by definition, have a correlation of 0. A population correlation is often represented by the symbol [math]\displaystyle{ \rho }[/math], and a sample correlation by [math]\displaystyle{ r }[/math].</ref>

count data

Data arising from counting, and which can therefore take only non-negative integer values.

covariance

Given two random variables X and Y, with expected values [math]\displaystyle{ E(X)=\mu }[/math] and [math]\displaystyle{ E(Y)=\nu }[/math], the expected value of random variable [math]\displaystyle{ (X - \mu) (Y - \nu) }[/math], written in statistical notation as [math]\displaystyle{ \operatorname{cov}(X, Y) }[/math]. The covariance is used for measuring correlation; it can be interpreted as the degree to which the two variables change simultaneously with each other or "co-vary".

D

data

data analysis

data set

A sample and the associated data points.

data point

A typed measurement — it can be a Boolean value, a real number, a vector (in which case it is also called a data vector), etc.

decision rule

decision theory

degrees of freedom

density estimation

dependence

dependent variable

descriptive statistics

design of experiments

deviation

discrete variable

dot plot

double counting

E

elementary event

An event which contains only a single outcome in the sample space; in a set of possibilities, a possibility that can occur in precisely one way. For example, when pulling a card from a standard deck of playing cards, 'pulling the jack of spades' is an elementary event (because there is only one jack of spades in the entire deck), while 'pulling a king or an ace' is not (because there are a combined four kings and four aces in the deck).

estimation theory

The branch of statistics concerned with estimating the values of parameters based on measured empirical data with a random component. The parameters describe an underlying physical setting in such a way that their values affect the distribution of the measured data; an estimator attempts to use the measurements to approximate the unknown parameters.

estimator

A function of the known data that is used to estimate an unknown parameter; an estimate is the result of the actual application of the function to a particular set of data. For example, the mean can be used as an estimator.

expected value

Also expectation, mathematical expectation, first moment, or simply mean or average.

The sum of the probabilities of each possible outcome of an experiment multiplied by their corresponding payoff or "value". Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. For example, the expected value of rolling a fair six-sided die is 3.5. The concept is, intuitively, a generalization of the weighted average of all possible outcomes of a particular procedure or experiment, and can be viewed as the arithmetic mean of a large number of independent realizations of the experiment. The expected value of random variable X is typically written as E(X) for the expectation operator, and [math]\displaystyle{ \mu }[/math] (mu) for the parameter.

experiment

Any procedure which can be infinitely repeated and which has a well-defined set of outcomes.

exponential family

event

A subset of the sample space of a procedure or experiment (i.e. a possible outcome) to which a probability can be assigned. For example, on rolling a die, "getting a three" is an event (with a probability of ​¹⁄₆ if the die is fair), as is "getting a five or a six" (with a probability of ​¹⁄₃).

F

factor analysis

factorial experiment

frequency

frequency distribution

frequency domain

frequentist inference

G

general linear model

generalized linear model

grouped data

H

histogram

An approximate graphical representation of the distribution of numerical data. A histogram displays this distribution by dividing the entire range of values into a series of consecutive, non-overlapping intervals and then counting how many instances of the dataset fall into each interval.

A histogram showing the distribution of tips given in a restaurant

I

independence

independent variable

interquartile range (IQR)

Also midspread, middle 50%, and H-spread.

A measure of the statistical dispersion or spread of a dataset, defined as the difference between the 25th and 75th percentiles of the data. To calculate the IQR, the dataset is divided into four rank-ordered even parts or quartiles, the boundaries between which, at the 25th, 50th, and 75th percentiles, are denoted [math]\displaystyle{ Q }[/math]₁, [math]\displaystyle{ Q }[/math]₂, and [math]\displaystyle{ Q }[/math]₃, respectively; the IQR = [math]\displaystyle{ Q }[/math]₃[math]\displaystyle{ - }[/math][math]\displaystyle{ Q }[/math]₁.

J

joint distribution

Given two random variables X and Y, the joint distribution of X and Y is the probability distribution of X and Y together.

joint probability

The probability of two events occurring together. The joint probability of A and B is written [math]\displaystyle{ P(A \cap B) }[/math] or [math]\displaystyle{ P(A, \ B) }[/math].

K

Kalman filter

kernel

kernel density estimation

kurtosis

A measure of the "tailedness" of the probability distribution of a real-valued random variable. There are different ways of quantifying, estimating, and interpreting kurtosis, but a common interpretation is that kurtosis represents the degree to which the shape of the distribution is influenced by infrequent extreme observations (outliers); in this case, higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations.

L

L-moment

law of large numbers (LLN)

A theorem according to which the average of the results obtained from performing the same experiment a large number of times should be close to the experiment's expected value, and tends to become closer to the expected value as more trials are performed. The law suggests that a sufficiently large number of trials is necessary for the results of any experiment to be considered reliable, and by extension that performing only a small number of trials may produce an incomplete or misleading interpretation of the experiment's outcomes.

A graph illustrating the law of large numbers: the observed results of repeatedly rolling a fair die (green line) quickly approach the theoretical expected value of a single roll (blue line) as the number of times the die is rolled increases

likelihood function

A conditional probability function considered a function of its second argument with its first argument held fixed. For example, imagine pulling a numbered ball with a number k from a bag of n balls, numbered 1 to n; a likelihood function for the random variable N could be described as the probability of pulling k given that there are n balls: the likelihood will be 1/n for n greater than or equal to k, and 0 for n smaller than k. Unlike a probability distribution function, this likelihood function will not sum up to 1 on the sample space.

loss function

likelihood-ratio test

M

M-estimator

marginal distribution

Given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X when information about Y is ignored.

marginal likelihood

marginal probability

The probability of a given event, ignoring any information about other events. The marginal probability of A is written P(A). Contrast conditional probability.

Markov chain Monte Carlo

mathematical statistics

maximum likelihood estimation

mean

1.  The expected value of a random variable.

2.  The arithmetic mean, i.e. the mathematical average of a set of numerical values, calculated by dividing the sum of the values by the number of values.

median

median absolute deviation

mode

moving average

Also moving mean and rolling mean.

A series of mathematical averages or means of different subsets of a larger data set, usually computed so as to understand trends in the data set over time.

multimodal distribution

multivariate analysis

multivariate kernel density estimation

multivariate random variable

A vector whose components are random variables on the same probability space.

mutual exclusivity

mutual independence

A collection of events is said to be mutually independent if for any subset of the collection, the joint probability of all events occurring is equal to the product of the joint probabilities of the individual events. Think of the result of a series of coin-flips. This is a stronger condition than pairwise independence.

N

nonparametric regression

nonparametric statistics

non-sampling error

normal distribution

normal probability plot

null hypothesis (H₀)

The statement being tested in a test of statistical significance; usually a statement of 'no effect' or 'no difference'. For example, in a test of whether light has an effect on sleep, the null hypothesis would be that light has no effect on sleep (i.e. sleep patterns are the same regardless of the lighting conditions). The null hypothesis is an expression of the expectation that the dependent variable will not change significantly as the independent variable is modified; statistical significance is measured and reported according to the degree to which this expectation is met. Contrast alternative hypothesis.

O

opinion poll

optimal decision

optimal design

outlier

P

p-value

pairwise independence

A set of random variables, any two of which are independent.

parameter

Any measured quantity of a statistical population that summarizes or describes an aspect of the population, e.g. a mean or a standard deviation; often a quantity to be estimated based on the corresponding quantity calculated by drawing random samples from the population. Can be a population parameter, a distribution parameter, or an unobserved parameter.

particle filter

percentile

pie chart

point estimation

power

prior probability

In Bayesian inference, prior beliefs or other information that is available before new data or observations are taken into account.

population parameter

See parameter.

posterior probability

The result of a Bayesian analysis that encapsulates the combination of prior beliefs or information (the prior probability) with observed data.

principal component analysis (PCA)

probability

probability density

The probability in a continuous probability distribution. For example, you can't say that the probability of a man being six feet tall is 20%, but you can say he has 20% of chances of being between five and six feet tall. Probability density is given by a probability density function. Contrast probability mass.

probability density function

The probability distribution for a continuous random variable.

probability distribution

A function that gives the probability of all elements in a given space; see List of probability distributions.

probability measure

The probability of events in a probability space.

probability plot

probability space

A sample space over which a probability measure has been defined.

Q

quantile

A particular point or value at which the range of a probability distribution is divided into continuous intervals with equal probabilities, or at which the observations in a sample are divided in the same way. The number of groups into which the range is divided is always one greater than the number of quantiles dividing them. Commonly used quantiles include quartiles (which divide a range into four groups), deciles (ten groups), and percentiles (one hundred groups). The groups themselves are termed halves, thirds, quarters, etc., though the terms for the quantiles are sometimes used to refer to the groups, rather than to the cut points.

quartile

A type of quantile which divides a range of data points into four groups, termed quarters, of equal size. For any quartile-divided dataset, there are exactly three quartiles or cut points that create the four groups. The first quartile ([math]\displaystyle{ Q }[/math]₁) is defined as the middle data point or value that is halfway between the smallest value (minimum) and the median of the dataset, such that 25 percent of the data lies below this quartile. The second quartile ([math]\displaystyle{ Q }[/math]₂) is the median itself, with 50 percent of the data below this point. The third quartile ([math]\displaystyle{ Q }[/math]₃) is defined as the middle value halfway between the median and the largest value (maximum) of the dataset, such that 75 percent of the data lies below this quartile. Because the data must be ordered from smallest to largest in order to compute them, quartiles are a type of order statistic.

A normal distribution divided by three quartiles into four quarters. Note that each of the quarters occupies the same total area under the curve, equal to 25 percent of the distribution.

quota sampling

R

random variable

A measurable function on a probability space, often real-valued. The distribution function of a random variable gives the probability of the different values of the variable. The mean and variance of a random variable can also be derived. See also discrete random variable and continuous random variable.

randomized block design

range

The length of the smallest interval which contains all the data.

recursive Bayesian estimation

regression analysis

repeated measures design

response variable

Any variable whose value is or is expected to be affected by an experimental treatment, or by changes in one or more other variables; e.g. cholesterol levels after following a particular diet for six months. Response variables are those that change or respond to some phenomenon under study. The term is often used interchangeably with dependent variable.

restricted randomization

robust statistics

round-off error

S

sample

That part of a population which is actually observed.

sample covariance

sample mean

The arithmetic mean of a sample of values drawn from a population, commonly denoted by [math]\displaystyle{ \overline{x} }[/math]. An example is the average test score of a subset of 10 students from a class. The sample mean is used as an estimator of the population mean, which in this example would be the average test score of all of the students in the class.

sample space

The set of possible outcomes of an experiment. For example, the sample space for rolling a six-sided die will be {1, 2, 3, 4, 5, 6}.

sampling

A process of selecting observations to obtain knowledge about a population. There are many methods to choose on which sample to do the observations.

sampling bias

sampling distribution

The probability distribution, under repeated sampling of the population, of a given statistic.

sampling error

scatter plot

scale parameter

significance level

simple random sample

Simpson's paradox

skewness

A measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Roughly speaking, a distribution has positive skew (right-skewed) if the higher tail is longer, and negative skew (left-skewed) if the lower tail is longer. Perfectly symmetrical distributions always have zero skewness, though zero skewness does not necessarily imply a symmetrical distribution.

The mean and median of a skewed distribution (left and right) may differ substantially from those of a symmetrical distribution (center) with zero skewness.

spaghetti plot

spectrum bias

standard deviation

The most commonly used measure of statistical dispersion. It is the square root of the variance, and is generally denoted with the lowercase Greek letter [math]\displaystyle{ \sigma }[/math] (sigma).

standard error

standard score

statistic

The result of applying a statistical algorithm to a data set. It can also be described as an observable random variable.

statistical dispersion

statistical graphics

statistical hypothesis testing

statistical independence

Two events are independent if the outcome of one does not affect that of the other (for example, getting a 1 on a single die roll does not affect the probability of getting a 1 on a second roll). Similarly, when we assert that two random variables are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other.

statistical inference

Inference about a population based on a random sample drawn from that population or, more generally, about a random process from its observed behavior during a finite period of time.

statistical model

statistical population

A set of entities about which statistical inferences are to be drawn, often based on random sampling. One can also talk about a population of measurements or values.

statistical dispersion

A measure of the diversity within a set of data, expressed by the variance or the standard deviation.

statistical parameter

A parameter that indexes a family of probability distributions.

statistical significance

statistics

Student's t-test

stem-and-leaf display

stratified sampling

survey methodology

survival function

survivorship bias

symmetric probability distribution

systematic sampling

T

test statistic

tidy data

Standard for structuring data such that "each variable is a column, each observation is a row, and each type of observational unit is a table".

time domain

time series

time series analysis

time series forecasting

treatments

Variables in a statistical study that are conceptually manipulable. For example, in a health study, following a certain diet is a treatment whereas age is not.

trial

Can refer to each individual repetition when talking about an experiment composed of any fixed number of them. As an example, one can think of an experiment being any number from one to n coin tosses, say 17. In this case, one toss can be called a trial to avoid confusion, since the whole experiment is composed of 17 ones.

trimmed estimator

type I and type II errors

U

unimodal probability distribution

units

In a statistical study, the objects to which treatments are assigned. For example, in a study examining the effects of smoking cigarettes, the units would be people.

V

variance

A measure of its statistical dispersion of a random variable, indicating how far from the expected value its values typically are. The variance of random variable X is typically designated as [math]\displaystyle{ \operatorname{var}(X) }[/math], [math]\displaystyle{ \sigma_X^2 }[/math], or simply [math]\displaystyle{ \sigma^2 }[/math]

W

weighted arithmetic mean

weighted median

X

XOR, exclusive disjunction

Y

Yates's correction for continuity, yules correction

Z

z-test

See also

• Notation in probability and statistics
• Probability axioms
• Glossary of experimental design
• List of statistical topics
• List of probability topics
• Glossary of areas of mathematics
• Glossary of calculus

External links

• "A Glossary of DOE Terminology", NIST/SEMATECH e-Handbook of Statistical Methods, NIST, http://www.itl.nist.gov/div898/handbook/pri/section7/pri7.htm, retrieved 28 February 2009 
• Statistical glossary, statistics.com, http://www.statistics.com/resources/glossary/, retrieved 28 February 2009 
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)