Glossary of probability and statistics

The following is a glossary of terms used in the mathematical sciences statistics and probability.

Statistics
Probability


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.  A feature of a sample that is not representative of the 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 0 and 1
binomial distribution
bivariate analysis
blocking
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, how will my headache feel if I take aspirin, versus if I do not take aspirin? Causal studies may be either experimental or observational.[1]
central limit theorem
central moment
characteristic function
chi-squared distribution
chi-squared test
cluster analysis
cluster sampling
complementary event
completely randomized design
computational statistics
concomitants
In a statistical study, concomitants are any variables whose values are unaffected by treatments, such as a unit’s age, gender, and cholesterol level before starting a diet (treatment).[1]
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 event B. Conditional probability is written P(A|B), and is read "the probability of A, given B"[2]
conditional probability distribution
confidence interval
In inferential statistics, a CI is a range of plausible values for some parameter, such as the population mean.[3] For example, based on a study of sleep habits among 100 people, a researcher may estimate that the overall population sleeps somewhere between 5 and 9 hours per night. This is different from the sample mean, which can be measured directly.
confidence level
Also known as a confidence coefficient, the confidence level indicates the probability that the confidence interval (range) captures the true population mean. For example, a confidence interval with a 95 percent confidence level has a 95 percent chance of capturing the population mean. Technically, this means that, if the experiment were repeated many times, 95 percent of the CIs would contain the true population mean.[3]
confounding
conjugate prior
continuous variable
convenience sampling
correlation
Also called correlation coefficient, a numeric measure of the strength of 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 have a correlation of 0. A population correlation is often represented by the symbol , while a sample correlation by .[2]
count data
Data arising from counting that can take only non-negative integer values
covariance
Given two random variables X and Y, with expected values and , covariance is defined as the expected value of random variable , and is written .[2] It is used for measuring correlation.

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's 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 with only one element. For example, when pulling a card out of a deck, "getting the jack of spades" is an elementary event, while "getting a king or an ace" is not
estimation theory
estimator
A function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. The mean can be used as an estimator
expected value
The sum of the probability of each possible outcome of the experiment multiplied by its payoff ("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 a six-sided die roll is 3.5. The concept is similar to the mean. The expected value of random variable X is typically written E(X) for the operator, and (mu) for the parameter.[2]
experiment
Any procedure that can be infinitely repeated and has a well-defined set of outcomes
exponential family
event
A subset of the sample space (a possible experiment's outcome), to which a probability can be assigned. For example, on rolling a die, "getting a five or a six" is an event (with a probability of one third if the die is fair)

F
factor analysis
factorial experiment
frequency
frequency distribution
frequency domain
frequentist inference

G
general linear model
generalized linear model
grouped data

H
histogram

I
Independence (probability theory)
independent variable
interquartile range

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 [2] or

K
Kalman filter
kernel
kernel density estimation
kurtosis
A measure of the infrequent extreme observations (outliers) of the probability distribution of a real-valued random variable. 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
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 the number k from a bag of n balls, numbered 1 to n. Then you could describe a likelihood function for the random variable N as the probability of getting k given that there are n balls : the likelihood will be 1/n for n greater 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 ignoring information about Y
marginal likelihood
marginal probability
The probability of an event, ignoring any information about other events. The marginal probability of A is written P(A). Contrast with conditional probability
Markov chain Monte Carlo
mathematical statistics
maximum likelihood estimation
mean
1.  The expected value of a random variable
2.  The arithmetic mean is the average of a set of numbers, or the sum of the values divided by the number of values
median
median absolute deviation
mode
moving average
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 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
The statement being tested in a test of statistical significance Usually the null hypothesis is a statement of 'no effect' or 'no difference'."[4] For example, if one wanted to test whether light has an effect on sleep, the null hypothesis would be that there is no effect. It is often symbolized as H0.

O
opinion poll
optimal decision
optimal design
outlier

P
p-value
pairwise independence
A pairwise independent collection of random variables is a set of random variables any two of which are independent
parameter
Can be a population parameter, a distribution parameter, an unobserved parameter (with different shades of meaning). In statistics, this is often a quantity to be estimated
particle filter
percentile
pie chart
point estimation
power
prior probability
In Bayesian inference, this represents 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 with observed data
principal component analysis
probability
probability density
Describes 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 with probability mass
probability density function
Gives 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
quartile
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 different results. We can also derive the mean and variance of a random variable
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
responses
In a statistical study, any variables whose values may have been affected by the treatments, such as cholesterol levels after following a particular diet for six months.[1]
restricted randomization
robust statistics
round-off error

S
sample
That part of a population which is actually observed
Sample mean and covariance
The arithmetic mean of a sample of values drawn from the population. It is denoted by .[2] An example is the average test score of a subset of 10 students from a class. 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. 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 (confusing the two is a common error)
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 written (sigma)[2]
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 one 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 from a random sample drawn from it or, more generally, about a random process from its observed behavior during a finite period of time
statistical interference
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
Statistical variability is a measure of how diverse some data is. It can be 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
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.[1]
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.[1]

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 , , or simply [2]

W
weighted arithmetic mean
weighted median

X
XOR, exclusive disjunction

Y
Yates's correction for continuity

Z
z-test

See also

References
  1. Reiter, Jerome (January 24, 2000). "Using Statistics to Determine Causal Relationships". American Mathematical Monthly. 107 (1): 24–32. doi:10.2307/2589374. JSTOR 2589374.
  2. "Probability and Statistics Symbols". Math Vault. Retrieved 2020-09-10.
  3. Pav Kalinowski. Understanding Confidence Intervals (CIs) and Effect Size Estimation. Association for Psychological Science Observer April 10, 2010. http://www.psychologicalscience.org/index.php/publications/observer/2010/april-10/understanding-confidence-intervals-cis-and-effect-size-estimation.html
  4. Moore, David; McCabe, George (2003). Introduction to the Practice of Statistics (4 ed.). New York: W.H. Freeman and Co. p. 438. ISBN 9780716796572.
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