95% of the area under the normal distribution lies within 1.96 standard deviations away from the mean.
The normal distribution, a very common probability density, useful because of the central limit theorem.
Continuous uniform distribution
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
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Probability distribution of only one random variable.

- Univariate distribution

Absolute deviation

- List of statistics articles

It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

- Studentized residual

A probability distribution can either be univariate or multivariate.

- Mathematical statistics

One way is by dividing by a measure of scale (statistical dispersion), most often either the population standard deviation, in standardizing, or the sample standard deviation, in studentizing (e.g., Studentized residual).

- Deviation (statistics)

Mathematical statistics

- List of statistics articles

Studentized residual

- List of statistics articles

Univariate distribution

- List of statistics articles

243 related topics

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Statistics

Discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

The normal distribution, a very common probability density, useful because of the central limit theorem.
Scatter plots are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.
Gerolamo Cardano, a pioneer on the mathematics of probability.
Karl Pearson, a founder of mathematical statistics.
A least squares fit: in red the points to be fitted, in blue the fitted line.
Confidence intervals: the red line is true value for the mean in this example, the blue lines are random confidence intervals for 100 realizations.
In this graph the black line is probability distribution for the test statistic, the critical region is the set of values to the right of the observed data point (observed value of the test statistic) and the p-value is represented by the green area.
The confounding variable problem: X and Y may be correlated, not because there is causal relationship between them, but because both depend on a third variable Z. Z is called a confounding factor.
gretl, an example of an open source statistical package

Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.

Probability distribution

Mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
The cdf of a discrete probability distribution, ...
... of a continuous probability distribution, ...
... of a distribution which has both a continuous part and a discrete part.
One solution for the Rabinovich–Fabrikant equations. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate.

List of statistical topics

Mathematics

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.

Standard score

Number of standard deviations by which the value of a raw score is above or below the mean value of what is being observed or measured.

Compares the various grading methods in a normal distribution. Includes: Standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores
The z score for Student A was 1, meaning Student A was 1 standard deviation above the mean. Thus, Student A performed in the 84.13 percentile on the SAT.
The z score for Student B was 0.6, meaning Student B was 0.6 standard deviation above the mean. Thus, Student B performed in the 72.57 percentile on the SAT.

In mathematical statistics, a random variable X is standardized by subtracting its expected value

Errors and residuals

The normal distribution, a very common probability density, useful because of the central limit theorem.

In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable).

The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals.

Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model).

Beta distribution

Family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by alpha and beta (β), that appear as exponents of the random variable and control the shape of the distribution.

An animation of the Beta distribution for different values of its parameters.
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Mode for Beta distribution for 1 ≤ α ≤ 5 and 1 ≤ β ≤ 5
Median for Beta distribution for 0 ≤ α ≤ 5 and 0 ≤ β ≤ 5
(Mean–Median) for Beta distribution versus alpha and beta from 0 to 2
Abs[(Median-Appr.)/Median] for Beta distribution for 1 ≤ α ≤ 5 and 1 ≤ β ≤ 5
Abs[(Median-Appr.)/(Mean-Mode)] for Beta distribution for 1≤α≤5 and 1≤β≤5
Mean for Beta distribution for 0 ≤ α ≤ 5 and 0 ≤ β ≤ 5
(Mean − GeometricMean) for Beta distribution versus α and β from 0 to 2, showing the asymmetry between α and β for the geometric mean
Harmonic mean for beta distribution for 0 < α < 5 and 0 < β < 5
Harmonic mean for beta distribution versus α and β from 0 to 2
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log geometric variances vs. α and β
log geometric variances vs. α and β
Ratio of Mean Abs.Dev. to Std.Dev. for Beta distribution with α and β ranging from 0 to 5
Ratio of Mean Abs.Dev. to Std.Dev. for Beta distribution with mean 0 ≤ μ ≤ 1 and sample size 0 < ν ≤ 10
Skewness for Beta Distribution as a function of variance and mean
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Excess Kurtosis for Beta Distribution as a function of variance and mean
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Re(characteristic function) symmetric case α = β ranging from 25 to 0
Re(characteristic function) symmetric case α = β ranging from 0 to 25
Re(characteristic function) β = α + 1/2; α ranging from 25 to 0
Re(characteristic function) α = β + 1/2; β ranging from 25 to 0
Re(characteristic function) α = β + 1/2; β ranging from 0 to 25
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:Mean, Median, Geometric Mean and Harmonic Mean for Beta distribution with 0 < α = β < 5
Solutions for parameter estimates vs. (sample) excess Kurtosis and (sample) squared Skewness Beta distribution
Inflection point location versus α and β showing regions with one inflection point
Inflection point location versus α and β showing region with two inflection points
Example of eight realizations of a random walk in one dimension starting at 0: the probability for the time of the last visit to the origin is distributed as Beta(1/2, 1/2)
Beta(1/2, 1/2): The arcsine distribution probability density was proposed by Harold Jeffreys to represent uncertainty for a Bernoulli or a binomial distribution in Bayesian inference, and is now commonly referred to as Jeffreys prior: p−1/2(1 − p)−1/2. This distribution also appears in several random walk fundamental theorems
Max (joint log likelihood/N) for beta distribution maxima at &alpha; = &beta; = 2
Max (joint log likelihood/N) for Beta distribution maxima at &alpha; = &beta; &isin; {0.25,0.5,1,2,4,6,8}
Fisher Information I(a,a) for α = β vs range (c − a) and exponent α = β
Fisher Information I(α,a) for α = β, vs. range (c − a) and exponent α = β
Beta(0,0): The Haldane prior probability expressing total ignorance about prior information, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure. As α, β → 0, the beta distribution approaches a two-point Bernoulli distribution with all probability density concentrated at each end, at 0 and 1, and nothing in between. A coin-toss: one face of the coin being at 0 and the other face being at 1.
Karl Pearson analyzed the beta distribution as the solution Type I of Pearson distributions

of mathematical statistics.

Variance

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In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean.

Median

Value separating the higher half from the lower half of a data sample, a population, or a probability distribution.

Finding the median in sets of data with an odd and even number of values
Geometric visualization of the mode, median and mean of an arbitrary probability density function
Comparison of mean, median and mode of two log-normal distributions with different skewness

A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace.

Median absolute deviation

Robust measure of the variability of a univariate sample of quantitative data.

The normal distribution, a very common probability density, useful because of the central limit theorem.

For a univariate data set X1, X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median

Probability density function

[[Image:Boxplot vs PDF.svg|thumb|350px|[[Box plot]] and probability density function of a normal distribution

Box plot and probability density function of a normal distribution N(0,&thinsp;σ2).
Geometric visualisation of the mode, median and mean of an arbitrary unimodal probability density function.

A probability density function is most commonly associated with absolutely continuous univariate distributions.