The normal distribution, a very common probability density, useful because of the central limit theorem.
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.
Scatter plots are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.
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.
Gerolamo Cardano, a pioneer on the mathematics of probability.
The cdf of a discrete probability distribution, ...
Karl Pearson, a founder of mathematical statistics.
... of a continuous probability distribution, ...
A least squares fit: in red the points to be fitted, in blue the fitted line.
... of a distribution which has both a continuous part and a discrete part.
Confidence intervals: the red line is true value for the mean in this example, the blue lines are random confidence intervals for 100 realizations.
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)?
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

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

- Probability distribution

Numerical descriptors include mean and standard deviation for continuous data (like income), while frequency and percentage are more useful in terms of describing categorical data (like education).

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

9 related topics

Alpha

The Poisson distribution, a discrete probability distribution.

Probability theory

Branch of mathematics concerned with probability.

Branch of mathematics concerned with probability.

The Poisson distribution, a discrete probability distribution.
The normal distribution, a continuous probability distribution.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion).

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.

A visual representation of selecting a simple random sample

Sampling (statistics)

A visual representation of selecting a simple random sample
A visual representation of selecting a random sample using the systematic sampling technique
A visual representation of selecting a random sample using the stratified sampling technique
A visual representation of selecting a random sample using the cluster sampling technique

In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population.

In this case, the 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the probability distribution of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel.

Example of samples from two populations with the same mean but different dispersion. The blue population is much more dispersed than the red population.

Statistical dispersion

Example of samples from two populations with the same mean but different dispersion. The blue population is much more dispersed than the red population.

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.

Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for 68.27 percent of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 percent; three standard deviations (light, medium, and dark blue) account for 99.73 percent; and four standard deviations account for 99.994 percent. The two points of the curve that are one standard deviation from the mean are also the inflection points.

Standard deviation

Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for 68.27 percent of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 percent; three standard deviations (light, medium, and dark blue) account for 99.73 percent; and four standard deviations account for 99.994 percent. The two points of the curve that are one standard deviation from the mean are also the inflection points.
Cumulative probability of a normal distribution with expected value 0 and standard deviation 1
Example of samples from two populations with the same mean but different standard deviations. Red population has mean 100 and SD 10; blue population has mean 100 and SD 50.
Percentage within(z)
z(Percentage within)
The standard deviation ellipse (green) of a two-dimensional normal distribution

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values.

The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance.

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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.

The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by.

Comparison of the arithmetic mean, median, and mode of two skewed (log-normal) distributions.

Mean

Comparison of the arithmetic mean, median, and mode of two skewed (log-normal) distributions.
Geometric visualization of the mode, median and mean of an arbitrary probability density function.

There are several kinds of mean in mathematics, especially in statistics.

The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution.

Examples of scatter diagrams with different values of correlation coefficient (ρ)

Pearson correlation coefficient

Examples of scatter diagrams with different values of correlation coefficient (ρ)
Several sets of (x, y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.
This figure gives a sense of how the usefulness of a Pearson correlation for predicting values varies with its magnitude. Given jointly normal X, Y with correlation ρ, (plotted here as a function of ρ) is the factor by which a given prediction interval for Y may be reduced given the corresponding value of X. For example, if ρ = 0.5, then the 95% prediction interval of Y|X will be about 13% smaller than the 95% prediction interval of Y.
Critical values of Pearson's correlation coefficient that must be exceeded to be considered significantly nonzero at the 0.05 level.

In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ― is a measure of linear correlation between two sets of data.

The population Pearson correlation coefficient is defined in terms of moments, and therefore exists for any bivariate probability distribution for which the population covariance is defined and the marginal population variances are defined and are non-zero.

Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.

Chi-squared distribution

Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.

In probability theory and statistics, the chi-squared distribution (also chi-square or χ 2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.

The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.

Probability mass function for the binomial distribution

Binomial distribution

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Probability mass function for the binomial distribution
Cumulative distribution function for the binomial distribution
Binomial probability mass function and normal probability density function approximation for n = 6 and p = 0.5

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).