The normal distribution, a very common probability density, useful because of the central limit theorem.
Several sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. The correlation reflects the noisiness 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.
Scatter plots are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.
Example scatterplots of various datasets with various correlation coefficients.
Gerolamo Cardano, a pioneer on the mathematics of probability.
Pearson/Spearman correlation coefficients between X and Y are shown when the two variables' ranges are unrestricted, and when the range of X is restricted to the interval (0,1).
Karl Pearson, a founder of mathematical statistics.
Anscombe's quartet: four sets of data with the same correlation of 0.816
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

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data.

- Correlation

These inferences may take the form of answering yes/no questions about the data (hypothesis testing), estimating numerical characteristics of the data (estimation), describing associations within the data (correlation), and modeling relationships within the data (for example, using regression analysis).

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

2 related topics

Alpha

Why-Because Graph of the capsizing of the Herald of Free Enterprise (click to see in detail).

Causality

Influence by which one event, process, state, or object (a cause) contributes to the production of another event, process, state, or object ( an effect) where the cause is partly responsible for the effect, and the effect is partly dependent on the cause.

Influence by which one event, process, state, or object (a cause) contributes to the production of another event, process, state, or object ( an effect) where the cause is partly responsible for the effect, and the effect is partly dependent on the cause.

Why-Because Graph of the capsizing of the Herald of Free Enterprise (click to see in detail).
Whereas a mediator is a factor in the causal chain (1), a confounder is a spurious factor incorrectly suggesting causation (2)
Used in management and engineering, an Ishikawa diagram shows the factors that cause the effect. Smaller arrows connect the sub-causes to major causes.

Alternative methods of structure learning search through the many possible causal structures among the variables, and remove ones which are strongly incompatible with the observed correlations.

Statistics and economics usually employ pre-existing data or experimental data to infer causality by regression methods.

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.