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	{{Statistics-stub}}		{{Statistics-stub}}
			<gallery>
			File:Correlation_coefficient.png\|Pearson correlation coefficient
			File:Correlation_examples2.svg\|Examples of correlation
			File:Regression_lines.png\|Regression lines
			File:Pearson_correlation_and_prediction_intervals.svg\|Pearson correlation and prediction intervals
			File:Critical_correlation_vs._sample_size.svg\|Critical correlation vs. sample size
			</gallery>

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'''Pearson correlation coefficient''', also known as '''Pearson's r''', is a measure of the strength and direction of association that exists between two continuous variables. It is a method of correlation: a statistical technique used to determine the degree to which two variables are related. The coefficient values range from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.

==Definition==
The Pearson correlation coefficient is defined as the covariance of the two variables divided by the product of their standard deviations. Mathematically, it is represented as:

\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

where:
* \(n\) is the number of data points,
* \(x\) and \(y\) are the variables,
* \(\sum\) denotes the summation.

==Interpretation==
The value of the Pearson correlation coefficient indicates the strength and direction of the linear relationship between two variables. A value close to 1 implies a strong positive relationship, a value close to -1 implies a strong negative relationship, and a value around 0 implies no linear relationship.

==Assumptions==
The calculation and interpretation of Pearson's r assume that:
* Both variables are normally distributed.
* The relationship between the variables is linear.
* The data is homoscedastic, meaning the variance within each variable is the same.

==Applications==
Pearson's r is widely used in the fields of [[statistics]], [[psychology]], [[medicine]], and [[social sciences]] to measure the linear relationship between variables. It is particularly useful in research studies that aim to determine the strength and direction of relationships among continuous variables.

==Limitations==
While Pearson's r is a powerful tool for measuring linear relationships, it has limitations:
* It can only measure linear relationships and may not accurately represent non-linear relationships.
* It is sensitive to outliers, which can significantly affect the coefficient value.
* It assumes that the relationship between variables is linear and may not accurately reflect the complexity of real-world data.

==See also==
* [[Spearman's rank correlation coefficient]]
* [[Kendall rank correlation coefficient]]
* [[Linear regression]]

==References==
<references/>

[[Category:Statistics]]
[[Category:Psychometrics]]

{{Statistics-stub}}

Pearson correlation coefficient - Revision history

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