Mathematics/Statistics/Correlation Coefficients - Revision history

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	== Pearson's Correlation Coefficient ==		== Pearson's Correlation Coefficient ==
			This is one of the most popular forms of calculating a correlation coefficient.

			The equation to calculate the '''Pearson Correlation Coefficient''' is
			<math>r = \frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{(n\sum x^2-(\sum x)^2)(n\sum y^2-(\sum y)^2)}}</math>
			Where
			* <math>\sum x</math> is the sum of all our original x values in our dataset.
			* <math>\sum y</math> is the sum of all our original y values in our dataset.
			* <math>\sum x^2</math> is the sum of all our x values, after squaring them first.
			* <math>\sum y^2</math> is the sum of all our y values, after squaring them first.
			* <math>\sum xy</math> is the sum of all our x and y pairs, after multiplying together first.

			For additional explanation, see [https://youtu.be/jBQz2RGxCek this youtube video].


	== Sample Correlation Coefficient ==		== Sample Correlation Coefficient ==

	The equation to calculate the '''Sample Correlation Coefficient''' is ~~the following.~~		The equation to calculate the '''Sample Correlation Coefficient''' is
	<math>r = \frac{1}{n - 1}\sum_{i=1}{n}(\frac{x_i-\bar{x}}{S_x})(\frac{y_i-\bar{y}}{S_y})</math>		<math>r = \frac{1}{n - 1}\sum_{i=1}{n}(\frac{x_i-\bar{x}}{S_x})(\frac{y_i-\bar{y}}{S_y})</math>
			Where
			* <math>x</math> is the [[Statistics/Core_Measurements#Mean\|mean]] of our x.
			* <math>y</math> is the [[Statistics/Core_Measurements#Mean\|mean]] of our y.
			* <math>S_y</math> is the [[Statistics/Core_Measurements#Standard Deviation\|standard deviation]] of our x.
			* <math>S_x</math> is the [[Statistics/Core_Measurements#Standard Deviation\|standard deviation]] of our y.

	For further explanation, see [https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/v/calculating-correlation-coefficient-r this Khan Academy video].		For further explanation, see [https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/v/calculating-correlation-coefficient-r this Khan Academy video].

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{{ Tip | This is often represented as '''r'''.}}

The '''Correlation Coefficient''' is a value that describes "how well can a straight line fit this data". It is similar to [[Statistics/Core Measurements#Covariance|covariance]] except that a correlation coefficient will always be between [-1, 1].

A value of exactly 1 indicates that there is a strong positive correlation between x and y. That is, as x increases, so does y. And as y increases, so does x.

A value of exactly -1 indicates that there is a strong negative correlation between x and y. That is, as x increases, y decreases. And as y increases, x decreases.

As values approach 0, it indicates a weaker and weaker correlation, with 0 indicating that there is absolutely no correlation between x and y.

There are a few ways to calculate a correlation coefficient.

== Pearson's Correlation Coefficient ==

== Sample Correlation Coefficient ==

The equation to calculate the '''Sample Correlation Coefficient''' is the following.
<math>r = \frac{1}{n - 1}\sum_{i=1}{n}(\frac{x_i-\bar{x}}{S_x})(\frac{y_i-\bar{y}}{S_y})</math>

For further explanation, see [https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/v/calculating-correlation-coefficient-r this Khan Academy video].