This is an online statistics calculator that can help in determining the correlation between two statistical variables x and y. Enter the set of x and y coordinates of the input points in the appropriate fields of this Correlation Coefficient Calculator and get the total number of (x,y) points and the coefficient of their correlation. You can paste the data copied from a spreadsheet or csv-file or input manually using comma, space or enter as separators.
Correlation coefficient formula
A correlation coefficient is a numerical measure of correlation or statistical relationship between two variables. Several types of correlation coefficient are known, but the most widely used of them is the Pearson correlation coefficient also referred to as Pearson’s \(r\), the Pearson product-moment correlation coefficient or the bivariate correlation. It is a measure of the linear correlation between two variables \(X\) and \(Y\).
The correlation coefficient \(r\) (sometimes also denoted \(R\)) is defined by the formula:
$$r= \frac { { S }_{ xy } }{ \sqrt { { S }_{ xx }{ \cdot S }_{ yy } } } ,$$
where
$${ S }_{ xy }=\sum _{ i=1 }^{ n }{ { x }_{ i }{ y }_{ i }-\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { x }_{ i } } } \sum _{ i=1 }^{ n }{ { y }_{ i } } ,$$
$${ S }_{ xx }=\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 }-\frac { 1 }{ n } } { \left( \sum _{ i=1 }^{ n }{ { x }_{ i } } \right) }^{ 2 } ,$$
$${ S }_{ yy }=\sum _{ i=1 }^{ n }{ { y }_{ i }^{ 2 }-\frac { 1 }{ n } } { \left( \sum _{ i=1 }^{ n }{ { y }_{ i } } \right) }^{ 2 }.$$
The correlation coefficient has the following characteristics.
1. The range of \(r\) is between -1 and 1, inclusive.
2. If \(r\) = 1, the data points fall on a straight line with positive slope.
3. If \(r\) = -1, the data points fall on a straight line with negative slope.
4. If \(r\) = 0, there is no linear relationship between the \(X\) and \(Y\) variables.
5. Correlation coefficient \(r\) is a measure of the linear association between the \(X\) and \(Y\) variables.
6. The value of \(r\) is unchanged if either \(X\) or \(Y\) is multiplied by a constant or if a constant is added.
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