Product Moment Correlation

Estimated Product moment correlation coefficient:

\(r=\frac{\sum{\left( x-\bar{x} \right)\left( y-\bar{y} \right)}}{\sqrt{\left\{ \sum{{{\left( x-\bar{x} \right)}^{2}}} \right\}\left\{ \sum{{{\left( y-\bar{y} \right)}^{2}}} \right\}}}=\frac{\sum{xy}-\frac{\sum{x}\sum{y}}{n}}{\sqrt{\left( \sum{{{x}^{2}}}-\frac{{{\left( \sum{x} \right)}^{2}}}{n} \right)\left( \sum{{{y}^{2}}}-\frac{{{\left( \sum{y} \right)}^{2}}}{n} \right)}}\)

Estimated regression line of \(y\) on \(x\):

\(y-\bar{y}=b\left( x-\bar{x} \right)\), where \(b=\frac{\sum{\left( x-\bar{x} \right)\left( y-\bar{y} \right)}}{\sum{{{\left( x-\bar{x} \right)}^{2}}}}\).

The value of \(r\) gives us an indication of:

1. The direction of correlation: Positive or negatively correlated.
2. The strength of linear correlation: The closer the absolute value of \(r\), \(\left| r \right|\), is to \(1\), the stronger the strength of the linear correlation.