# How correlation coefficient is Independent of Units of

How correlation coefficient is Independent of Units of Measurement?
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Correlation:
Correlation a measure which indicates the “go-togetherness” of two data sets. It can be denoted as r. The value of correlation coefficient lies between –1 and +1. The positive 1 indicates that the two data sets are perfect and both are in same direction. The negative 1 indicates that the two data sets are perfect and both are in opposite direction. It will be zero when there is no relationship between the two data sets.
Correlation coefficient - r:
The Karl Pearson’s product-moment correlation coefficient or simply, the Pearson’s correlation coefficient is a measure of the strength of a linear association between two variables and is denoted by r or ${r}_{xy}$.
The coefficient of correlation ${r}_{xy}$ between two variables x and y for the bivariate data set $\left({x}_{i},{y}_{i}\right)$ for i=1,2,3…N is given below:
${r}_{xy}=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n\left(\sum {x}^{2}\right)-{\left(\sum x\right)}^{2}\right]×\left[n\left(\sum {y}^{2}\right)-{\left(\sum y\right)}^{2}\right]}}$
The Pearson’s product-moment correlation does not take into consideration whether a variable has been classified as a dependent or independent variable. It treats all variables equally. A change of scale and origin does not affect the value of r. Indeed, the calculations of Pearson’s correlation coefficient were designed in such that the units of measurement do not affect the calculation. This allows the correlation coefficient to be comparable and not influenced by the units of the variables used.
From the above said points, it is clear that, ${r}_{xy}={r}_{yx}$. In the formula for r, if we exchange the symbols x and y, then the result is same. This is because the formula is not dependent on the symbols.
It measures the degree of linear relationship between the variables. When we construct the correlation coefficient, the units of measurements that are used, cancel out. Thus, the correlation coefficient is independent of units of measurement.