Karl Pearson’s Correlation Coefficient

Corelation is a statistical measure is a coefficient which describes the size and direction of the relation between two or more variables. Two variables are said to be corelated if the change in one variable affects the change in the other variable. Karl Pearson’s Correlation Coefficient between two variables $X$ and $Y$ is given by.

Here $\bar{X}$ is the average

Variation of a random variable $X$ is $E(X-\mu {)}^2$ Therefore,

$$r=E[(X-\bar{X})(Y-\bar{Y})]$$

where $r$ is the coefficient of correlation it has to be a number without unit so we divide

$$r=\frac{E[(X-\bar{X})(Y-\bar{Y})]}{{\sigma }_x{\sigma }_y}$$ $$r=\frac{E[(X-\bar{X})(Y-\bar{Y})]}{(E(X^2)-E(X{)}^2)(E(Y^2)-E(Y{)}^2)}$$ $$r=\frac{\sum (X-\bar{X})(Y-\bar{Y})}{\sqrt{\sum E(X-X^2)E(Y-Y^2)}}$$ $$r=\frac{\frac{\sum XY}{N}-\bar{X}\bar{Y}}{\sqrt{\left(\frac{\sum X^2}{N}-{\bar{X}}^2\right)\left(\frac{\sum Y^2}{N}-\overline{Y^2}\right)}}$$ $$r=\frac{\sum xy}{\sqrt{\left(\sum X^2\right)\left(\sum Y^2\right)}}$$

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Positive Correlation

Two variables are said to be positively correlated if they deviate to the same direction. Examples are height & weight, income & expenditure. https://www.investopedia.com/ask/answers/032515/what-does-it-mean-if-correlation-coefficient-positive-negative-or-zero.asp

Negative Correlation

Two variables are said to be negatively correlated if they deviate in the opposite direction. Example volume and pressure of a perfect gas, price and demand.

Uncorrelation

Two variable are said to be uncorrelated or statistically independent if there is no relation

Implication from the value of $r$

Using Karl Pearson’s coefficient we can conclude the following

• if $r=1$ then correlation is perfectly positive
• if $r=-1$ then th5e correlation is perfectly negative
• if $r=0$ then variables are uncorrelated and for values lying between $0to1$
• if $r\in (0,1)$ then correlation is perfectly positive
• if $r\in (-1,0)$ then the correlation is perfectly negative
• if $r=0$ then variables are uncorrelated- if $r\in (0,1)$ then the correlation is positive

References

Information

• date: 2025.03.22
• time: 18:45