Calculate the Karl Pearson’s correlation coefficient between the marks (out of 10) in statistics and mathematics of 6 students.

$\begin{array}{|lcccccc|}\hline \text{ Student }& 1& 2& 3& 4& 5& 6\\ \text{ Statistics }& 7& 4& 6& 9& 3& 8\\ \text{ Mathematics }& 8& 5& 4& 8& 3& 6\\ \hline\end{array}$

1. Correlation and its types:

It is the study about finding the linear relationship between two variables

(i) Simple correlation:

(ii) Partial correlation

(iii) Multiple correlation.

2. Methods to find correlation:

(i) Scatter diagram

(ii) Karl Pearson's product moment correlation coefficient

(iii) Spearman's Rank correlation coefficient

(iv) Yule's coefficient of Association

3. Scatter diagram and types of correlation:

A scatter diagram is the simplest way of the diagrammatic representation of bivariate data.

(i) Positive correlation:

If the plotted points in the plane form a band and they show the rising trend from the lower left hand corner to the upper right hand corner.

(ii) Negative correlation:

If the plotted points in the plane form a band and they show the falling trend from the upper left hand corner to the lower right hand corner.

(iii) Uncorrelated:

If the plotted points spread over in the plane then the two variables are uncorrelated.

(iv) Perfect positive correlation:

If all the plotted points lie on a straight line from lower left hand corner to the upper right hand corner.

(v) Perfect negative correlation:

If all the plotted points lie on a straight line falling from upper left hand corner to lower right hand corner.

4. Karl Pearson's coefficient correlation:

$r\left(X,Y\right)=\frac{n{\sum }_{i=1}^n{x}_i{y}_i-{\sum }_{i=1}^n{x}_i{\sum }_{i=1}^n{y}_i}{\sqrt{n{\sum }_{i=1}^n{x}_i^2-\left({\sum }_{i=1}^n{x}_i\right)}\sqrt{n{\sum }_{i=1}^n{y}_i^2-{\left({\sum }_{i=1}^n{y}_i\right)}^2}}$

5. Spurious correlation:

It means an association extracted from correlation coefficient that may not exist in reality.

6. Spearman’s rank correlation coefficient:

$\rho =1-\frac{6{\sum }_{i=0}^{*}{D}_i^2}{n\left(n^2-1\right)},D_i=R_{1i}-R_{2i}$

7. Spearman’s rank correlation coefficient for repeated ranks:

$\rho =1-\frac{\sum D_i^2+\frac{1}{12}\left(m_1^3-m_1\right)+\frac{1}{12}\left(m_2^3-m_2\right)+\dots }{n\left(n^2-1\right)}$

8. Yule's coefficient:

$Q=\frac{\left(AB\right)\left(\alpha B\right)-\left(A\beta \right)\left(\alpha B\right)}{\left(AB\right)\left(\alpha B\right)+\left(A\beta \right)\left(\alpha B\right)}$