Scatter Plot and Karl Pearson's Correlation Cofficient

Scatter Plot

Graphs having plots between two variables are called scatter plots. The values of one variable are stored on the X-axis, whereas the values of another variable are kept on the Y-axis. The diagram produced by displaying these pairs of X and Y data is called a scatter diagram. The two factors under consideration are said to be connected if the depicted dots demonstrate an upward or downward trend. There is a significant relationship between the dots if they are close together and show a trend of either growing or falling values.

Correaltion:

A statistical tool called correlation is used to assess how closely two or more variables are related.

Types of Correlation

• Positive correlation
• Negative correlation
• Linear correlation
• Non-linear correlation
• Partial correlation
• Multiple correlationn

Measures of Correlation

• Graphic method or scatter diagram method
• Karl Pearson's correlation coefficient
• Sparkman's correlation coefficient

Graphic method or scatter diagram method:

It is the most straightforward technique for examining correlation between two variables. This method maintains the values of one variable in the X-axis and another variable in the Y-axis. The diagram created by locating these pairs of X and Y values is referred to as a scatter diagram. The two factors under consideration are said to be connected if the depicted dots demonstrate an upward or downward trend. There is a significant relationship between the dots if they are closely spaced and show a trend of either growing or decreasing value.

Karl Pearson's Correlation Coefficient

Karl Pearson's correlation coefficient simply assesses the degree of linear relationship between two variables. Karl Pearson's correlation coefficient between X and Y is typically represented by the symbols rxy, r(X,Y), or just r. It is also known as a correlation, the simple correlation coefficient, or the product moment correlation coefficient. This is how it is explained:

r=\{\frac{COV(X,Y)}{\{sqrt{var(X) \}{sqrt{var(Y)}\}

where, COV(X,Y) is read as covarience between X and Y. This measures the simultraneous changes between two variables.

and COV(X,Y) = 1/n\{sum{(X - \{overline{X})\} }\} (Y -\{overline{Y}\})

=1/n\{sum{XY - \{overline{X}\} \{overline{Y}\}}\}

Properties of Karl Pearson's Correlation Coefficient

• Correlation cofficient (r) lies between -1 to +1.
• Correlation cofficient (r) is the geomatric mean between two regression coefficients i.e. r= + -\{sqrt{byx.bxy}\} where, byx = regression coefficient of regression line of Y on X, bxy =regression coefficient of regression line of xon X
• Correlation cofficient is independent of change of origin as well as scale.
• Correlation cofficient is a relative stastical measures.
• Two independent variables are uncorrelated but the converse may not be true i.e. uncorrelated variables may not be independent.

Example:

Calculate the coefficient of correlation for the following data:

Solution:

we have,\{\overline{X}\} =\{\frac{\{\sum{X}}{n}\} =\{\frac{20}{5}\} = 4

\{\overline{Y}\} =\{\frac{\{\sum{Y}}{n}\} =\{\frac{55}{5}\= 11

now correlation coefficient, r =\{\frac{\{sum{xy}\} }{ \{sqrt{x2}\} \{sqrt{y2}\} }\} = \{\frac{21}{\{sqrt{10}\} \{sqrt{46}\} }\} =0.98

therefore, r= 0.98. this shows that there is almost perfect positive correlation between X and Y.

References

Chaudary, A.K. (2061).Business statistics. kathmandu:Bhundipuran Prakshan

Dhakal Bashanta (2014).Business Statistics,Buddha academic publisher