scatter plot and karl Pearson's correlation cofficient

scatter plot

scatter plots are graphs with plots between two variable.one variable values are kept in X-axis and another variable's values are kept in Y-axis.the diagram formed , by plotting these pairs of values of X and Y, is known as scatter diagram.if the plotted dots show some trend of upward or downward, then the two considering variables are said to be correlated.if the dots are close together and follows some trend of either increasing or decreasing,then there is a strong relation between them.

correaltion

correlation is a statistical device designed to measure the degree of association between two or more variables.

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 sipmlest method of studying correlation between two variables.in this method,one variable's values are kept in X-axis and another variable's values are kept in Y-axis.the diagram formed,by plotting these pairs of values of X and Y , is known as scatter diagram.if the plotted dots show some trend of upward or downward, then the two considering variables are said to be correlated.if the dots are close together and follows some trend of either increasing or decreasing, then there is a strong relation between them.

karl Pearson's correlation coefficient

karl Pearson's correlation cofficient measures a degree of association between two variables only to extent to which it is linear. let X and Y are two variables, karl Pearson's correlation coefficient between X and Y is generally denoted by r_xy or r(X,Y) or simply r only. it is also called product moment correlation coefficient or simple correlation cofficient or simply a correlation.it is defined as follows:

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{b_yx.b_xy}\}

where, b_yx = regression coefficient of regression line of Y on X

b_xy =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