Correlation Dr Mahamood Usman Khan Ph D IIT ISM Dhanbad M Phil M Sc AMU Correlation If the change in one variable affect the change in other variable then they are said to be correlated It measures the strength of relationship between two variables It is also known as the measure of association Correlation or Karl Pearson Correlation Coefficient or Simple Correlation is defined as yx SS x Cov r xy y or xy xy y x Various Formulas to Calculate Correlation Coming from formula Example 1 Weight and Systolic Blood Pressure of 10 patient are given below Find product moment correlation coefficient between Wt and SBP Example 2 Variance and Standard deviation Variance It measures the variability between the observations from mean n x x 2 i 2 x i 1 n 1 It gives the average area of spread of a set of observations from their average value Standard deviation SD It is the square root of the variance x SD Var x x It measures the dispersion of the set of values from their mean SD is always ve Covariance It measures the joint variability between two variables cov yx n i 1 x i yx i y n 1 When X and Y cov x y ve When X and Y cov x y ve When no constant relationship cov x y 0 Scatter diagrams of Correlation Y Y Y Y Y Y X X X Positive correlation Negative correlation No correlation If X and Y deviate in same direction that is the increase decrease in one results in corresponding increase decrease in other For example height and weight of group of persons If X and Y deviate in opposite direction that is the increase decrease in one results in corresponding decrease increase in other For example pressure and volume of gas The two variables X and Y is said to be uncorrelated if and increase decrease in one results in no change in other For example weight and IQ Degree of Correlation 1 yx Limit of Correlation 1 If the correlation is 1 then it is called as the perfect negative correlation and if it is 1 then it is called as perfect positive correlation If the correlation is 1 then it is called as the perfect negative correlation and if it is 1 then it is called as perfect positive correlation Variance vs Covariance Measures of Variability Variance It measures the variability between the observations from mean It gives the average area of spread of a set of observations from their average value Covariance It measures the joint variability between two variables cov yx 2 x x i 2 x n 1 n i 1 n i 1 x i yx i y n 1 Correlation VS Covariance Correlation Provides the strength of relationship between two variables Range 1 yx 1 Covariance Does not itself provide the information about the strength of the relationship between the variables Range yx Rank Correlation Associated with ordinal data For example i Marks obtained by the students of a class in Maths and Physics ii The beauty contest Spearman s formula for the rank correlation coefficient is n 6 1 i 2 nn d 2 i 1 1 where Rx i Ry i Ry i d Rx i Xof Yof i Rank Rank Example Marks of ten students in Mathematics and Statistics are given below Calculate Spearman coefficient of rank correlation Repeated Rank Here D i d i Tied observations Tied Rank Common Rank Average Rank Example Types of Correlation Nature Behaviour Variable Positive variables deviate in same direction Linear Simple correlation between x y Negative variables deviate in opposite direction Non linear Curvelinear Partial controlling the variables other the x y Zero no correlation Multiple one to many others Other behaviours The correlation coefficient is independent of change of origin and scale Two independent variables are uncorrelated But the converse may not true Hence two uncorrelated variables need not necessarily independent Linear Non Linear Correlation Linear correlation When the amount of change in one variable tends to bear a constant ratio to the amount of change in the other Non linear If the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable Partial Correlation Multiple Correlation Summary Solved Example Partial Multiple Correlation A study was conducted to know role of hours spent on revision and anxiety level on exam score The following data were recorded Compute i The correlation between exam score and hours of revision after controlling the effect of anxiety level ii The correlation between exam scores and anxiety level after controlling the effect of hours of revision iii The joint effect of hours of revision and anxiety level on exam score Exercise 1 Calculate the followings i All possible simple correlations ii All possible partial correlations iii All possible multiple correlations Calculate the followings i All possible simple correlations ii All possible partial correlations iii All possible multiple correlations Example 3 Calculate the followings i All possible simple correlations ii All possible partial correlations iii All possible multiple correlations Example 4 Calculate the followings i All possible simple correlations ii All possible partial correlations iii All possible multiple correlations Non Sense Correlation The number of runs scored by a batsman increase with an increase in the consumption of fertilizer in the local market The number of flights space is increasing with a decrease in the population of tigers The correlation between the shoe size and intelligent level So on Conclusions Shows the amount strength of relationship present Can be used to make predictions about the variables under study Can be used in many places including natural settings libraries etc Easier to collect co relational data Thank You