This slide describe the stepwise methods of hand calculation of Pearson correlation coefficient. it involves the hypothesis making and testing. Two methods are explained, one with covariance and second with direct formula. The formula derivation is also explained and at the last the graphic presentation is also given to show the line of fitness and direction of the correlation.

1. THE MANUAL CALCULATION OF
PEARSON’S CORRELATION
COEFFICIENT
DR. Rajeev Kumar,
M.S.W (TISS, Mumbai), M.Phil. (CIP, Ranchi), UGC-JRF, Ph.D., (IIT
Kharagpur)

2. THE FOUNDATION OF PEARSON'S CORRELATION

3. THE RELATION BETWEEN COVARIANCE AND
CORRELATION
• A covariance is the relation between two continuous variables.
• When covariance is divided by the product of standard deviations of x and y, it
gives the correlation coefficient. Covariance is the unstandardized while
correlation is the standardized.

4. THE DERIVATION OF PEARSON'S CORRELATION
FORMULA

5. EXERCISE-1: Basic understanding of correlation
formula
1. What will happen to the correlation coefficient, when the standard deviation of
(x) and (y) decreases?
2. What will happen to the correlation coefficient, if the standard deviation of (x)
and (y) increases?
3. What will be the impact on the value of standard deviation, if sample increase
and decrease?

6. A PRACTICAL EXAMPLE
• An examination was conducted among 11 students of M.Sc. of research
methodology. We have the theory marks (20) and practical marks (10).
• We are interested to know, whether students are having the equal command in
theoretical and practical of research method? Whether there is any association or
relation between their theoretical and practical knowledge of research
methodology? Based on this assumption, we made two hypotheses.
• Ha: There will be a significant association between theory and practical marks of
students
• H0: There will be no significant association between the marks of theory and
practical examination.
• To test these hypotheses, we need to calculate the Pearson’s correlation.

7. THE METHOD NO-1
Calculation through covariance and standard
deviation of both x and y

8. STEP-1: Arrange the data in table and calculate their
means

9. STEP-2: Subtract means of (X) and (Y) from X and Y

10. STEP-3: Obtain the value of subtraction

11. STEP-4: Multiply the (X-X bar) and (Y-Ybar) and do the summation
of that multiplication

12. STEP-5: Now calculate the standard deviation of x
(marks in theory)

13. Step-6: Now calculate the standard deviation of Y
(marks in practical)

14. STEP-7: Now calculate the covariance. (note down the
value of summation of step-4: multiplication of (X-X bar)
and (Y-Ybar) and its summation

15. STEP-8: Now calculate the Pearson’s correlation
coefficient

16. THE SECOND METHOD
• In this method, we will use direct formula, which we had derived from the
relationship of covariance and correlation.

17. Now we have the readymade formula.

18. STEP-4: As we have already done in the previous
method up to Step-4

19. STEP-5: Do the square of (X-Xbar) and its
summation.

20. STEP-6: Do the square of (Y-Ybar) and its
summation

21. STEP-7: NOW PUT ALL THE VALUE INTO THE
FORMULA

22. NOW LOOK AT THE TABLE OF CRITICAL VALUES

23. INTERPRETATION OF THE RESULT
• Pearson’s correlation coefficient (r) = 0.93
• Sample size =n =11, degree of freedom (df) = n-2 = 11-2 =9
• The critical value of Pearson Correlation Coefficient (p≤ 0.05) =
0.602
• The critical value of Pearson Correlation Coefficient (p≤ 0.01) =
0.735
• Since the (r = 0.93) is greater than 0.602 and 0.735, therefore,
correlation coefficient (r=0.93) is statistically significant at (p≤ 0.01)
with confidence interval 99.99%

24. THE GRAPHIC PRESENTATION OF THE
CORRELATION

25. THE FINAL RESULT
• Since the (r = 0.93) is greater than 0.602 and 0.735, therefore, correlation coefficient
(r=0.93) is statistically significant at (p≤ 0.01) with confidence interval 99.99%
• Ha: There will be a significant association between theory and practical marks of students
• H0: There will be no significant association between the marks of theory and practical
examination.
• Now, there is a highly significant positive correlation between the marks of theory and
practical examinations of M.Sc. Students. Therefore, the alternative hypothesis is accepted
and the null hypothesis is rejected.
• It is proved that students have equal knowledge in theory and practical and they are
applying their theoretical knowledge in practical also.

26. Thanks for your attention!
In the next session, we will learn about the Spearman’s
rank Correlation
keep learning !!!