Pearson’s Product Moment Correlation- Numerical

How to calculate the Pearson's Product Moment Coefficient of Correlation?

Step 1: Organize Data

• Organize the data into a table – test scores reflected in columns X and Y

• Subject	Test A (X)	Test B (Y)	X - Meanx	Y - Meany	(X - Meanx)2	(Y - Meany)2	(X - Meanx) x (Y - Meany)
  A	1	2	-7	-5.5	49	30.25	38.5
  B	6	3	-2	-4.5	4	20.25	9
  C	7	5	-1	-2.5	1	6.25	2.5
  D	3	6	5	-1.5	25	2.25	7.5
  E	11	6	3	-1.5	9	2.25	-4.5
  F	9	8	1	0.5	1	0.25	0.5
  G	7	10	-1	2.5	1	6.25	-2.5
  H	11	10	3	2.5	9	6.25	7.5
  I	14	12	6	4.5	36	20.25	27
  J	11	13	3	5.5	9	30.25	16.5
  Total	80	75			144	124.5	102

Step 2: Compute means of X and Y

• Mean 𝑋 = (∑𝑋)/𝑛 = 80/10 = 8
• Mean 𝑌 = (∑𝑌)/𝑛 = 75/10 = 7.5

Step 3: Calculate Deviations

• Compute deviation of each score from its mean (see table)

Step 4: Calculate squares of deviation.

• Calculate squares of deviation.
• Then calculate sum of these deviations. (see table)

Step 5: Calculate standard deviations for X and Y.

• 𝑆_x = √((∑(𝑋 − Mean 𝑋)^2 ) / 𝑛) = √(144/10) = 3.79
• 𝑆_y = √((∑(𝑌 − Mean 𝑌)^2 ) / 𝑛) = √(124.5/10) = 3.53

Step 6: Calculate correlation coefficient

• Pearson’s Product-Moment correlation cofficient, r:
• r = ∑((𝑋 − Mean 𝑋) x (Y − Mean Y))/(n x Sx x Sy) = 102/(10 x 3.53 x 3.79) = 0.762

Step 7: Interpretation of Result

• r = 0.762. This indicates a high positive correlation between the scores of the two tests. A student who scores well in one subject can be expected to score high in the other test too.