It is named after Karl Pearson who developed the correlational method to do agricultural research.
designated by the Greek letter rho ()
The product moment part of the name comes from the way in which it is calculated by summing up the products of the deviations of the scores from the mean.
A correlation is a number between -1 and 1 that measures the degree of association between two variables (call them X and Y).
A positive value for the correlation implies a positive association
A negative value for the correlation implies a negative or inverse association
3 The formula for the Pearson correlation Suppose we have two variables X and Y with means XBAR and YBAR respectively and standard deviations SX and SY respectively. The correlation is computed as as the sum of the product of the Z-scores for the two variables divided by the number of scores. 4 If we substitute the formulas for the Z-scores into this formula we get the following formula for the Pearson Product Moment Correlation Coefficient which we will use as a definitional formula. The numerator of this formula says that we sum up the products of the deviations of a subjects X score from the mean of the Xs and the deviation of the subjects Y score from the mean of the Ys. This summation of the product of the deviation scores is divided by the number of subjects times the standard deviation of the X variable times the standard deviation of the Y variable 5
When will a correlation be positive
Suppose that an X value was above average and that the associated Y value was also above average. Then the product would be the product of two positive numbers which would be positive.
If the X value and the Y value were both below average then the product above would be of two negative numbers which would also be positive.
Therefore a positive correlation is evidence of a general tendency that large values of X are associated with large values of Y and small values of X are associated with small values of Y.
When will a correlation be negative
Suppose that an X value was above average and that the associated Y value was instead below average. Then the product would be the product of a positive and a negative number which would make the product negative.
If the X value was below average and the Y value was above average then the product above would be also be negative.
Therefore a negative correlation is evidence of a general tendency that large values of X are associated with small values of Y and small values of X are associated with large values of Y.
7 Interpretation of the correlation coefficient The correlation coefficient measures the strength of a linear relationship between two variables. The correlation coefficient is always between -1 and 1. The closer the correlation is to /-1 the closer to a perfect linear relationship. Here is to interpret correlations. -1.0 to -0.7 strong negative association. -0.7 to -0.3 weak negative association. -0.3 to 0.3 little or no association. 0.3 to 0.7 weak positive association. 0.7 to 1.0 strong positive association. 8
Lets calculate the correlation between Reading (X) and Spelling (Y) for the 10 students. There is a fair amount of calculation required as you can see from the table below. First we have to sum up the X values (55) and then divide this number by the number of subjects (10) to find the mean for the X values (5.5). Then we have to do the same thing with the Y values to find their mean (10.3).
9 Formula We then calculate The correlation we obtained was -.36 showing us that there is a small negative correlation between reading and spelling. The correlation coefficient is a number that can range from -1 (perfect negative correlation) through 0 (no correlation) to 1 (perfect positive correlation). 10 The computational formula for the Pearsonian r is
By looking at the formula we can see that we need the following items to calculate r using the raw score formula
The number of subjects N
The sum of each subjects X score times the Y score summation XY
The sum of the X scores summation X
The sum of the Y scores summation Y
The sum of the squared X scores summation X squared
The sum of the squared Y scores summation Y squared
11 (No Transcript) 12 In we plug each of these sums into the raw score formula we can calculate the correlation coefficient We can see that we got the same answer for the correlation coefficient (-.36) with the raw score formula as we did with the definitional formula. 13 GRACIAS!
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