Pearson Product-Moment Correlation

Pearson Product-Moment Correlation

Pearson product-moment correlation is the most widely used in statistics to measure the degree of the relationship between the linear related variables. The Pearson correlation would require both variables to be normally distributed. Correlation refers to the departure of two random variables from independence.

For example, in the stock market, if we want to measure how two products related to each other. Pearson is used to measure the degree of relationship between the two products.

Correlation Coefficient

The correlation coefficient is defined as the covariance divided by the standard deviation of the variables. The following formula is used to calculate the Pearson correlation:

Other Formula for Correlation Coefficient

Pearson’s product-moment correlation or simply correlation coefficient or Pearson is a measure of the linear strength of the association between two variables. It is founded by Karl Pearson. The value of the correlation coefficient varies between and . When the value of the correlation coefficient lies around , then it is said to be a perfect degree of association between two variables. As the value of the correlation coefficient goes closer to zero, the relationship between the two variables will be weaker.

Perfect Positive Correlation

Perfect Negative Correlation

Positive Correlation

Negative Correlation

Zero Correlation

Correlation Coefficient and Strength of Relationships

Coefficient Relationship

0.00 No correlation, no relationship

Very low correlation, almost negligible relationship

Slight correlation, definite but small relationship

Moderate correlation, substantial relationship

High correlation, marked relationship

Very high correlation, very dependable relationship

Perfect correlation, perfect relationship

Test of Significance

A test of significance for the coefficient of correlation may be used to find out if the computed Pearson’s could have occurred in a population in which the two variables are related or not. The test static follows the with degrees of freedom. The significance is computed using the formula of

where t = t test for correlation coefficientr = correlation coefficientN = number of paired samples

Assumptions in Pearson Product-Moment Correlation test

Subjects are randomly selected.

Both populations are normally distributed

Procedure for Pearson Product-Moment Correlation Test

1. Set up the hypothesis:

(The correlation in the population is zero) (The correlation in the population is

different from zero)where is the correlation in the

population2. Set the level of significance.

Procedure for Pearson Product-Moment Correlation Test

3. Calculate the degrees of freedom ( and determine the critical value of .4. Calculate the Pearson’s .5. Calculate the value of and determine the statistical decision for hypothesis testing:

If , do not reject .If , reject .

6. State the conclusion.

Procedure for Pearson Product-Moment Correlation Test

When the null hypothesis has been rejected for a specific significance level, there are possible relationships between X and Y variables:

1. There is a direct cause-and-effect relationship between the two variables.

2. There is a reverse cause-and-effect relationship between the two variables.

3. The relationship between the two variables may be caused by the third variable.

4. There maybe a complexity of interrelationship among many variables.

5. The relationship between the two variables may be coincidental.

Example:

The owner of a chain of fruit shake stores would like to study the correlation between atmospheric temperature and sales during the summer season. A random sample of 12 days is selected with the results given as follows:

DAY TEMPERATURE TOTAL SALES

1 147

2 76 143

3 78 147

4 84 168

5 90 206

6 83 155

7 93 192

8 94 211

9 97 209

10 85 187

11 88 200

12 82 150

Solution:Step 1: Graph the scatter plot.

75 80 85 90 95 100140

150

160

170

180

190

200

210

220

TO

TA

L S

AL

ES

TEMPERATURE

Step 2: State the hypotheses

(There is no correlation between atmospheric temperature and total sales of fruit shake)

(There is a correlation between atmospheric temperature and total sales of fruit shake)

The level of significance is

Step 3: Compute for the value of Pearson’s , degrees of freedom and critical values of t.

DAY X Y X2 Y2 XY

1 79 147 6,241 21,609 11,613

2 76 143 5,776 20,449 10,868

3 78 147 6,084 21,609 11,466

4 84 168 7,056 28,224 14,112

5 90 206 8,100 42,436 18,540

6 83 155 6,889 24,025 12,865

7 93 192 8,649 36,864 17,856

8 94 211 8,836 44,521 19,834

9 97 209 9,409 43,681 20,273

10 85 187 7,225 34,969 15,895

11 88 200 7,744 40,000 17,600

12 82 150 6,724 22,500 12,300

TOTAL 1029 2115 88,733 380,887 183,222

Degrees of freedom

Critical values of t

The coefficient of correlation, , between the atmospheric temperature and total sales indicates a very high positive correlation – that is an increased in atmospheric temperature is highly associated with the increased in total sales of fruit shake.

Decision Rule

In order to make decision on the significant relationship, we need to determine the value of t.

Since the computed t-value of 8.00 is greater than the tabular value of 2.288 at level of significance of 0.05, we would need to reject the null hypothesis.

Conclusion

Since the null hypothesis has been rejected, we can conclude that there is evidence that shows significant association between the atmospheric temperature and the total sales of fruit shake.