Here is the example and how to calculate correlation coefficient [r] by hand. Note: we refer to the correlation coefficient mathematically as: Suppose you want to determine if a relationship exists between the number of hours that you spend hitting golf balls at the driving range before playing a round of golf, and the score that you have for 18 holes each time. To collect your data, you go to the driving range, hit some balls, record the hours spent, and then immediately play 18 holes and record your score. You do this six times to build a data set. The purpose of this is to see if a relationship exists between spending time at the driving range and your golf score. In other words, you want to know if spending more time at the range is related to a better golf game (lower score). Here are the results of your data collection. Hours on the Practice Range Score for 18 holes Hours multiplied by score Hours squared Score squared x y xy x2 y2 2 95 190 4 9025 3 91 273 9 8281 2 94 188 4 8836 4 87 348 16 7569 3 88 264 9 7744 2 93 186 4 8649 Totals 16 = ?x 548 = ?y 1449 =?xy 46 = ?x2 50104 = ?y2 Using the information from the table, we can compute the correlation coefficient by hand using the following formula. Note: refers to the number of observations. In this case it is six, as we went to the range, then played 18 holes, and recorded the data six times. After completing the calculations, we can see that the correlation between our two variables (hours on the range, and golf score) is: -0.925. Remember that the strength of a correlation between variables is measured by how close is to +1.0 or -1.0. A value of +1.0 represents a perfect positive correlation, and a value of -1.0 represents a perfect negative correlation. The closer that is to 0, the more likely that no relationship exists. So, in our example a correlation coefficient of -0.925 indicates a pretty strong relationship between the two variables,...Here is the example and how to calculate correlation coefficient [r] by hand.

Note: we refer to the correlation coefficient mathematically as: 

Suppose you want to determine if a relationship exists between the number of hours that you spend hitting golf balls at the driving range before playing a round of golf, and the score that you have for 18 holes each time.  To collect your data, you go to the driving range, hit some balls, record the hours spent, and then immediately play 18 holes and record your score.  You do this six times to build a data set.  The purpose of this is to see if a relationship exists between spending time at the driving range and your golf score.  In other words, you want to know if spending more time at the range is related to a better golf game (lower score).  Here are the results of your data collection.

 

 

Hours on the Practice Range Score for 18 holes Hours multiplied by score Hours squared Score squared   

x y xy x2 y2   

2 95 190 4 9025   

3 91 273 9 8281   

2 94 188 4 8836   

4 87 348 16 7569   

3 88 264 9 7744   

2 93 186 4 8649   

Totals 16 = ∑x 548 = ∑y 1449 =∑xy 46 = ∑x2 50104 = ∑y2   

   

 

Using the information from the table, we can compute the correlation coefficient  by hand using the following formula.  Note:  refers to the number of observations.  In this case it is six, as we went to the range, then played 18 holes, and recorded the data six times.

 

 

 

 

 

 

 

After completing the calculations, we can see that the correlation between our two variables (hours on the range, and golf score) is: -0.925.  

Remember that the strength of a correlation between variables is measured by how close  is to +1.0 or -1.0.  A value of +1.0 represents a perfect positive correlation, and a value of -1.0 represents a perfect negative correlation.  The closer that  is to 0, the more likely that no relationship exists.

So, in our example a correlation coefficient of -0.925 indicates a pretty strong relationship between the two variables, as this number is a near-perfect correlation of -1.0.  Now that we know the correlation coefficient, we can test the significance of the relationship by using a hypothesis test.

Class:  What does a correlation of -0.925 tell us about our variables.  What generalizations can we make before our hypothesis test?  Why should we move forward with a hypothesis test? If we want to see if the correlation coefficient is statistically significant from 0, what would the hypothesis statement look like?