12.3 Correlation

The purpose of simple linear regression is to build a linear model that can be used to make predictions of the [latex]y[/latex] variable for a given value of the [latex]x[/latex] variable. Of course, we want the model to give us good predictions—there is no point in using a model that gives bad or inaccurate predictions. But how can we tell if the linear model will provide accurate predictions? As we have seen, we can examine the scatter diagram for a set of data to get a sense of the strength and direction of the linear relationship between the independent variable [latex]x[/latex] and the dependent variable [latex]y[/latex]. But we would like a numerical measure of the strength and direction of the linear relationship we observe on the scatter diagram. This numerical measure is called the correlation coefficient.

The correlation coefficient was developed by Karl Pearson in the early 1900s and is sometimes referred to as Pearson’s correlation coefficient. Denoted by [latex]r[/latex], the correlation coefficient is a numerical measure of the strength and direction of the linear relationship between the independent variable [latex]x[/latex] and the dependent variable [latex]y[/latex]. Although there is an algebraic formula to find the value of [latex]r[/latex], we will perform the calculation using the built-in function in Excel.

Interpreting the Correlation Coefficient

What does the value of [latex]r[/latex] tell us?

• The value of the correlation coefficient [latex]r[/latex] is always a number between [latex]-1[/latex] and [latex]1[/latex].
• Values of [latex]r[/latex] close to [latex]1[/latex] or [latex]-1[/latex] indicate a strong linear relationship between [latex]x[/latex] and [latex]y[/latex].  If [latex]r=1[/latex], then there is a perfect, positive correlation between [latex]x[/latex] and [latex]y[/latex], in which case the points on the scatter diagram would all lie on a straight line that rises from left to right. If [latex]r=-1[/latex], then there is a perfect, negative correlation between [latex]x[/latex] and [latex]y[/latex], in which case the points on the scatter diagram would all line on a straight line that falls from left to right.
• Values of [latex]r[/latex] close to [latex]0.5[/latex] or [latex]-0.5[/latex] indicate a moderate linear relationship between [latex]x[/latex] and [latex]y[/latex].
• Values of [latex]r[/latex] close to [latex]0[/latex] indicate a weak linear relationship between [latex]x[/latex] and [latex]y[/latex].  If [latex]r=0[/latex], then there is no correlation between [latex]x[/latex] and [latex]y[/latex].

What does the sign of [latex]r[/latex] tell us?

• A positive value of [latex]r[/latex] means that the points on the scatter diagram have the general tendency to rise from left to right. In other words, when [latex]x[/latex] increases, [latex]y[/latex] tends to increase, and when [latex]x[/latex] decreases, [latex]y[/latex] tends to decrease.
• A negative value of [latex]r[/latex] means that the points on the scatter diagram have the general tendency to fall from left to right. In other words, when [latex]x[/latex] increases, [latex]y[/latex] tends to decrease, and when [latex]x[/latex] decreases, [latex]y[/latex] tends to increase.

	Positive Correlation	Negative Correlation
Strong Correlation
Moderate Correlation
Weak Correlation

Correlation versus Causation

The correlation coefficient only measures the correlation between two variables, not causation. A strong correlation between two variables does not mean that changes in one variable actually cause changes in the other variable. The correlation coefficient can only tell us that changes in the independent variable and dependent variable are related.  In general, remember that “correlation does not equal causation.”

Video: “Using Excel to calculate a correlation coefficient || interpret relationship between variables” by Matt Macarty [5:22] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

Exercises

1. For each scatter diagram shown below, determine if the value of the correlation coefficient indicates a strong, moderate, or weak linear relationship and a positive or negative linear relationship.

   1. 

   2. 

   3. 

   Click to see Answer

   1. strong, positive
   2. moderate, negative
   3. weak, positive

    

2. What does an [latex]r[/latex] value of [latex]0[/latex] mean?
   
   Click to see Answer

   There is no linear relationship between the independent and dependent variables.

    

3. Interpret each of the following values of [latex]r[/latex].
   1. [latex]r=0.67[/latex]
   2. [latex]r=-0.12[/latex]
   3. [latex]r=-0.93[/latex]
   Click to see Answer

   1. There is a moderate, positive linear relationship between the independent and dependent variables.
   2. There is a weak, negative linear relationship between the independent and dependent variables.
   3. There is a strong, negative linear relationship between the independent and dependent variables.

    

4. In a random sample of ten professional athletes, the number of endorsements the player has and the amount of money (in millions of dollars) the player earns are recorded in the table below.
   

   Player	Number of Endorsements	Money Earned (in millions)
   1	0	2
   2	3	8
   3	2	7
   4	1	3
   5	5	13
   6	5	12
   7	4	9
   8	3	9
   9	0	3
   10	4	10

   1. Calculate the correlation coefficient.
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]0.9786[/latex]
   2. There is a strong, positive linear relationship between the number of endorsements and money earned.

    

5. The table below gives the percentage of workers who are paid hourly rates for the years 1979 to 1992.
   

   Year	Percent of Workers Paid Hourly Rates
   1979	61.2
   1980	60.7
   1981	61.3
   1982	61.3
   1983	61.8
   1984	61.7
   1985	61.8
   1986	62.0
   1987	62.7
   1990	62.8
   1992	62.9

   1. Find the correlation coefficient
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]0.9448[/latex]
   2. There is a strong, positive linear relationship between the year and the percentage of workers paid an hourly rate.

    

6. The table below contains real data for the first two decades of AIDS cases.
   

   Year	Number of AIDS Cases
   1981	319
   1982	1,170
   1983	3,076
   1984	6,240
   1985	11,776
   1986	19,032
   1987	28,564
   1988	35,447
   1989	42,674
   1990	48,634
   1991	59,660
   1992	78,530
   1993	78,834
   1994	71,874
   1995	68,505
   1996	59,347
   1997	47,149
   1998	38,393
   1999	25,174
   2000	25,522
   2001	25,643
   2002	26,464

   1. Calculate the correlation coefficient.
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]0.4526[/latex]
   2. There is a moderate, positive linear relationship between the year and the number of AIDS cases.

    

7. Recently, the annual number of driver deaths per [latex]100,000[/latex] for the selected age groups was as follows:
   

   Age	Number of Driver Deaths per [latex]100,000[/latex]
   17.5	38
   22	36
   29.5	24
   44.5	20
   64.5	18
   80	28

   1. Find the correlation coefficient.
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]-0.5787[/latex]
   2. There is a moderate, negative linear relationship between age and the number of driver deaths per [latex]100,000[/latex].

    

8. The table below shows the life expectancy for an individual born in the United States in certain years.
   

   Year of Birth	Life Expectancy
   1930	59.7
   1940	62.9
   1950	70.2
   1965	69.7
   1973	71.4
   1982	74.5
   1987	75
   1992	75.7
   2010	78.7

   1. Find the correlation coefficient
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]0.9614[/latex]
   2. There is a strong, positive linear relationship between year and life expectancy.

    

9. The height (sidewalk to roof) of notable tall buildings in America is compared to the number of stories of the building (beginning at street level).
   

   Height (in feet)	Number of Stories
   1,050	57
   428	28
   362	26
   529	40
   790	60
   401	22
   380	38
   1,454	110
   1,127	100
   700	46

   1. Find the correlation coefficient
   2. Interpret the correlation coefficient.
   Click to see Answer

   1. [latex]0.9436[/latex]
   2. There is a strong, positive linear relationship between number of stories and height.

    

10. The following table shows data on average per capita wine consumption and heart disease rate in a random sample of 10 countries.
    

    Per Capita Yearly Wine Consumption in Liters	Per Capita Death from Heart Disease
    2.5	221
    3.9	167
    2.9	131
    2.4	191
    2.9	220
    0.8	297
    9.1	71
    2.7	172
    0.8	211
    0.7	300

    1. Find the correlation coefficient
    2. Interpret the correlation coefficient.
    Click to see Answer

    1. [latex]-0.8359[/latex]
    2. There is a strong, negative linear relationship between per capita yearly wine consumption and per capita deaths from heart disease.

     

11. The following table consists of one student athlete’s time (in minutes) to swim 2000 meters and the student’s heart rate (beats per minute) after swimming on a random sample of 10 days.
    

    Swim Time	Heart Rate
    34.12	144
    35.72	152
    34.72	124
    34.05	140
    34.13	152
    35.73	146
    36.17	128
    35.57	136
    35.37	144
    35.57	148

    1. Find the correlation coefficient
    2. Interpret the correlation coefficient.
    Click to see Answer

    1. [latex]-0.1236[/latex]
    2. There is a weak, negative linear relationship between swim time and heart rate.

     

“12.4 Correlation” and “12.8 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.