Linear correlation of bivariate data AI HL

IB Mathematics: Applications and Interpretation HL · Unit 4

Lesson 4.4 · Linear correlation of bivariate data

Use paired data and scatter diagrams to describe and measure linear relationships.

God has set eternity in our hearts

Ecclesiastes 3:11 (NIV) says that God “has also set eternity in the human heart.”

When we study data, we look for patterns and relationships between quantities. The Bible reminds us that our hope reaches beyond what we can measure in this life. Christians look forward to eternal life with God, trusting in his promises and the hope found through Jesus Christ.

Learning objectives

By the end of this lesson, you should be able to:

identify bivariate data and explain why observations must remain paired;
draw and interpret a scatter diagram;
describe the direction and strength of a linear association;
calculate and interpret Pearson’s product-moment correlation coefficient, $r$ ;
identify possible outliers and explain their effect on correlation; and
explain why correlation does not necessarily mean causation.

1. Bivariate data

Bivariate data consist of paired observations of two variables. Each pair $(x,y)$ describes two measurements made on the same object, person or occasion.

Examples include:

hours spent studying and examination score for each student;
height and mass for each person;
daily temperature and ice cream sales; and
age of a car and its resale value.

Keep each pair together. If one variable is sorted without keeping its matching value, the relationship between the variables is lost.

Example · Keeping pairs together

If a student studies for 4 hours and scores 78%, the observation is the pair $(4,78)$ . The study time and score belong to the same student and should be entered in the same row of a table or the same point on a graph.

2. Scatter diagrams

A scatter diagram displays each paired observation as a point. Usually, the explanatory or input variable is placed on the horizontal axis and the response variable on the vertical axis. Label both axes and include units.

The upward pattern in this example suggests a positive linear association.

Describing association

Describe a scatter diagram by commenting on:

Direction: positive, negative or no clear linear association;
Strength: strong, moderate or weak; and
Form: approximately linear or curved, with any possible outliers noted.

Positive association: larger values of $x$ tend to be paired with larger values of $y$ .
Negative association: larger values of $x$ tend to be paired with smaller values of $y$ .
No linear association: the points show no clear upward or downward linear pattern.

3. Pearson’s product-moment correlation coefficient

Pearson’s product-moment correlation coefficient, usually written as $r$ , measures the direction and strength of the linear association between two quantitative variables.

$\boxed{-1\leq r\leq1}$

$r$ close to 1 indicates a strong positive linear association;
$r$ close to $-1$ indicates a strong negative linear association;
$r$ close to 0 indicates little linear association; and
$r=1$ or $r=-1$ indicates a perfect linear relationship.

The value of $r$ has no units. It is a number that summarises the linear pattern in the paired data.

Worked example · Calculating $r$

For five observations, the paired values are:

$x$	1	2	3	4	5
$y$	2	3	5	4	6

Using technology gives $r=0.900$ . This indicates a strong positive linear association: as $x$ increases, $y$ tends to increase.

The coefficient describes the pattern in these data. It does not prove that a change in $x$ causes a change in $y$ .

Use both the graph and $r$ . First inspect the scatter diagram. A correlation coefficient alone may hide curvature or the influence of an unusual point.

4. Calculating correlation using a TI-84 Plus CE

Press STAT, choose EDIT, and enter the $x$ -values in $L_1$ .
Enter the matching $y$ -values in $L_2$ , in the same order.
To display $r$ , enable the diagnostics: press 2nd, then 0 to open the catalogue. Select DiagnosticsOn and press ENTER twice.
Press STAT, move to CALC, and choose LinReg(ax+b).
Enter $L_1,L_2$ and select Calculate. Read the value of $r$ in the results.

Menu names can vary slightly by calculator operating system. Check that the lists contain the paired values before calculating.

5. Important limitations

Correlation does not prove causation

Two variables can be associated without one causing the other. A third variable may influence both, or the association may be coincidental. To argue that one variable causes another, more evidence and a suitable investigation design are needed.

Pearson’s $r$ measures linear association

A value of $r$ near zero does not always mean that the variables are unrelated. The scatter diagram may show a curved pattern that Pearson’s coefficient does not describe well.

Outliers can affect correlation

A single unusual point can make a linear relationship appear stronger, weaker or different in direction. Look for possible outliers on the scatter diagram and consider whether they are genuine observations or recording errors.

Example · A misleading conclusion

Ice cream sales and the number of people swimming may both increase during warmer months. This association does not show that buying ice cream causes people to swim. Temperature may influence both variables.

IB-style practice · Skill questions

A researcher records hours of revision and examination scores for each student.
1. Explain why the observations are bivariate data.
2. Which variable would usually be placed on the horizontal axis?
[2 marks]

Show solution

(a) Each student provides a paired observation: revision time and examination score. (b) Hours of revision, as the explanatory variable, would usually be placed on the horizontal axis.
A scatter diagram has points close to a straight line that slopes downwards from left to right. Describe the association. [2 marks]

Show solution

There is a strong negative linear association.
The correlation coefficient between two variables is $r=-0.84$ . Interpret this value. [2 marks]

Show solution

There is a strong negative linear association between the variables. As one tends to increase, the other tends to decrease. This does not establish causation.
The value of Pearson’s correlation coefficient for two variables is $r=0.06$ . Does this prove that there is no relationship between the variables? Explain. [2 marks]

Show solution

No. It indicates little linear association, but the scatter diagram could show a curved or other non-linear relationship.
A researcher finds a strong positive correlation between the number of hours people spend outdoors and the number of cold drinks they buy. Explain why this does not prove that spending time outdoors causes people to buy cold drinks. [2 marks]

Show solution

A third variable, such as hot weather, may increase both time spent outdoors and cold-drink purchases. Correlation alone cannot establish cause and effect.

IB-style practice · Application

A group of students records the number of hours they revise for a test and their test scores.

Revision time, $x$ (hours)	1	2	3	4	5
Test score, $y$ (%)	42	50	57	65	71

Draw a scatter diagram for the data. [3 marks]
Describe the direction and form of the association. [2 marks]
Use technology to calculate Pearson’s correlation coefficient, to three significant figures. [2 marks]
Interpret the value of $r$ in context. [2 marks]
Explain why the data do not prove that extra revision causes a higher test score. [2 marks]

Show worked solution

(a) Plot the five pairs $(1,42),(2,50),(3,57),(4,65),(5,71)$ , with revision time on the horizontal axis and test score on the vertical axis.

(b) The points show a strong, positive, approximately linear association.

(c) Technology gives $r\approx\boxed{0.999}$ .

(d) There is a very strong positive linear association between revision time and test score in this group of students.

(e) Other factors, such as prior knowledge, attendance or study methods, may affect scores. The observed association by itself does not prove that revision time caused the higher scores.

Reflection · Hope beyond what we can measure

Correlation helps us notice how two measured quantities vary together. It cannot measure the fullness of God’s promises or explain everything about a person’s life. Christians hold to the hope of eternal life with God, trusting in the love revealed through Jesus Christ.

Ecclesiastes 3:11 (NIV) — God “has also set eternity in the human heart.”

Lesson 4.4 · Linear correlation of bivariate data

God has set eternity in our hearts

Learning objectives

1. Bivariate data