Question

What value does \(r\) assume if all the data points fall on the same straight line in these cases? a. The line has positive slope. b. The line has negative slope.

Short answer

Answer: For a perfect positive correlation where the line has a positive slope, the value of 'r' is +1. For a perfect negative correlation where the line has a negative slope, the value of 'r' is -1.

Step-by-step solution

a. Line with positive slope

If all the data points fall on the same straight line and the line has a positive slope, the value of 'r' will be +1. This implies a strong, direct (positive) relationship between the two variables. A perfect positive correlation signifies that if one variable increases, the other variable will also increase.

b. Line with negative slope

If all the data points fall on the same straight line and the line has a negative slope, the value of 'r' will be -1. This implies a strong, indirect (negative) relationship between the two variables. A perfect negative correlation signifies that if one variable increases, the other variable will decrease.

Key concepts

Positive Correlation

Imagine watching someone blowing up a balloon. As they pump more air, the balloon grows bigger. This is what we call a positive correlation: as one variable increases, the other one does too.

When we talk about the Pearson correlation coefficient, denoted as 'r', values close to +1 indicate such a relationship. If all data points in a scatterplot fall on a straight line with an upward slope—imagine the rising path of a hiking trail—that's a sign of perfect positive correlation, where 'r' equals +1. In real-world terms, think about the relationship between studying time and test scores; generally, as one goes up, so does the other.

Negative Correlation

Now, think of a car using up its tank of gas. The more you drive, the less fuel you have. This represents a negative correlation: one variable increases while the other decreases.

In terms of the Pearson correlation coefficient, a perfect negative correlation is marked by an 'r' value of -1. On a graph, data points creating a line with a downward slope—like sliding down a playground slide—reflect a perfect negative correlation. An everyday example could be the relationship between the temperature outside and the amount of natural gas used for heating; as the outside temperature drops, the gas usage typically increases, showing that the variables move in opposite directions.

Correlation and Causality

It's tempting to look at correlation and jump to causality—the idea that one event is the result of the occurrence of the other event—assuming that 'if two things correlate, one must cause the other'. However, it's crucial to understand that correlation does not imply causality.

Always consider the possibility of coincidence, underlying factors, or third variables affecting the correlation. For instance, ice cream sales and shark attacks are positively correlated; they both increase during summer months, but eating ice cream doesn't cause shark attacks. There's a lurking variable: the summer season, which leads to more people swimming in the ocean and more ice cream being consumed.

Linear Relationship

In both positive and negative correlations described above, we were considering a linear relationship between two variables. A linear relationship is one in which the change in one variable is associated with a proportional change in another variable.

Imagine plotting data points on a graph. If these points can be closely approximated by a straight line, you've found a linear relationship. The steeper the line, the stronger the relationship. The Pearson correlation coefficient 'r' measures the strength and direction of this linear relationship. Perfect linear relationships, where all points fall exactly on a line, result in an 'r' value of +1 or -1, but in the real world, it's normal for relationships to be less than perfect, leading to 'r' values somewhere between -1 and +1.