Correlation Coefficient
The correlation coefficient, represented as r, is a value between -1 and 1 that measures the strength of a linear relationship between two variables. In Grade 7 math, students learn that r > 0 indicates a positive (increasing) relationship, while r < 0 suggests a negative (decreasing) relationship. When r is close to 1 or -1, data points form an almost perfect line with minimal scatter. For example, the relationship between distance driven and fuel used shows strong positive correlation (r = 0.98), while candle burning time versus remaining height demonstrates strong negative correlation (r = -0.99). This concept from Chapter 4: Linear Functions in the OpenStax Algebra and Trigonometry textbook helps students interpret scatter plot data and understand variable relationships.
Key Concepts
Property The correlation coefficient is a value, $r$, between $ 1$ and $1$. $r 0$ suggests a positive (increasing) relationship $r < 0$ suggests a negative (decreasing) relationship The closer the value is to 0, the more scattered the data. The closer the value is to 1 or $ 1$, the less scattered the data is.
Examples The relationship between the distance a car is driven and the amount of fuel used would have a strong positive correlation, with an $r$ value close to 1, like $r = 0.98$. Data comparing the number of hours a candle has been burning to its remaining height would show a strong negative correlation, with an $r$ value close to 1, like $r = 0.99$. A scatter plot of a person's height versus the last digit of their phone number would have a correlation coefficient very close to 0.
Explanation The correlation coefficient, $r$, is a score from 1 to 1 that measures how strong a linear relationship is. A score near 1 or 1 means the data points form an almost perfect line, while a score near 0 means no line.
Common Questions
What does a correlation coefficient of 0 mean?
A correlation coefficient close to 0 means there is no linear relationship between the two variables, and the data points appear scattered with no clear pattern. An example would be comparing a person's height to the last digit of their phone number.
What is the range of the correlation coefficient r?
The correlation coefficient r always falls between -1 and 1. Values cannot exceed this range regardless of the data being analyzed.
What does a positive correlation coefficient indicate?
A positive correlation coefficient (r > 0) indicates an increasing relationship where both variables move in the same direction. The distance a car travels and fuel used is an example with r = 0.98.
What is an example of negative correlation?
Candle burning time compared to remaining candle height shows strong negative correlation with r = -0.99. As burning time increases, the candle height decreases.
How do you know if a correlation is strong or weak?
The closer r is to 1 or -1, the stronger the linear relationship and the less scattered the data. Values near 0 indicate weak correlation with more scattered data points.
What does the correlation coefficient measure in 7th grade math?
The correlation coefficient measures how strong a linear relationship is between two variables. It tells you whether data points form an almost perfect line or show no pattern at all.