What type of relationship exists between two variables, if all points on the scatter diagram appear to fall on a straight line going upward from left to right?
Perfect positive correlation
Understanding Correlation in Scatter Diagrams
A scatter diagram is a graphical representation used to show the relationship between two variables. Each point on the diagram represents a pair of values for these two variables. The pattern of the points helps us understand the type and strength of the relationship, known as correlation.
Types of Correlation
Based on how the points are scattered on the diagram, we can identify different types of correlation:
- Positive Correlation: If the points generally show an upward trend from left to right. This means as one variable increases, the other variable also tends to increase.
- Negative Correlation: If the points generally show a downward trend from left to right. This means as one variable increases, the other variable tends to decrease.
- Zero or No Correlation: If the points are scattered randomly with no apparent trend. This means there is no linear relationship between the variables.
Perfect Correlation Explained
Correlation can also be described by its strength. A strong correlation means the points cluster closely around a line, while a weak correlation means the points are more spread out. The strongest possible linear relationship is called perfect correlation.
- Perfect Positive Correlation: Occurs when all the points on the scatter diagram fall exactly on a straight line that goes upward from left to right. In this case, there is a perfect, direct linear relationship between the two variables. The correlation coefficient (\(r\)) is exactly \(+1\).
- Perfect Negative Correlation: Occurs when all the points on the scatter diagram fall exactly on a straight line that goes downward from left to right. There is a perfect, inverse linear relationship. The correlation coefficient (\(r\)) is exactly \(-1\).
The question describes a scenario where all points on the scatter diagram appear to fall on a straight line going upward from left to right. This specific visual pattern is the defining characteristic of a perfect positive correlation. Because all points lie *exactly* on the line, the relationship is perfect, and because the line goes upward from left to right, the relationship is positive.
| Scatter Plot Pattern | Type of Correlation | Correlation Coefficient (\(r\)) |
|---|---|---|
| Points on a straight line going up | Perfect Positive | \(+1\) |
| Points on a straight line going down | Perfect Negative | \(-1\) |
| Points trending up but scattered | Positive (Strong/Moderate/Weak) | Between \(0\) and \(+1\) |
| Points trending down but scattered | Negative (Strong/Moderate/Weak) | Between \(-1\) and \(0\) |
| Points scattered randomly | Zero / No Correlation | Close to \(0\) |
Therefore, based on the description given in the question, the type of relationship that exists is a perfect positive correlation.
Revision Table: Correlation Coefficient
| Value of \(r\) | Strength and Direction |
|---|---|
| \(+1\) | Perfect Positive |
| Between \(0.7\) and \(1\) | Strong Positive |
| Between \(0.3\) and \(0.7\) | Moderate Positive |
| Between \(0\) and \(0.3\) | Weak Positive |
| \(0\) | No Linear Correlation |
| Between \(-0.3\) and \(0\) | Weak Negative |
| Between \(-0.7\) and \(-0.3\) | Moderate Negative |
| Between \(-1\) and \(-0.7\) | Strong Negative |
| \(-1\) | Perfect Negative |
Additional Information on Scatter Diagrams and Correlation
While scatter diagrams and correlation analysis are powerful tools for understanding relationships between variables, it's important to remember a few key points:
- Correlation is Not Causation: A strong correlation between two variables does not automatically mean that one variable causes the other. There might be a third variable influencing both, or the relationship could be coincidental.
- Linear Relationships Only: The correlation coefficient (\(r\)) and the types of correlation discussed (positive, negative, perfect) specifically measure the strength and direction of *linear* relationships. Variables might have a strong non-linear relationship (e.g., curved), but the linear correlation coefficient could be low.
- Outliers: Extreme points (outliers) on a scatter diagram can significantly affect the correlation coefficient and the apparent relationship. It's often useful to examine outliers and their potential impact.
- Variables' Units: The correlation coefficient is a standardized measure and does not have units, making it easy to compare relationships between different pairs of variables.
Important Questions from Correlation Analysis - Teaching
An experiment consists of tossing a coin 20 times. Such an experiment is performed 50 times. The number of heads and the number of tails in each experiment are noted. What is the correlation coefficient between the two?
Find the coefficient of correlation if given, cov(x, y) = 420, \(\sigma_x^2 = 484\) , and \(\sigma_y^2 = 441\)
