Graph of the relation s
the graph of the relation s is shown below. The plotted points represent paired observations \((x,y)\) and display the association between an explanatory variable \(x\) and a response variable \(y\).
Association described by the scatterplot
Direction: The association is positive; the cloud of points rises from left to right.
Form: The pattern is approximately linear; curvature is not visually dominant.
Strength: The points cluster tightly around a straight line, consistent with a strong linear correlation (|r| close to 1).
Data values consistent with the graph
A concrete representation of relation \(s\) is a set of paired observations \((x,y)\) matching the plotted locations. The table below lists the values used to construct the displayed graph.
| Observation | x | y |
|---|---|---|
| 1 | 1 | 4 |
| 2 | 2 | 5 |
| 3 | 3 | 8 |
| 4 | 4 | 8 |
| 5 | 5 | 10 |
| 6 | 6 | 14 |
| 7 | 7 | 13 |
| 8 | 8 | 17 |
| 9 | 9 | 18 |
| 10 | 10 | 22 |
Correlation coefficient and regression line
Linear correlation is quantified by Pearson’s correlation coefficient \(r\), which standardizes the sample covariance:
\[ r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}, \quad S_{xx} = \sum (x_i - \bar{x})^2, \quad S_{yy} = \sum (y_i - \bar{y})^2, \quad S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}). \]
The least-squares regression line for predicting \(y\) from \(x\) has the form \(\hat{y} = b_0 + b_1 x\) with \(b_1 = S_{xy}/S_{xx}\) and \(b_0 = \bar{y} - b_1 \bar{x}\). For the values in relation \(s\), \(\bar{x} = 5.5\), \(\bar{y} = 11.9\), \(S_{xx} = 82.5\), \(S_{xy} = 158.5\), and \(S_{yy} = 314.9\).
\[ b_1 = \frac{158.5}{82.5} \approx 1.921, \qquad b_0 = 11.9 - 1.921 \cdot 5.5 \approx 1.333, \qquad \hat{y} \approx 1.33 + 1.92 \cdot x. \]
\[ r = \frac{158.5}{\sqrt{82.5 \cdot 314.9}} \approx 0.983, \] which aligns with the tight, upward scatter in the graph and supports the description “strong positive linear correlation.”
Interpretation and limitations
A strong correlation indicates a strong linear association in the observed data, while leaving causal interpretation unresolved. The regression line supports prediction of average \(y\) at a given \(x\); individual observations vary around the line by their residuals. Visual screening for outliers remains important because a single extreme point can substantially change both \(r\) and the fitted slope.