Why Choosing a Summary Starts With the Scatterplot
A correlation coefficient is a compact numerical summary, but a compact summary can conceal important features of the data. Before calculating one, inspect the scatterplot: the shape of the point cloud determines whether Pearson’s correlation coefficient, \(r\), is a sensible description. In particular, \(r\) summarizes the direction and strength of a linear association, not every kind of relationship between two variables.
The goal is not to find a number for every pair of quantitative variables. It is to decide whether one number will represent the observed pattern fairly, then state what that number describes. A useful justification links the plotted pattern to the purpose of \(r\): the points show a roughly straight-line trend, and there is no prominent feature that would make a single linear summary misleading.
A sensible use of \(r\) requires paired quantitative observations: each \(x_i\) must correspond to the \(y_i\) measured on the same unit. The scatterplot should show a pattern that is approximately straight rather than strongly curved, and it should not contain an unusual point or distinct group structure that makes one overall coefficient unrepresentative. These are reasons to inspect the data before relying on a summary, not mechanical rules that guarantee a perfect fit.
The coefficient is symmetric: swapping the roles of \(x\) and \(y\) leaves its value unchanged. Its sign describes whether larger values of one variable tend to accompany larger or smaller values of the other. Its magnitude describes the strength of the linear pattern, interpreted in context. It does not give a slope, identify a cause, or guarantee that a straight line is useful for prediction.
A Short Justification of the Formula
The formula can be understood as an average product of standardized deviations. If \(s_x\) and \(s_y\) are the sample standard deviations of the two variables, standardize each observation by subtracting its sample mean and dividing by its sample standard deviation. The average product of these standardized values is
By the sample standard deviation formula, \(s_x=\sqrt{\sum (x_i-\bar{x})^2/(n-1)}\), with the corresponding expression for \(s_y\). Substituting these expressions makes the factors of \(n-1\) cancel:
This shows why \(r\) is dimensionless: deviations in each variable are scaled by that variable’s own spread. It also explains why the sign comes from whether paired deviations tend to have the same sign or opposite signs. The calculation, however, does not decide whether a linear summary is appropriate; the scatterplot does that work.
A Practical Decision Process
A defensible correlation analysis makes the choice and the reason visible. The following sequence keeps the numerical calculation in its proper place.
Check that each plotted point represents a pair of quantitative measurements on the same observational unit.
Look for an approximately straight trend. If the points form a clear curve, a single linear coefficient can fail to describe the relationship.
Look for unusual points or distinct clusters that could dominate or obscure the overall summary.
Give \(r\), its direction and strength in context, and a brief reason the scatterplot supports using it.
Worked Examples
Worked Example: A Roughly Linear Positive Pattern
In a hypothetical exercise, five students each complete a study session, and a tutor records the study time and a later quiz score. The paired data are:
| Study time (hours) | Quiz score (points) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 5 |
| 4 | 4 |
| 5 | 6 |
Choose the summary. The scatterplot would show an overall upward pattern, with the points reasonably close to a straight trend and no isolated point or separate cluster. Pearson’s \(r\) is appropriate as a concise summary of the linear association.
The means are \(\bar{x}=3\) hours and \(\bar{y}=4\) points. The deviations from the means are \((-2,-1,0,1,2)\) for study time and \((-2,-1,1,0,2)\) for quiz score. Thus,
so
Report in context. For these five hypothetical students, study time and quiz score have a strong positive linear association: longer study sessions tended to accompany higher scores. The coefficient is unitless; the variables and their units belong in the interpretation. This descriptive result alone does not show that additional study time caused a higher score.
Worked Example: A Strong Curve That Pearson’s r Misses
A hypothetical sensor is tested at five input settings, with its output recorded in arbitrary output units. The input settings and outputs are:
| Input setting | Output |
|---|---|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
Choose the summary. The scatterplot is U-shaped, not approximately straight. Pearson’s \(r\) is not an appropriate stand-alone description of the relationship. Its calculation is still possible, but interpreting it as a measure of the overall relationship would be misleading.
Here \(\bar{x}=0\) and \(\bar{y}=2\). The products of paired deviations are \((-2)(2)=-4\), \((-1)(-1)=1\), \((0)(-2)=0\), \((1)(-1)=-1\), and \((2)(2)=4\). Their sum is zero. Also, the sums of squared deviations are
Therefore,
The value \(r=0\) says there is no linear association in this particular dataset. It does not mean that the input and output have no relationship: the scatterplot shows a clear curved pattern. A careful report would describe the U-shape and explain that a single Pearson correlation does not capture it.
Worked Example: A Roughly Linear Negative Pattern
For five hypothetical days, an analyst records the outdoor temperature and the heating energy used in a small test building:
| Temperature (°C) | Heating energy (kWh) |
|---|---|
| 1 | 10 |
| 2 | 8 |
| 3 | 9 |
| 4 | 5 |
| 5 | 3 |
Choose the summary. The scatterplot would show a generally descending, roughly straight pattern. The third observation departs somewhat from a perfectly regular decline, but it is not an isolated point that reverses the overall pattern. Pearson’s \(r\) is a reasonable summary of the negative linear association in these five observations.
The means are \(\bar{x}=3\) °C and \(\bar{y}=7\) kWh. The temperature deviations are \((-2,-1,0,1,2)\); the energy deviations are \((3,1,2,-2,-4)\). The cross-products sum to
The squared deviations sum to \(10\) for temperature and \(34\) for heating energy. Thus,
Report in context. In these five hypothetical observations, outdoor temperature and heating energy have a strong negative linear association: higher temperatures tended to accompany lower heating energy use. The association describes the paired measurements; it does not by itself establish why heating energy differed.
How to Defend the Choice
A good report combines the coefficient with a sentence about the graph. For example: “The scatterplot shows a roughly straight, downward pattern with no prominent isolated point, so Pearson’s correlation is a reasonable summary; \(r=-0.922\), indicating a strong negative linear association between outdoor temperature in degrees Celsius and heating energy in kilowatt-hours in these observations.” This statement identifies the decision, its visual basis, the coefficient, and the variables with units.
There is no universal numerical cutoff that makes a correlation “strong” in every setting. The same magnitude can have different practical importance across contexts, and the scatterplot still matters. A value near zero may coexist with a pronounced curve; a large magnitude may summarize a clear straight-line trend, but it does not prove that a line captures every feature of the data.
Common Mistake
Wrong reasoning: “The variables are quantitative, so I should calculate Pearson’s \(r\); if the result is close to zero, there is no relationship.” Quantitative variables are necessary for this calculation, but they are not enough to justify treating \(r\) as a useful summary. A curved pattern can cancel positive and negative contributions, producing a small coefficient despite a strong relationship.
Correction: Inspect the scatterplot first. If its pattern is approximately straight and not dominated by unusual observations or separate clusters, report \(r\) as a summary of the linear association. If the plot is clearly curved or otherwise poorly represented by one overall linear summary, describe that feature instead of presenting \(r\) as a complete account.
Check Your Understanding
For each question, explain your reasoning rather than giving only a number or label.
- A scatterplot of two quantitative variables shows a pronounced inverted U-shape. Is Pearson’s \(r\) a suitable stand-alone summary? Why?
- A dataset contains paired measurements, and the scatterplot shows a roughly straight upward trend without a prominent outlier. What should a report say about the suitability and meaning of \(r\)?
- What does \(r=0\) establish about linear association, and what does it not establish about all possible relationships?
- Why must the measurements be paired before calculating Pearson’s correlation?
- A report gives a correlation between temperature in degrees Celsius and energy use in kilowatt-hours. Which units belong to \(r\), and which belong in the contextual description?