What the Limits and Sign of \(r\) Tell You
In “What the Correlation Coefficient Measures,” you learned that \(r\) summarizes the direction and strength of the linear association between two quantitative variables. This tutorial looks more closely at three basic properties: \(r\) must be between \(-1\) and \(1\), its sign gives the direction of the linear pattern, and its endpoints describe perfect straight-line relationships.
These properties are useful checks when you interpret a calculated value. A reported correlation of \(1.3\), for example, cannot be correct. A value of \(-0.8\) says something different from \(0.8\), even though the two values have the same distance from zero. And \(r=0\) does not automatically mean that the variables have no relationship at all.
Why \(r\) Cannot Be Outside \([-1,1]\)
The formula from “What the Correlation Coefficient Measures” combines standardized deviations. A standardized deviation tells how far an observation is from its variable’s mean, measured in standard deviations. Write the standardized deviations for the \(i\)th observation as \(a_i=(x_i-\bar{x})/s_x\) and \(b_i=(y_i-\bar{y})/s_y\). The correlation formula can then be written as:
The two lists of standardized deviations each have a sum of squares equal to \(n-1\). In other words, \(\sum a_i^2=n-1\) and \(\sum b_i^2=n-1\). The Cauchy–Schwarz inequality says that the absolute value of the sum of the products of two lists cannot be greater than the product of their lengths:
Substituting the two sums of squares gives an upper limit of \(n-1\) for \(\left|\sum a_i b_i\right|\). Dividing by \(n-1\), as the formula for \(r\) does, gives \(|r|\leq 1\). This is why a valid correlation cannot be less than \(-1\) or greater than \(1\). The limit is built into the formula, not a convention chosen for convenience.
The endpoints occur when the two lists of standardized deviations match exactly or are opposites. If each \(b_i=a_i\), the standardized values line up in the same direction and \(r=1\). If each \(b_i=-a_i\), they line up in opposite directions and \(r=-1\). In the original variables, these cases mean that every point falls exactly on a straight line with, respectively, a positive or negative slope.
How the Sign of \(r\) Works
Recall from “Describing Direction in a Scatterplot” that direction is described by moving from left to right. The formula for \(r\) connects this visual idea to the data: it multiplies each observation’s standardized \(x\)-deviation by its standardized \(y\)-deviation, then combines those products.
When an observation is above both variable means or below both means, its two deviations have the same sign and their product is positive. When it is above one mean and below the other, the deviations have opposite signs and their product is negative. If the positive products outweigh the negative products overall, \(r\) is positive and the linear pattern tends upward. If the negative products outweigh the positive products, \(r\) is negative and the pattern tends downward.
This is an overall summary, not a rule that every individual point must follow. A positive \(r\) does not require every observation to be above both means or below both means; it means the combined pattern of paired deviations produces a positive value. Likewise, a negative \(r\) describes the overall direction, not every pair of observations considered separately.
What the Three Special Values Look Like
At \(r=1\), the scatterplot points lie exactly on a straight line that rises from left to right. At \(r=-1\), they lie exactly on a straight line that falls from left to right. These are called perfect positive and perfect negative linear associations. “Perfect” means there is no scatter around the line; it does not mean the relationship is important, causal, or useful for every purpose.
At \(r=0\), the positive and negative cross-products balance in the correlation formula. There is no linear association summarized by \(r\) in those data. However, the points could still follow a clear curve or some other non-linear pattern. As emphasized in “Association Versus Linear Association,” zero linear association is not the same as no association.
Values between the endpoints describe degrees of linear association. For instance, \(r=0.6\) and \(r=-0.6\) have the same magnitude, so each indicates the same degree of linear strength, but their directions differ. In practice, describing a value as weak, moderate, or strong depends on the setting and the scatterplot; there is no universal cutoff for those labels.
Worked Example: Exact Positive Linear Association
Worked Example: Exact Positive Linear Association
A fictional greenhouse activity records the number of hours a lamp is used and a plant’s measured growth, in centimeters, across four observations.
| Lamp hours, \(x\) | Growth, \(y\) (cm) |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
Question: Calculate \(r\) and explain what the value says about the pattern.
The means are \(\bar{x}=(1+2+3+4)/4=2.5\) hours and \(\bar{y}=(3+5+7+9)/4=6\) cm. The \(x\)-deviations are \(-1.5,-0.5,0.5,1.5\), and the \(y\)-deviations are \(-3,-1,1,3\). The cross-products sum to \(4.5+0.5+0.5+4.5=10\). The squared \(x\)-deviations sum to \(5\), and the squared \(y\)-deviations sum to \(20\).
Here the sign is positive, and the value is the largest possible positive correlation. Each additional lamp hour in these observations goes with exactly 2 cm more measured growth, so the four plotted points lie on the same rising straight line. The correlation itself is unitless; the 2 cm per hour describes the line’s rate of change, not the value of \(r\). These invented observations show a pattern, not proof that lamp use caused the growth.
Worked Example: Exact Negative Linear Association
Worked Example: Exact Negative Linear Association
For a fictional delivery-route exercise, a student records distance remaining to a destination and the number of minutes already spent on the route. The small set below is constructed to show a perfect negative line.
| Minutes, \(x\) | Distance remaining, \(y\) (km) |
|---|---|
| 0 | 10 |
| 1 | 8 |
| 2 | 6 |
| 3 | 4 |
Question: Find \(r\) and interpret its sign and endpoint value.
The means are \(\bar{x}=(0+1+2+3)/4=1.5\) minutes and \(\bar{y}=(10+8+6+4)/4=7\) km. The \(x\)-deviations are \(-1.5,-0.5,0.5,1.5\), and the \(y\)-deviations are \(3,1,-1,-3\). The cross-products sum to \(-4.5-0.5-0.5-4.5=-10\). The squared deviations sum to \(5\) for \(x\) and \(20\) for \(y\).
The negative sign indicates that larger elapsed times go with smaller remaining distances. The value \(-1\) means the four points lie exactly on a falling straight line, with no scatter around it. In this particular constructed set, the distance remaining drops by exactly 2 km per minute. That rate comes from the data’s line; \(r=-1\) communicates the perfect negative linear pattern, not the rate or a claim about all delivery routes.
Worked Example: Zero Correlation with a Curved Pattern
Worked Example: Zero Correlation with a Curved Pattern
A fictional sensor exercise records a signed calibration offset, \(x\), and an error measure, \(y\), for five settings. The errors are smallest near the target setting and larger on either side.
| Offset, \(x\) | Error measure, \(y\) |
|---|---|
| \(-2\) | 8 |
| \(-1\) | 3 |
| 0 | 2 |
| 1 | 3 |
| 2 | 8 |
Question: What is \(r\), and what feature would a correlation-only description miss?
The means are \(\bar{x}=0\) and \(\bar{y}=(8+3+2+3+8)/5=4.8\). The \(x\)-deviations are \(-2,-1,0,1,2\), and the \(y\)-deviations are \(3.2,-1.8,-2.8,-1.8,3.2\). The cross-products sum to \(-6.4+1.8+0-1.8+6.4=0\). The squared \(x\)-deviations sum to \(10\), and the squared \(y\)-deviations sum to \(10.24+3.24+7.84+3.24+10.24=34.8\).
The linear association is zero: the positive and negative cross-products balance. But the observations clearly form a U-shaped pattern. The error measure tends to be larger as the signed offset moves farther from zero, on either side. Therefore, saying “there is no relationship” would be wrong. The accurate statement is that \(r\) finds no linear association in these data, while the scatterplot reveals a curved association.
Common Mistakes and AP Exam Tips
- Reporting a value outside the range. Any \(r\) below \(-1\) or above \(1\) signals a calculation, entry, or reporting error. Correlation is undefined if one variable has no variation; it is not assigned a value of zero.
- Using the sign to describe strength. The sign tells direction. The distance from zero tells linear strength. For example, \(-0.7\) is negative in direction and has the same magnitude as \(0.7\).
- Calling \(r=0\) “no association.” Say “no linear association” unless the scatterplot supports a broader conclusion. A curved relationship may be evident even when \(r=0\).
- Calling a correlation of \(1\) or \(-1\) merely “strong.” Those exact endpoints mean every point lies on a straight line. A value close to an endpoint indicates a close linear pattern, but not necessarily a perfect one.
- Confusing perfect correlation with causation. Even when \(r=1\) or \(r=-1\) in observed data, the calculation alone does not establish that one variable causes the other to change.
- Forgetting context. A full-credit interpretation names both quantitative variables and explains what tends to happen as one increases. State the direction and, when appropriate, describe the scatterplot’s form rather than giving only a number.
On an AP response, keep the pieces separate: the sign describes direction, the magnitude describes the strength of the linear pattern, and the scatterplot shows form and unusual features. At \(r=0\), qualify the conclusion as “no linear association.” At \(r=1\) or \(r=-1\), specify that the observed points fall exactly on a rising or falling straight line.
Check Your Understanding
Use the range, sign, and endpoint properties to answer each question. Include a brief explanation where appropriate.
- A calculator display reports \(r=1.08\). What should you conclude about that reported value?
- What does \(r=-1\) mean about the direction and form of the observed pattern?
- In a scatterplot of commute time and distance from work, \(r=-0.65\). What does the sign tell you about the linear direction?
- Why is “there is no relationship” too broad a conclusion when \(r=0\)?
- What happens to the definition of \(r\) if every observed \(x\)-value is identical?