From a Scatterplot to a Correlation Value
In “What the Correlation Coefficient Measures” and “Properties of \(r\): Range and Sign,” you learned that \(r\) summarizes the direction and strength of the linear association between two quantitative variables. Now use those properties in reverse: given several correlation values and scatterplots, decide which value best describes each plot.
A useful match starts with direction. A plot that generally rises from left to right must have a positive correlation; a plot that generally falls must have a negative correlation. Next, compare strength: points that stay close to a straight-line pattern generally have a larger \(|r|\), while points more widely scattered around a linear trend generally have an \(r\) closer to zero.
Read the Sign, Then Compare the Magnitude
The sign separates upward and downward linear patterns. If the points tend to rise as you move from left to right, match a positive \(r\). If they tend to fall, match a negative \(r\). A value such as \(r=-0.3\) is negative even though its magnitude is fairly close to zero: the sign still tells you the overall linear direction.
Once direction is accounted for, compare the absolute values. For example, \(|0.9|>|0.5|>|-0.3|\). The first value describes the strongest linear association of these three, the second describes a less closely organized linear pattern, and the third describes a weaker linear association. The sign of the third value means its weak trend is downward.
These comparisons are relative, not universal cutoffs. A value such as \(0.5\) is not automatically “moderate” in every setting, and there is no single threshold that defines a strong correlation in all circumstances. In a matching question, use the plots provided as the comparison: which pattern is most tightly linear, which is less tight, and which shows the most scatter around a straight trend?
| What you see in the plot | What to look for in \(r\) |
|---|---|
| The overall pattern rises from left to right. | A positive value. |
| The overall pattern falls from left to right. | A negative value. |
| Points cluster closely around a straight trend. | A magnitude closer to 1. |
| Points show substantial scatter around a straight trend. | A magnitude closer to 0. |
| The points follow a clear curve rather than a straight trend. | Do not judge association strength from \(r\) alone; \(r\) summarizes linear association. |
Keep form in view, as in “Recognizing Linear and Nonlinear Form.” Correlation describes linear association. A plot can show a clear, strong curved relationship and still have a correlation near zero, especially when the curve is balanced around its center. So do not automatically assign a large \(|r|\) to the plot that looks most organized overall. Ask whether its organized pattern is straight.
What Correlation Does Not Show You
A correlation value does not tell you the slope of the pattern. Two plots can have the same \(r\) even if one rises steeply and the other rises gently. Correlation is unitless, and its magnitude describes how closely the points follow a linear pattern—not how many units the response changes for each unit of the explanatory variable.
The value of \(r\) also does not replace a careful look at unusual features. A point far from the rest of the linear pattern can affect the correlation, as discussed in “Spotting Outliers in Bivariate Data.” When matching plots, notice whether one has an unusual point and consider whether the overall pattern is still approximately linear. Do not assume the same correlation must fit every plot that has the same general direction.
Finally, axis scales can change how steep or visually stretched a plot appears, as you learned in “Effect of Axis Scale on Appearance.” Focus on the direction and how closely points follow a straight pattern, not on the apparent steepness. The correlation summarizes the data’s linear association; it is not a visual rating of slope.
Worked Example: Matching \(0.9\), \(0.5\), and \(-0.3\)
Worked Example: Matching \(0.9\), \(0.5\), and \(-0.3\)
Three fictional scatterplots compare weekly practice time with a skill score. Each plot has a roughly linear pattern, but the direction and amount of scatter differ. Match the plots to \(r=0.9\), \(r=0.5\), and \(r=-0.3\).
| Plot | Visual description |
|---|---|
| A | Points form a narrow band rising from left to right. |
| B | Points generally fall from left to right, with considerable scatter around the trend. |
| C | Points generally rise from left to right, but are more scattered around the trend than in Plot A. |
Solution: Plot A has a positive direction and the tightest straight-line pattern. It matches \(r=0.9\), which is positive and close to 1. Plot B has a negative direction, so it must match the only negative value, \(r=-0.3\). Its substantial scatter is also consistent with a relatively weak linear association. Plot C rises, so it needs a positive value; its pattern is less tight than Plot A’s. It matches \(r=0.5\).
A quick check confirms that each value fits both features of its plot. The sign agrees with the direction, and the relative magnitudes agree with the amount of linear scatter: \(|0.9|>|0.5|>|-0.3|\). The negative sign on \(-0.3\) describes direction; it does not make that correlation stronger than \(0.5\).
Worked Example: Match Direction Before Strength
Worked Example: Match Direction Before Strength
A fictional set of plots shows daily outdoor temperature and the number of hot drinks sold at a small market. Match the descriptions below to \(r=-0.85\), \(r=0.15\), and \(r=0.7\).
| Plot | Visual description |
|---|---|
| D | A tight, nearly straight band slopes downward. |
| E | A loose pattern has a slight upward tendency, with no close band around a line. |
| F | A fairly clear upward linear trend is visible, but the points are not as tightly grouped as in Plot D. |
Solution: Plot D falls from left to right and follows a line very closely. It matches \(r=-0.85\): the negative sign indicates a downward direction, and the large magnitude indicates a strong linear association. Plot E has a slight upward tendency and substantial scatter. It matches \(r=0.15\), a positive value close to zero. Plot F rises more clearly and consistently than Plot E, so it matches \(r=0.7\), a positive value with a larger magnitude.
Notice that comparing absolute values alone is not enough at the start. Both \(-0.85\) and \(0.7\) have fairly large magnitudes, but only one is negative. Direction identifies which value can match Plot D. Then the relative closeness of the points to a line distinguishes the strong downward pattern from the weaker upward one.
Worked Example: A Curved Plot Is Not the Strongest Linear Plot
Worked Example: A Curved Plot Is Not the Strongest Linear Plot
An environmental science class makes two fictional scatterplots of a measurement against distance from a reference point. Plot G is a clear U-shape, with small response values near the middle and larger values toward both ends. Plot H rises in a roughly straight band, with some visible scatter. The correlation values to match are \(r=0.02\) and \(r=0.72\).
Solution: Plot H matches \(r=0.72\). Its points rise from left to right and stay reasonably close to a straight-line pattern, so a positive correlation with a moderately large magnitude is suitable. Plot G matches \(r=0.02\), which is very close to zero. The curve is strong as a nonlinear pattern, but its two sides do not create a strong overall linear trend. The correlation summarizes linear association, not how clearly the points follow any shape.
It would be a mistake to assign \(0.72\) to Plot G simply because the U-shape is easy to see. That value would indicate a fairly strong positive linear association, whereas the U-shaped plot has no clear overall upward straight-line trend. The plot still shows a relationship; the point is that \(r\) does not summarize its curved form well.
Worked Example: Separate Strength from Steepness
Worked Example: Separate Strength from Steepness
Two fictional plots compare the number of minutes spent charging a device with its battery level. Plot J rises gently, and its points stay close to a straight line. Plot K rises much more steeply, but its points are widely scattered around the overall line. Which plot should receive \(r=0.88\), and which should receive \(r=0.35\)?
Solution: Plot J should receive \(r=0.88\), and Plot K should receive \(r=0.35\). Both plots rise, so both correlations are positive. Plot J has points closer to a straight-line pattern, so it should have the larger magnitude. Plot K’s steeper appearance does not make its correlation larger: its wider scatter indicates a weaker linear association.
The match is based on how closely observations follow a straight trend, not on how quickly the response appears to change. A steep pattern can have a modest correlation, and a gentle pattern can have a strong one. To describe how quickly the battery level changes per minute, one would need information about the line’s slope; \(r\) does not provide that rate.
Common Mistakes and AP Exam Tips
- Ignoring the sign. A plot that rises from left to right cannot match a negative \(r\). State the direction first, then compare magnitudes.
- Using the signed values as a strength ranking. Strength comparisons use \(|r|\). For example, compare the magnitudes of \(-0.8\) and \(0.6\), not just the fact that one number is negative.
- Confusing steepness with strength. The steepest-looking plot does not necessarily have the largest \(|r|\). Look at scatter around the straight trend.
- Calling a clear curve a strong linear association. A curved pattern may be strong overall but not linear. Match \(r\) to the straight-line trend, not to how organized the curve looks.
- Treating rough labels as fixed cutoffs. There is no universal boundary at which a correlation becomes “strong.” Explain the comparison using the plots and the relative sizes of \(|r|\).
- Forgetting unusual points. An outlier can change the appearance and strength of a linear association. Describe what the plot shows rather than matching by direction alone.
For full-credit communication, name the sign and the visual evidence: for example, “This plot matches \(r=-0.3\) because its overall linear trend is downward, and its substantial scatter indicates a weaker linear association than the tighter plots.” This explanation addresses both direction and strength without treating \(r\) as a measure of slope.
Check Your Understanding
For each item, explain how the direction and strength of the linear pattern guide your answer.
- One plot rises tightly from left to right; another rises but has much more scatter. Which should match \(r=0.92\), and which should match \(r=0.4\)?
- A plot slopes downward with a fairly close linear pattern. Would \(r=0.75\) or \(r=-0.75\) be a better match? Why?
- A scatterplot has a clear inverted U-shape and little overall upward or downward linear trend. Should it receive \(r=0.8\) or \(r=0.05\)? Explain what the correlation does and does not summarize.
- Two plots both rise from left to right. One is steep but widely scattered; the other is gentle but tightly linear. Which is more likely to have the larger \(|r|\)?
- A student says a plot must have a strong correlation because its points follow a steep line. Identify the mistake in that reasoning.