Introduction
A scatterplot can show a clear overall tendency even when its points do not sit exactly on a line or curve. To describe the strength of the relationship, ask how closely the points follow that overall pattern. Points that stay close to it indicate a stronger relationship; points spread more widely around it indicate a weaker one.
Strength is a visual judgment, not a calculation with a universal cutoff. The words weak, moderate, and strong summarize how much scatter there is relative to the pattern. They do not tell us whether the relationship is positive or negative, how steep it is, or whether one variable causes the other.
Read Closeness Around the Pattern
First identify the overall pattern, then judge the scatter around it. For a roughly linear pattern, look at how far points tend to fall above or below the general straight-line trend. For a curved pattern, judge their closeness to the curve instead. Do not decide strength just by looking for points that form a straight line: a close-fitting curve can also show a strong relationship.
A useful description of the comparison is “spread around the pattern,” rather than simply “spread of the points.” A data set can cover a wide range of values and still show a strong relationship if the points stay close to a clear trend across that range. Conversely, points can occupy a narrow range but show little relationship if there is no consistent pattern.
Strength also differs from direction and steepness. Direction tells whether larger values of one variable tend to accompany larger or smaller values of the other. Steepness describes how quickly the plotted values change across the graph. Neither one, by itself, tells how tightly the observations follow the pattern.
Worked Example: Comparing Three Positive Linear Patterns
Suppose three hypothetical scatterplots show weekly practice time \(x\), in hours, and a quiz score \(y\), in points, for five students each. In all three plots, the broad pattern is positive. To make the visual comparison concrete, use the same reference pattern, \(y_g=50+8x\), and calculate each point’s vertical departure from it:
Here \(y_g\) is a reference value on the guide pattern at that student’s practice time. The signed departure \(d\) says whether the observed score is above or below the guide; its absolute size, \(|d|\), describes the vertical distance. This is a descriptive aid for these examples, not a formal strength statistic.
For \(x=1,2,3,4,5\), the guide scores are \(58,66,74,82,90\). In the first plot, the hypothetical observed scores are \(59,65,75,81,91\). For example, at \(x=1\), \(d=59-58=1\) point; at \(x=2\), \(d=65-66=-1\) point. The departures across all five points are \(+1,-1,+1,-1,+1\) points. The observations stay very close to the positive linear pattern, so this plot shows a strong relationship.
In the second plot, the scores are \(68,55,83,70,102\). The departures are \(68-58=10\), \(55-66=-11\), \(83-74=9\), \(70-82=-12\), and \(102-90=12\) points. There is still an overall upward tendency, but the observations sit noticeably farther from the guide than in the first plot. This is a reasonable description of a moderate positive relationship.
In the third plot, the scores are \(40,86,58,98,76\). The departures are \(40-58=-18\), \(86-66=20\), \(58-74=-16\), \(98-82=16\), and \(76-90=-14\) points. The points are much more dispersed around the positive tendency, so the relationship is weak compared with the other two. The classification comes from comparing closeness to the pattern, not from whether every successive score increases.
These labels are judgments, not boundaries produced by the departure calculations. The reference pattern helps make the comparison visible, but there is no universal rule saying that a particular number of score points makes a relationship weak or strong.
Why Dispersion Matters
The connection between closeness and strength can be understood by considering how useful the overall pattern would be for describing an individual observation. Imagine a pattern guide that gives a typical \(y\)-value for each \(x\)-value. At a particular \(x\), a point’s vertical departure from that guide is the observed value minus the guide value:
In this expression, \(x_i\) and \(y_i\) are the coordinates of point \(i\), \(g(x_i)\) is the guide’s value at \(x_i\), and \(d_i\) is the point’s signed vertical departure. If two plots share a similar overall pattern but one has departures that are generally smaller in absolute value, its points lie closer to that pattern. For example, a departure of \(1\) hour is closer to the guide than a departure of \(3\) hours because \(1<3\). Repeatedly smaller departures mean less vertical scatter around the pattern, which is why the relationship is judged stronger.
This reasoning supports a visual comparison; it does not turn every scatterplot into a formal calculation. In practice, the overall pattern may itself be only approximate, and its shape may not be summarized by one simple formula. The key is to compare how the points cluster around that pattern rather than to treat a hand-drawn guide as an exact model.
Worked Example: Judging a Moderate Negative Pattern
In a hypothetical set of observations, \(x\) is a person’s late-evening screen time in hours and \(y\) is that night’s sleep duration in hours. Five plotted pairs are \((0,8.3)\), \((1,7.2)\), \((2,7.4)\), \((3,5.0)\), and \((4,6.0)\). The overall tendency is downward: larger screen-time values generally accompany shorter sleep durations.
For a comparison, suppose a visual guide for that downward tendency gives \(y_g=8.5-0.7x\) hours. The guide values at \(x=0,1,2,3,4\) are \(8.5,7.8,7.1,6.4,5.7\) hours. Applying \(d=y-y_g\), the departures are:
The points follow the downward tendency, but their vertical departures are not uniformly tiny; one is \(1.4\) hours below the guide. A measured description is a moderate negative relationship: the overall direction is apparent, while the points show meaningful scatter around it. This describes an association in these hypothetical observations; it does not establish that screen time causes a change in sleep duration.
Strength Does Not Require a Straight Pattern
The same idea applies when the overall pattern is curved. Judge the points’ closeness to the curve, not their distance from a straight line. Otherwise, a plot with a strong curved relationship could be incorrectly described as weak simply because a line would not follow its shape.
Worked Example: A Strong Curved Pattern
Imagine hypothetical daily observations from one building, with outdoor temperature \(T\) in degrees Celsius and energy use \(y\) in kilowatt-hours. Suppose the overall pattern is U-shaped: energy use tends to be higher on cooler and warmer days and lower at moderate temperatures. A guide curve has values \(24,16,10,16,24\) kilowatt-hours at temperatures \(10,15,20,25,30\) degrees, respectively.
Suppose the observed energy-use values at those five temperatures are \(24.5,15.0,10.7,15.5,23.2\) kilowatt-hours. Their departures from the guide are \(24.5-24=0.5\), \(15.0-16=-1.0\), \(10.7-10=0.7\), \(15.5-16=-0.5\), and \(23.2-24=-0.8\) kilowatt-hours. Each point lies close to the corresponding value on the U-shaped pattern. The relationship is therefore strong in this example, even though the pattern is not a straight line. The description concerns the observed association and, by itself, does not show that temperature caused the energy use.
Common Mistakes
- Confusing strength with direction. “Positive” and “negative” describe direction. Either kind of relationship can be weak, moderate, or strong depending on how closely points follow the pattern.
- Confusing strength with steepness. A steep trend can have considerable scatter, and a shallow trend can have little scatter. Slope alone does not determine strength.
- Looking for a straight line in every plot. If the points follow a curve closely, judge their closeness to that curve rather than calling the relationship weak because it is not linear.
- Treating labels as fixed numerical categories. There is no universal cutoff that separates weak from moderate or moderate from strong in a visual judgment. Explain the observed closeness in context.
- Claiming causation from a close pattern. A strong association describes how two variables vary together. It does not, on its own, establish that changes in one variable cause changes in the other.
Check Your Understanding
Use the ideas in this tutorial to answer the following questions.
- Two scatterplots show similar positive linear patterns. In one, points stay close to the trend; in the other, they are widely dispersed around it. Which has the stronger relationship, and why?
- Can a relationship be strong if its overall pattern is curved? Explain how you would judge its strength.
- A plot has a steep downward trend with substantial scatter. Does the steepness alone mean the relationship is strong? Explain.
- For a point with observed value \(y=12\) and guide value \(g(x)=15\), calculate its signed departure \(d=y-g(x)\). What does the sign indicate?
- A strong association is observed between two variables. What does that say about causation, and what does it not establish?