Why Inspect the Scatterplot First?
Pearson’s correlation coefficient, \(r\), compresses the relationship between two quantitative variables into one number. That number can be useful, but it does not show the shape of the data, reveal separate groups, or identify unusual observations. Before reporting \(r\), inspect the scatterplot: a picture can show whether a single correlation gives a fair summary of the observed relationship.
The checks in this tutorial do not impose extra mathematical requirements for calculating \(r\). Rather, they help determine whether interpreting one value of \(r\) is informative. A small \(r\), for example, can accompany a strong curved pattern, while a large pooled \(r\) can be driven by separation between groups rather than by a strong relationship within either group.
Each point should represent one observational unit, with its value of one variable on the horizontal axis and its paired value of the other variable on the vertical axis. The plot should identify the variables and their units. Correlation is symmetric, so swapping the axes does not change \(r\); clear axis placement still helps a reader understand what was measured.
Check the Form: Is One Linear Summary Suitable?
Pearson’s \(r\) describes the direction and strength of a linear association. A roughly straight, upward pattern is consistent with positive correlation; a roughly straight, downward pattern is consistent with negative correlation. If the pattern bends, a single linear summary may conceal a clear relationship. In particular, a curved pattern can have a correlation near zero even when the variables are strongly related.
Worked Example: A Clear Curve With Zero Correlation
Suppose a hypothetical set of five observations has \(x\)-values \(-2,-1,0,1,2\) and \(y\)-values \(4,1,0,1,4\). The plotted points form a symmetric U-shape. Calculate \(r\) and decide whether it summarizes the visible pattern well.
The sample means are \(\bar{x}=0\) and \(\bar{y}=2\). Using the formula for Pearson’s correlation,
The cross-product sum is \( (-2)(2)+(-1)(-1)+(0)(-2)+(1)(-1)+(2)(2)=-4+1+0-1+4=0 \). The sum of squared \(x\)-deviations is \(4+1+0+1+4=10\). The sum of squared \(y\)-deviations is \(4+1+4+1+4=14\). Therefore,
The calculated correlation is zero, but the plot is not a cloud without structure: it shows a pronounced U-shape. Reporting only \(r=0\) would hide the nonlinear pattern. Describe the curve rather than treating zero correlation as evidence of no relationship.
A scatterplot need not lie exactly on a straight line for \(r\) to be useful. Real data commonly show scatter around a general linear trend. The practical question is whether a straight-line description captures the main shape, not whether every point falls on a line.
Check for Clusters and Outliers
A cluster is a group of points that is relatively separate from other points in the plot. Clusters may correspond to meaningful categories, such as different locations or populations. A pooled \(r\), calculated after combining all observations, can reflect differences between groups and may not describe the relationship within any group. If groups are visible or known, identify them and consider whether the pooled value needs qualification.
An outlier is an observation that lies unusually far from the general pattern. Because \(r\) depends on distances from the means, an unusual point can substantially change its value. A point that looks unusual deserves investigation, not automatic deletion: check for a recording or measurement error, and retain a valid observation unless there is a defensible reason to exclude it. When it is important, explain how the correlation changes with and without it.
Worked Example: An Outlier Changes the Correlation
A hypothetical set of five observations follows the exact pattern \(x=1,2,3,4,5\) and \(y=2,4,6,8,10\). A sixth valid observation is added: \((6,0)\). Compare the correlation before and after adding it.
For the first five observations, every \(y\)-value equals \(2x\), so the points lie exactly on an increasing straight line and \(r=1\). For all six observations, \(\bar{x}=3.5\) and \(\bar{y}=5\). The needed sums can be calculated from the data: \(\sum x_i^2=1+4+9+16+25+36=91\), \(\sum y_i^2=4+16+36+64+100+0=220\), and \(\sum x_i y_i=2+8+18+32+50+0=110\).
Thus the centered sums are \(S_{xx}=91-6(3.5)^2=17.5\), \(S_{yy}=220-6(5)^2=70\), and \(S_{xy}=110-6(3.5)(5)=5\). Substituting into the correlation formula gives
The sixth point changes the correlation from \(1\) to about \(0.143\). The scatterplot makes the reason visible: \((6,0)\) is far from the original upward pattern. If it is a valid observation, the changed value is part of the data’s story. Report the sensitivity rather than silently removing the point.
Worked Example: Separate Clusters Produce a Strong Pooled Correlation
Consider two hypothetical groups, each with six paired observations. Group A has \(x=1,2,3,4,5,6\) and \(y=1,2,3,3,2,1\). Group B has \(x=7,8,9,10,11,12\) and \(y=6,7,8,8,7,6\). Group B’s arrangement is Group A’s shifted six units on the \(x\)-axis and five units on the \(y\)-axis. Compare the pooled pattern with the patterns within each group.
For Group A, \(\bar{x}=3.5\) and \(\bar{y}=2\). Its centered \(x\)-values are \(-2.5,-1.5,-0.5,0.5,1.5,2.5\), and its centered \(y\)-values are \(-1,0,1,1,0,-1\). Their cross-product sum is \(2.5+0-0.5+0.5+0-2.5=0\). Group B has the same within-group arrangements after shifting both variables by constants, so its within-group cross-product sum is also zero. Each group therefore has \(r=0\).
For all 12 observations combined, \(\bar{x}=6.5\) and \(\bar{y}=4.5\). Within the groups, the sums of squared \(x\)-deviations add to \(17.5+17.5=35\), and the sums of squared \(y\)-deviations add to \(4+4=8\). The two group means for \(x\) are 3 units below and above the pooled mean; for \(y\), they are 2.5 units below and above. The between-group contributions to \(S_{xx}\), \(S_{yy}\), and \(S_{xy}\) are therefore \(108\), \(75\), and \(90\), respectively. Including the within-group contributions gives \(S_{xx}=143\), \(S_{yy}=83\), and \(S_{xy}=90\). Hence,
The pooled correlation is strongly positive even though neither group has a within-group linear association. The overall value reflects the separation between groups. A scatterplot colored by group would expose the two clusters; reporting only the pooled \(r\) would be incomplete.
Check the Range and the Axes
A restricted range means the observed values cover only a limited part of the values relevant to the question. When a study includes only a narrow band of \(x\)-values, the observed correlation describes that band; it may not represent the association across a broader population or range. Restriction can make a relationship appear weaker than it would over a wider range, but the scatterplot alone cannot establish what the unobserved data would look like. State the range sampled and avoid extending the conclusion beyond it.
Axis labels are part of the evidence a reader needs to interpret a plot. Each axis should name its variable, give units where applicable, and show a scale with understandable tick marks. Check that the plotted values fall within the stated limits and that the tick spacing is consistent. A narrow or truncated axis can make visual changes seem dramatic; it does not change the calculated \(r\), but it can affect the impression created by the graph. A scatterplot does not always need to start at zero, so the key is to show and label the scale honestly.
Worked Example: A Restricted Range and an Unclear Axis
A hypothetical report examines study hours and quiz scores for students who study between 4 and 6 hours per week. The scatterplot shows a modest upward pattern, and the horizontal axis is labeled only “Hours,” with no units. What can the plot support, and what should be checked before reporting the correlation?
First, the range is only 4–6 hours per week. The observed \(r\) can summarize the linear association among the sampled students within that range, but it does not establish the association for students studying 0–20 hours per week. Values outside the observed range are not shown, so a broader pattern cannot be inferred from this plot alone.
Second, “Hours” is incomplete because it does not say what was measured or over what period. A clearer label is “Study time (hours per week).” The vertical axis should likewise name the quiz score and its units or scale, such as “Quiz score (points out of 50).” Check that tick marks match the plotted values and that any shortened axis range is visible. If the plot still shows a roughly straight pattern with no separate clusters or influential points, \(r\) may be reported as a summary—but with the sampled range and population clearly stated.
A Practical Pre-Reporting Checklist
A scatterplot diagnosis can be organized as a short sequence. Use it before writing a numerical result so that the interpretation reflects what the paired observations actually show.
Each point should correspond to one observational unit with both measurements recorded.
Decide whether the pattern is reasonably straight or clearly curved; \(r\) is a linear summary.
Ask whether clusters or individual observations may be driving the pooled correlation.
State which values are represented and keep conclusions within that range.
Name both variables, include units, and make the scale and limits clear.
Why the Checks Matter
A correlation is not a substitute for looking at data. It is a compact numerical summary whose usefulness depends on context and pattern. The form check guards against treating a curve as no relationship. The cluster check guards against interpreting a pooled association as a within-group pattern. The outlier check alerts the reader to sensitivity, while the range and axis checks clarify what the data cover and how they are displayed. These checks support a careful report; they do not by themselves establish causation.
A common mistake is to calculate \(r\), see that it is near zero, and conclude that the variables are unrelated. The U-shaped example shows why that reasoning fails: its correlation is zero even though the points follow a clear curve. The correction is to inspect the scatterplot and describe the form before interpreting \(r\). Likewise, a high correlation does not guarantee a single consistent pattern, as the two-cluster example demonstrates.
Check Your Understanding
Use the diagnostic ideas to answer each question.
- A scatterplot has a clear inverted-U shape, but the reported \(r\) is close to zero. Why is “there is no relationship” an inadequate conclusion?
- Two visibly separate groups have a strong pooled correlation, while each group separately has a correlation near zero. What should be investigated before interpreting the pooled value?
- One point lies far from an otherwise tight linear pattern. What checks should be made before deciding whether to include it in the report?
- A study includes only people whose ages fall between 20 and 25 years. What limitation should accompany a correlation between age and a health measurement?
- What information should the horizontal and vertical axis labels provide for a scatterplot of daily exercise and resting heart rate?