Introduction
When two measurements are recorded for each individual, the information lies not only in the separate measurements but also in which two values belong together. For example, a student’s study time and quiz score form a pair because both measurements describe that same student. A scatterplot displays these pairs as points, making it possible to see the data in their original paired form.
The central task here is to construct the plot correctly. Each point must represent one individual, and its horizontal and vertical coordinates must be that individual’s two recorded values. The variable chosen to help explain or predict the other is placed on the horizontal axis; the response variable is placed on the vertical axis.
Paired Quantitative Data
The phrase “on the same individual” is essential. If one list records study time for several students and another list records quiz scores for those same students, the data are paired only when it is known which score belongs to which student. Knowing the two separate lists of values without knowing their matching does not recover the paired observations.
In many questions, one variable is designated as the explanatory variable: the variable used to organize or help explain differences in the other variable. The other is the response variable, the outcome being measured. The explanatory variable goes on the horizontal axis, often called the \(x\)-axis; the response variable goes on the vertical axis, often called the \(y\)-axis. This arrangement is a plotting convention, not proof that one variable causes the other. The choice should follow the question and how the data were collected.
Both variables must be quantitative for the standard scatterplot described here. A value such as 6.5 hours or 82 points has a numerical scale on which locations and differences are meaningful. A category such as “morning” or “evening” is not itself a quantitative coordinate for this kind of plot.
From a Data Table to Points
A paired data table should keep each individual’s measurements together in one row. If there are \(n\) individuals, there are \(n\) pairs and therefore \(n\) plotted points. Some different individuals may have identical values for both variables, so two points can fall at the same location. The point still represents two observations, even if one marker covers the other.
A clear plot also needs an informative horizontal and vertical axis. Label each axis with the variable name and its measurement units, if it has units. Choose number scales that include the observed values and provide enough room to place points accurately. A plotted coordinate is meaningful only when the reader can determine what each axis measures and how its numerical scale is marked.
Confirm that each row describes one individual and contains both quantitative measurements for that individual.
Use the question to identify the explanatory variable for the horizontal axis and the response variable for the vertical axis.
Write the variable names and units, and mark numerical scales that include the observed values.
For each individual, use the two values from that same row to place one point.
Worked Example: Plotting Study Time and Quiz Score
Suppose, hypothetically, that an instructor records hours spent studying and quiz score for five students. The question is whether study time helps explain variation in quiz scores. Study time is the explanatory variable \(x\); quiz score is the response variable \(y\). Both are quantitative, and each row contains both measurements for one student, so a scatterplot is appropriate.
| Student | Study time, \(x\) (hours) | Quiz score, \(y\) (points out of 100) |
|---|---|---|
| A | 1 | 62 |
| B | 2 | 70 |
| C | 3 | 68 |
| D | 4 | 81 |
| E | 5 | 76 |
Label the horizontal axis “Study time (hours)” and the vertical axis “Quiz score (points out of 100).” Choose scales that include 1 through 5 hours and scores from 62 through 81 points; for example, the horizontal scale can run from 0 to 6 hours and the vertical scale from 50 to 100 points.
Use the paired coordinates row by row. Student A contributes \((1,62)\), because A studied for 1 hour and scored 62 points. The remaining plotted points are B: \((2,70)\), C: \((3,68)\), D: \((4,81)\), and E: \((5,76)\). Thus five students contribute five points. The score of 68 belongs with 3 hours because those two values are in the same row; it must not be matched with another student’s study time.
The plotted points show the recorded data, not a claim that more study time caused a particular score. The plot’s construction preserves what was measured for each student; conclusions about the relationship require careful interpretation.
Why the Pairing Must Be Preserved
A scatterplot is not made by placing the values from one list beside an independently sorted version of the other list. Pairing is part of the data. Changing which \(y\)-value is matched to an \(x\)-value can move points to different locations and change what the plot displays, even when the separate lists of \(x\)-values and \(y\)-values remain unchanged.
This can be seen directly from the definition. Suppose the recorded observations are \((x_1,y_1),\ldots,(x_n,y_n)\). The plotted point for individual \(i\) must have coordinates \((x_i,y_i)\). If the response values are reassigned using a permutation \(\pi\), the plotted points become \((x_i,y_{\pi(i)})\). The separate collection of response values is the same because a permutation only reorders it, but the point for an individual can change whenever the reassigned response value differs from that individual’s original value.
This is why the data table’s rows, identifiers, or other matching information must remain intact. If response values happen to repeat, reassignment does not necessarily change every plotted coordinate. For an individual with \(y_{\pi(i)}=y_i\), the point remains at \((x_i,y_i)\), even if the source of the matched value has changed. The key is to use the actual recorded pairing, not to assume that a reassigned value must always produce a different coordinate.
Worked Example: Reordering Values Without Losing the Data Lists
Imagine a hypothetical set of four practice sessions. For each session, record practice time \(x\) in minutes and trial time \(y\) in seconds. The explanatory variable is practice time, and the response variable is trial time.
| Session | Practice time, \(x\) (minutes) | Trial time, \(y\) (seconds) |
|---|---|---|
| A | 20 | 35 |
| B | 22 | 39 |
| C | 18 | 43 |
| D | 24 | 31 |
The recorded points are A: \((20,35)\), B: \((22,39)\), C: \((18,43)\), and D: \((24,31)\). The separate practice-time list is 20, 22, 18, 24 minutes, and the separate trial-time list is 35, 39, 43, 31 seconds.
Now suppose someone reassigns the trial times by swapping the first two values, while leaving the practice times in their original rows. The resulting points are \((20,39)\), \((22,35)\), \((18,43)\), and \((24,31)\). The trial-time list is still 35, 39, 43, 31 seconds, just in a different order, and the practice-time list is unchanged. Two plotted pairs, \((18,43)\) and \((24,31)\), match the recorded sessions; the first two do not.
The reassignment therefore preserves both separate lists but changes the pairing information and the plotted locations for sessions A and B. A scatterplot of the original observations must use the original coordinates, not the reassigned ones. In this example all response values differ, so the reassigned \(y\)-values for A and B produce different coordinates. If two response values were equal, exchanging them might leave one or more plotted coordinates unchanged, but that would not justify discarding the original session-to-measurement matching.
Axis Roles and Scales in a Second Example
The plotting steps apply to measurements with different units as well. The axis roles are determined by the question, while the units describe how each coordinate is measured. Make sure that the labels and plotted values use the same units; for instance, a value recorded in centimeters should not be plotted using a scale labeled in millimeters unless it is first converted.
Worked Example: Plotting Light Exposure and Plant Height
In a hypothetical classroom activity, students record the daily light exposure and height of each of six seedlings. The question asks whether recorded light exposure can help describe seedling height. Let \(x\) be light exposure in hours per day and \(y\) be seedling height in centimeters. Both variables are quantitative, and every row matches the two measurements for one seedling, so each row can be represented by a point.
| Seedling | Light exposure, \(x\) (hours per day) | Height, \(y\) (cm) |
|---|---|---|
| 1 | 4 | 8.2 |
| 2 | 6 | 9.1 |
| 3 | 5 | 8.7 |
| 4 | 8 | 10.4 |
| 5 | 7 | 9.8 |
| 6 | 6 | 9.5 |
Put “Light exposure (hours per day)” on the horizontal axis and “Seedling height (cm)” on the vertical axis. For a possible scale, mark the horizontal axis from 0 to 10 hours per day and the vertical axis from 7 to 11 cm. The bounds include all the recorded values: the smallest and largest exposure values are 4 and 8 hours, and the smallest and largest heights are 8.2 and 10.4 cm.
Translate each row directly into a coordinate: seedling 1 gives \((4,8.2)\); seedling 2 gives \((6,9.1)\); seedling 3 gives \((5,8.7)\); seedling 4 gives \((8,10.4)\); seedling 5 gives \((7,9.8)\); and seedling 6 gives \((6,9.5)\). There are six rows and therefore six observations to plot. Seedlings 2 and 6 share an \(x\)-value of 6 hours, but they have different heights, so their points have the same horizontal coordinate and different vertical coordinates.
No conversion is needed because the table and axes use hours per day and centimeters consistently. Also, the data are measurements on seedlings, not a randomized experiment described in enough detail to establish a causal effect. The scatterplot construction itself makes no causal claim.
A Short Justification: Why There Is One Point per Individual
The one-point-per-individual rule follows from the definition of paired data. For each individual \(i\), the table provides exactly one explanatory measurement \(x_i\) and one response measurement \(y_i\). Together those values identify one location in the coordinate plane, \((x_i,y_i)\). Repeating that operation for \(i=1,\ldots,n\) gives the complete set of \(n\) paired observations:
If the data are instead split into two lists and independently sorted, the resulting operation combines values according to their new positions in those lists rather than according to the individual they describe. That does not follow the definition of \((x_i,y_i)\), so it can create pairings that were not recorded. Keeping each row intact is therefore not a formatting preference; it is required to display the measured paired observations faithfully.
Common Mistakes and Why the Plot Matters
- Plotting two separately sorted lists. Sorting each variable independently can match one person’s \(x\)-value with another person’s \(y\)-value. Keep the original rows together.
- Putting the response variable on the horizontal axis by default. The convention is explanatory variable horizontally and response variable vertically. Identify the roles from the question before plotting.
- Leaving off units or unclear labels. A reader needs to know what the numbers measure. Label the axes with both the variable and its units, such as “Height (cm).”
- Treating a scatterplot as evidence of causation. A plot displays paired measurements. By itself, it does not show that changing one variable causes a change in the other.
- Assuming every individual must have a visibly separate marker. Two individuals can have identical measured values and hence share one plotted location. The underlying data still contain two observations.
A scatterplot is useful because it preserves two measurements together for each observational unit in a compact visual display. Correct construction is the necessary first step before describing or analyzing a relationship. A plot based on mismatched pairs may look orderly, but it does not represent the recorded data.
Check Your Understanding
Answer each question using the definitions and construction rules above.
- A dataset records the distance run and the time taken by each runner. What information must be known before constructing a scatterplot, and what does one point represent?
- A researcher asks whether daily screen time helps explain nightly sleep duration. Which variable belongs on each axis, and what should the axis labels include?
- Why can independently sorting the two variable columns create a scatterplot that does not represent the recorded individuals?
- Six observations are recorded, but two individuals have exactly the same values for both variables. How many observations are in the data, and where do those two observations appear on a scatterplot?
- Suppose two response values are equal and are exchanged between individuals. Must the affected plotted coordinates change? Explain using the ordered-pair definition.