Introduction
A response value can look unusual for two different reasons: it may be large or small compared with all the other responses, or it may sit far from the pattern among observations with similar predictor values. In a relationship plot, the second comparison is usually the more useful one. If response values tend to rise as the predictor rises, a high response at a high predictor value may be ordinary, while a less extreme response at a low predictor value may be far from the surrounding pattern.
The goal is not to label every distant point as an error. An unusual-looking observation may be a valid and informative result. The plot identifies a value worth checking; the original record and context help determine what it means.
Unusual Relative to What?
The comparison should account for the relationship’s direction and form. If the points follow a rising curve, compare an observation with the curve near its predictor value, not with the response average for the entire plot. A LOESS smoother or a binned summary can help make the pattern easier to see; each is a visual reference, not a guarantee of what an individual response “should” be.
One way to describe a point’s position is to name a reference response \(m(x_i)\) for its predictor value \(x_i\), then calculate its vertical departure:
Here \(y_i\) is the observed response, and \(d_i\) is a signed difference in response units. A positive value means the point is above the reference; a negative value means it is below. The reference may be read approximately from a smooth pattern or specified by a simple local comparison. This calculation is a descriptive aid: it does not require that a regression model has been fitted, and it does not establish a universal cutoff for unusualness.
The reason to compare locally can be shown by adding and subtracting the overall response mean \(\bar{y}\):
The first term describes how the pattern’s reference response at \(x_i\) differs from the overall mean. The second is the point’s vertical departure from that pattern. A response can be far from the overall mean because the relationship itself predicts higher or lower responses in that part of the plot, even when the point is close to its local pattern.
How to Locate and Assess a Departure
Start by reading the broad direction and form of the scatterplot. Then compare a potentially unusual response with nearby points in predictor value and with the local pattern. Ask whether the point is clearly separated vertically, whether nearby observations show similar departures, and whether the vertical spread changes across the predictor range. When spread changes, the same absolute distance may look more striking in a narrow-spread region than in a wide-spread region. A fixed distance rule should not replace inspection of the plot.
Find the observation and note its predictor value and response units. Do not judge its response in isolation from the rest of the plot.
Use nearby observations and, where helpful, the plotted smoother or binned summary to describe the local pattern.
One point far from nearby responses is an isolated departure. Several points departing in a similar way may indicate broader structure that a single-point explanation cannot capture.
Verify the source value, units, row identity, and measurement context. If the record is valid, retain it and describe the departure cautiously.
An isolated point and a systematic pattern call for different questions. For an isolated point, first check whether the value was transcribed or measured correctly and whether it belongs to the stated individual and units. For repeated departures, examine whether the plotted pattern may be changing form or whether some relevant measurement or recording condition differs across observations. Neither pattern alone establishes a cause.
Worked Example: A Response Far Above Its Local Pattern
Consider hypothetical measurements of recovery time, in minutes, against hours of a training activity. A smooth visual summary gives approximate reference responses at the observed predictor values. One observation has a recovery time of 74 minutes at 6 hours. Is it unusual relative to the pattern?
| Hours of activity, \(x_i\) | Observed recovery time, \(y_i\) (minutes) | Reference \(m(x_i)\) (minutes) | Departure \(d_i\) (minutes) |
|---|---|---|---|
| 2 | 43 | 45 | -2 |
| 3 | 49 | 47 | 2 |
| 4 | 50 | 51 | -1 |
| 5 | 57 | 55 | 2 |
| 6 | 74 | 58 | 16 |
| 7 | 62 | 61 | 1 |
| 8 | 66 | 65 | 1 |
For the observation at 6 hours, the departure is:
The other listed departures range from -2 to 2 minutes, while this point is 16 minutes above its reference. In this hypothetical plot, it is a clear candidate for an unusual response value. The calculation describes its position; it does not show that the record is wrong or explain why recovery time was high.
The next step is to check the original record for the 6-hour observation: confirm that the response was recorded as 74 minutes, that the units are minutes, and that the response and predictor belong to the same observation. If the record is verified, report it as a valid point far above the displayed pattern unless further information supports a more specific interpretation.
Worked Example: A High Response That Follows the Trend
In another hypothetical data set, a study records hours spent practicing and a performance score. The response at 6 hours is the largest score in the table. Does that make it unusual relative to the relationship?
| Practice hours, \(x_i\) | Score, \(y_i\) | Reference \(m(x_i)\) | Departure \(d_i\) |
|---|---|---|---|
| 1 | 52 | 53 | -1 |
| 2 | 59 | 59 | 0 |
| 3 | 64 | 65 | -1 |
| 4 | 72 | 71 | 1 |
| 5 | 76 | 77 | -1 |
| 6 | 83 | 83 | 0 |
The score of 83 is the largest observed response, but its vertical departure from the reference pattern is:
The observations show a rising relationship, so higher scores occur at higher practice hours. The score of 83 follows that pattern rather than standing far above it. It is extreme in the list of response values, but it is not unusual relative to comparable predictor values. This distinction prevents a high response from being flagged merely because it is the maximum.
Worked Example: Several Departures Suggest Broader Structure
A hypothetical plot relates a device setting to a measured response, in units. A smoother provides the reference values below. The last three observations are all above that reference. Calculate their departures and decide whether the plot suggests one isolated unusual response.
| Setting, \(x_i\) | Observed response, \(y_i\) (units) | Reference \(m(x_i)\) (units) | Departure \(d_i\) (units) |
|---|---|---|---|
| 1 | 11 | 10 | 1 |
| 2 | 12 | 12 | 0 |
| 3 | 15 | 14 | 1 |
| 4 | 15 | 16 | -1 |
| 5 | 24 | 18 | 6 |
| 6 | 27 | 20 | 7 |
| 7 | 30 | 22 | 8 |
For the observations at settings 5, 6, and 7, the arithmetic is:
These are not just one point separated from the pattern: three consecutive observations are above it, with positive departures that increase from 6 to 8 units. That repeated structure is a reason to inspect the overall relationship and measurement context, rather than treating each value as an unrelated anomaly. Check whether the recorded values and units are correct and whether some measurement condition differs for these observations. If the records are accurate, describe the group of departures as structure not captured by the displayed reference; do not discard the points simply because they are inconvenient.
Why the Record Check Matters
A scatterplot can reveal that a value deserves attention, but it cannot determine whether it came from a transcription mistake, an unusual but valid outcome, or a feature of the measurement process. Checking the record is therefore part of interpreting the graph, not an optional step after deciding what the point means.
For a candidate observation, confirm that the plotted row corresponds to the intended individual, that the response and predictor were paired correctly, and that the values use the stated units and definitions. Compare the plotted value with its original source when available. A verified observation that departs from the pattern should generally remain visible in the analysis. If a data correction is supported by the source record, correct it transparently rather than silently deleting the point.
The earlier material on changing vertical spread is also relevant: points may naturally be more dispersed in one part of the predictor range than another. Visual distance must be judged in context. A departure that looks large in a tightly packed region may be less exceptional where responses are naturally more spread out. The plot supports a careful description, not a universal numerical threshold.
Common Mistakes
- Using the overall response range as the only comparison. The largest response may follow an increasing pattern. Compare with responses at similar predictor values.
- Calling a point an error because it is far away. Distance from a pattern identifies a value to investigate; it does not prove a recording mistake. Check the source record.
- Assuming every unusual point is isolated. Several nearby departures in the same direction may indicate broader structure. Inspect the surrounding observations before explaining one point on its own.
- Applying one distance cutoff everywhere. Vertical spread can differ across a plot. Consider the local spread and state any judgment as visual and descriptive.
- Removing a valid observation to make the pattern look cleaner. Keep verified data and explain the unusual departure; do not exclude a point solely because it does not fit the surrounding pattern.
Check Your Understanding
Use the ideas in this tutorial to answer the following questions.
- A response is the largest in a data set, but it lies close to the pattern among observations with similar predictor values. Is it necessarily an unusual response relative to the relationship? Explain.
- A point has an observed response of 41 units and a local reference value of 35 units. Calculate its signed vertical departure and state what the sign means.
- Why might several consecutive points above a smoother call for a different interpretation from one point above it?
- Name two details to verify in the original record before interpreting an unusual-looking response.
- Why should the same fixed vertical distance not automatically be treated as equally unusual everywhere in a plot with changing spread?