Whiskers and Flagged Points in Modified Boxplots

What the Whiskers and Separate Points Show

A boxplot gives a compact view of a quantitative distribution. Its box marks the first and third quartiles, and a line inside the box marks the median. In a modified boxplot, the whiskers and any points plotted separately add another distinction: they show which observed values lie within a specified range and which lie beyond it.

The distinction matters because a whisker does not necessarily reach the smallest or largest value in the data, and it does not usually end at a calculated fence. Instead, it reaches the most extreme observed value that is not beyond a fence. Values beyond a fence are plotted individually, often as dots. Those dots are commonly called flagged points or potential outliers.

Definition: The interquartile range is $IQR=Q_3-Q_1$, where $Q_1$ is the first quartile and $Q_3$ is the third quartile. In a modified boxplot, the lower and upper fences are $$ Q_1-1.5(IQR) \qquad \text{and} \qquad Q_3+1.5(IQR). $$ An observed value below the lower fence or above the upper fence is flagged and plotted separately. The lower whisker reaches the smallest observed value at or above the lower fence; the upper whisker reaches the largest observed value at or below the upper fence. A value exactly on a fence is not beyond it, so it is not flagged by this rule.

The fences are reference values calculated from the quartiles; they need not be values that occurred in the data. The whiskers, by contrast, end at actual observations. For that reason, a whisker can stop well inside its fence: there may simply be no observed value between the whisker endpoint and the fence. A separate point also represents an observed value, not the fence itself.

How the Fence Rule Works

The interquartile range measures the distance from $Q_1$ to $Q_3$, containing the middle half of the ordered data. The modified-boxplot rule places each fence 1.5 interquartile ranges beyond the nearer quartile. The lower fence is below $Q_1$, and the upper fence is above $Q_3$. The rule therefore uses the middle half of the data to set a reference range for identifying unusually distant observations.

The distance from $Q_1$ down to the lower fence is $1.5(IQR)$, and the distance from $Q_3$ up to the upper fence is also $1.5(IQR)$. Since the quartiles themselves are $1(IQR)$ apart, the full distance between the two fences is $4(IQR)$:

\bigl[Q_3+1.5(IQR)\bigr]-\bigl[Q_1-1.5(IQR)\bigr] =(Q_3-Q_1)+3(IQR)=4(IQR).

This calculation explains the symmetry of the rule around the box: each fence extends the same multiple of the IQR beyond its corresponding quartile. It does not say that the data are symmetric, nor does it guarantee that any observations will lie beyond the fences. The rule is a convention for displaying observations relative to the quartiles.

Read the quartiles.
Identify $Q_1$ and $Q_3$ from the boxplot or its accompanying summary.

Calculate the IQR and fences.
Subtract $Q_1$ from $Q_3$, then extend $1.5(IQR)$ below $Q_1$ and above $Q_3$.

Classify the observations.
Values below the lower fence or above the upper fence are plotted separately. Values on or between the fences are non-flagged.

Locate the whisker endpoints.
Find the smallest and largest non-flagged observed values. These, not the fences, are the whisker endpoints.

Worked Examples

Worked Example: One Flagged Point and Two Whiskers

Suppose a hypothetical set of eight measurements, in units, is $2, 8, 10, 12, 12, 14, 16, 34$. For the quartile convention used to make this boxplot, the reported quartiles are $Q_1=9$ units and $Q_3=15$ units. Identify the fences, the whisker endpoints, and any separately plotted point.

First calculate the interquartile range:

IQR=Q_3-Q_1=15-9=6\text{ units}.

The lower fence is $9-1.5(6)=9-9=0$ units. The upper fence is $15+1.5(6)=15+9=24$ units. An observation must be below 0 or above 24 units to be flagged. The observed value 2 is above the lower fence, so it is not flagged. The values 8, 10, 12, 12, 14, and 16 are also between the fences. The value 34 is above 24, so it is flagged and appears as a separate point.

The smallest non-flagged observation is 2 units, and the largest is 16 units. Thus, the lower whisker ends at 2 and the upper whisker ends at 16. The upper whisker does not extend to the fence at 24, because 24 is not an observed value; nor does it extend to the sample maximum, 34, because that value is plotted separately. The display has one flagged point, not two.

Worked Example: Flagged Points on Both Sides

A hypothetical set of eight measurements, in units, is $0, 8, 9, 10, 12, 13, 14, 30$. Its boxplot reports $Q_1=8.5$ units and $Q_3=13.5$ units. Determine which observations are flagged and where the whiskers end.

The IQR is $13.5-8.5=5$ units. The lower fence is $8.5-1.5(5)=1$ unit, and the upper fence is $13.5+1.5(5)=21$ units. The observation 0 is below 1, and 30 is above 21. These two observations are plotted individually beyond the fences.

The other observations, 8, 9, 10, 12, 13, and 14, lie between the fences. The smallest and largest of those non-flagged observations are 8 and 14, so the whiskers extend to 8 and 14 units. Notice that the lower whisker endpoint is not the minimum observation (0), and the upper whisker endpoint is not the maximum observation (30). Both extremes are shown separately.

Worked Example: No Separately Plotted Points

Consider hypothetical measurements of $10, 18, 20, 24, 26, 28, 30,$ and $39$ minutes. A boxplot made using a consistent quartile convention reports $Q_1=19$ minutes and $Q_3=29$ minutes. Find the fences and describe the whiskers.

The IQR is $29-19=10$ minutes. The lower fence is $19-1.5(10)=4$ minutes, and the upper fence is $29+1.5(10)=44$ minutes. Every measurement, from 10 through 39 minutes, is between 4 and 44 minutes. There are no flagged points.

Because all observations are non-flagged, the smallest and largest observations are also the whisker endpoints: 10 minutes and 39 minutes. The whiskers happen to reach the full observed range in this example. That is a result of these data, not a general feature of modified boxplots.

Reading the Display Carefully

A useful interpretation names the different parts without treating them as interchangeable. The box shows the interval from $Q_1$ to $Q_3$; the median line marks the median. The whiskers reach the most extreme observations that remain within the fences. Separately plotted points show observations beyond a fence. The fence locations define the rule, but are not automatically visible marks on the boxplot.

Part of the display	What it represents	What it does not necessarily represent
Lower or upper fence	A calculated reference value based on a quartile and the IQR	An observed measurement or a whisker endpoint
Whisker endpoint	The most extreme observed value that is not beyond a fence	Necessarily the data minimum or maximum
Separately plotted point	An observed value below the lower fence or above the upper fence	The fence itself

The axis scale is essential for reading approximate values, but the drawing alone may not reveal exact measurements. If several observations are flagged, a plot may show each one separately; points with equal or very similar values can overlap visually. When exact counts or values matter, consult the data or a numerical summary rather than estimating from overlapping marks.

The modified boxplot rule gives a consistent way to distinguish the whisker range from observations beyond that range. It does not, by itself, explain why a measurement is far from the rest, and a plot should not be used to claim more than it displays. A contextual description can report that a value is beyond a fence without assigning a cause to it.

Common Mistakes

Calling a fence a whisker endpoint. The fence is calculated; the whisker reaches an observed value. If no observation occurs at the fence, the two locations differ.
Assuming whiskers always reach the minimum and maximum. In a modified boxplot, observations beyond the fences are plotted separately, so a whisker can stop short of an extreme value.
Calling every point near a whisker a flagged point. A point is flagged by its position relative to the numerical fences, not by how far it looks from the box in a drawing.
Reporting the rule as “at least 1.5 IQR away from the box.” The lower and upper checks use the fences $Q_1-1.5(IQR)$ and $Q_3+1.5(IQR)$, respectively. Compare the observation with those values.
Treating a value on a fence as beyond it. Under the rule stated here, only values strictly below the lower fence or strictly above the upper fence are flagged.

Key takeaway: In a modified boxplot, calculate the fences from $Q_1$, $Q_3$, and the IQR. Whiskers end at the most extreme observed values that are not beyond those fences; observations beyond a fence are displayed separately.

Check Your Understanding

Use the fence rule and distinguish calculated fences from observed whisker endpoints.

A boxplot has $Q_1=20$ and $Q_3=32$. Calculate its IQR and both fences.
If a data set has minimum 5, lower fence 7, and next-smallest observation 9, where does its lower whisker end? Which observation is plotted separately?
Explain why a whisker endpoint is not necessarily equal to the corresponding fence.
For $Q_1=4$, $Q_3=10$, and an observation equal to 19, is that observation beyond the upper fence? Show the calculation.
In a modified boxplot, what does a separate point represent, and what should you avoid claiming from that display alone?