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Adjacent values
Definition
Adjacent values are the most extreme observed data values that remain within the inner fences of a conventional Tukey-style box plot. The lower adjacent value is the smallest observation at or above the lower fence; the upper adjacent value is the largest observation at or below the upper fence.
What it means
Adjacent values determine where the whiskers end in this box-plot convention. They are actual observations, whereas fences are calculated thresholds that need not coincide with any observation. Points beyond the fences are displayed separately as potentially unusual values.
The word "adjacent" does not mean the observations immediately beside the median or successive values in the dataset. It refers to the observations nearest the fences from inside. Other box-plot conventions use different whisker rules, so the chart's definition matters whenever its endpoints are interpreted.
How it works
First calculate the lower and upper quartiles, or the hinges used by the chosen implementation. Their difference is the interquartile range, abbreviated IQR. With the usual multiplier of 1.5, the lower inner fence is Q1 - 1.5 times IQR and the upper inner fence is Q3 + 1.5 times IQR.
Find the smallest observation that is not below the lower fence and the largest observation that is not above the upper fence. These are the adjacent values. Draw whiskers from the box to those observations and show observations beyond the fences individually.
Quartile and hinge algorithms can differ, particularly for small samples. Software can also allow a different multiplier or use minimum-to-maximum whiskers. Record these choices when reproducing a chart. A fence rule identifies observations for attention; it does not determine whether their underlying measurements are incorrect or whether they should be excluded.
Three practical examples
1. A support analyst uses adjacent values to mark the whisker endpoints of a response-time distribution while displaying unusually long waits separately.
2. A production team compares box plots across machines, checking that all plots use the same quartile algorithm and fence multiplier before interpreting their spreads.
3. A data-quality reviewer examines observations outside the whiskers to distinguish recording errors from valid but uncommon cases that still belong in the dataset.
Think of it this way
Think of putting two boundary markers along a line of numbered stepping stones. The adjacent values are the outermost stones still between the markers. The markers can fall in gaps between stones, so their positions need not equal the positions of the stones used as endpoints.
Worked example
Consider the original ordered dataset 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30. Use the median-of-halves quartile convention: the lower half is 1 through 6 and the upper half is 7, 8, 9, 10, 11, 30.
Q1 is (3 + 4)/2 = 3.5 and Q3 is (9 + 10)/2 = 9.5. The IQR is 6, making the inner fences 3.5 - 9 = -5.5 and 9.5 + 9 = 18.5. The lower adjacent value is 1 and the upper adjacent value is 11. The value 30 lies beyond the upper fence and is plotted separately.
Notice that neither whisker ends at a fence: there is no observation at -5.5 or 18.5. The sample minimum is an adjacent value here, but the sample maximum is not. The median is (6 + 7)/2 = 6.5, which belongs inside the box and plays a different role from either whisker endpoint.
Illustrative case study
Fictional case study: Queuebridge displays support-resolution times using box plots. A manager reads the top whisker as the maximum time and concludes that no customer waited longer. The analyst explains that the whisker ends at the upper adjacent value and that several longer waits appear as separate points.
The team adds a short chart note describing the whisker convention and makes the underlying observations available for review. It investigates the long waits without automatically deleting them, because some reflect genuine complex requests.
This fictional case illustrates how chart interpretation changes when adjacent values are distinguished from extrema. It does not establish an acceptable service threshold or show that every point beyond a fence is a process failure.
Common misconceptions and mistakes
A frequent mistake is drawing whiskers directly to the calculated fences even when the convention requires observed adjacent values. Another is assuming that whiskers always show the minimum and maximum. Some charts do use that convention, but a fenced box plot generally does not.
An outside point is not automatically an error, and an inside point is not automatically accurate. Skewed distributions can generate many outside points without data corruption. Neither the fences nor adjacent values are confidence limits, specification limits or universal boundaries separating acceptable from unacceptable performance.
Business use
Adjacent values help teams communicate distribution shape compactly while retaining visibility of extreme observations. They can support comparisons of service times, transaction sizes or experimental measurements, provided the plots use consistent definitions and comparable units.
Include sample sizes and investigate major differences in data coverage before comparing groups. A box plot alone can hide clusters and changes over time, so complement it with appropriate detailed views when those features matter. Keep operational targets separate from statistical fences. A customer wait beyond a service commitment may occur inside the whiskers, while a valid high-value transaction may appear outside them.
People also ask
Can an adjacent value equal a fence?
Yes. An observation exactly on the fence is included under the usual inclusive rule.
Are adjacent values always the sample minimum and maximum?
No. They equal those extrema only when the extrema lie within the corresponding fences.
Why do two programs show different whisker endpoints?
They may calculate quartiles or hinges differently, use different fence multipliers, or implement different box-plot conventions. Compare the settings and data handling before treating the difference as an error.
Check your understanding
The upper fence is 24 and the largest observations are 19, 23 and 29. What is the upper adjacent value?
Answer: 23. It is the largest observation that does not exceed 24. The fence remains 24, and 29 is shown separately under the stated convention.
Related terms
- Box plot
- Whiskers
- Quartile
- Inner fences
- whisker
- Tukey hinge
- interquartile range
- inner fence
- outlier and five-number summary
Learn next
build a small box plot manually, keeping a separate list of quartiles, fences, adjacent values and outside observations. Then reproduce it in software and explain any difference by checking the quartile and whisker conventions.
More in Statistics and probability
- Additive effect - An additive effect is a modeled change in the expected response from changing one input factor that does not depend on the settings of other input factors.
- Additive law of probability - The additive law of probability states that the probability of one of several mutually exclusive events occurring is the sum of their individual probabilities.
- Additive model - An additive model represents an outcome as a sum of component contributions.
- Adjusted correlation ( r adj ) - Adjusted correlation, in the source glossary's regression sense, is a correlation measure corrected for the number of predictors relative to the sample size.
- Adjusted means - Adjusted means are model-based group averages calculated after accounting for specified covariates or other predictors.
- Adjusting or controlling for a variable - Adjusting or controlling for a variable means accounting for that variable when assessing the relationship between other variables.
Sources
Primary sources: Howell, Statistical Methods for Psychology glossary, Adjacent values: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html
NIST/SEMATECH, What are outliers in the data?: https://www.itl.nist.gov/div898/handbook/prc/section1/prc16.htm
NIST Dataplot, Box Plot: https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/boxplot.htm
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