High outliers are common in data that cannot go below zero but can grow without limit: salaries, house prices, response times, insurance claims and city populations. The upper fence gives a principled cut-off for “unusually large” in such data, based on the third quartile and the interquartile range rather than the mean, which the large values would inflate.
To compute it, start at Q3 and add 1.5 times the IQR. A value above the result lies further above the box than one and a half box-widths and is drawn as a separate point on a box plot. Increasing the multiplier to 3 gives the outer fence, which flags only extreme values.
For strongly right-skewed data the 1.5 rule may flag many legitimate large values; in that case a larger multiplier or a log transformation is often more sensible. The calculator lets you change k, shows the high outliers on their own and gives the data without them for a quick before-and-after comparison.
How to use the Upper Fence Calculator
- Data set
- Fence multiplier (k) — 1.5 is the standard (Tukey) value; use 3 to flag only extreme outliers.
- Quartile method
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How it works
Sort the data, compute Q1 and Q3 with the chosen method and the IQR. Add k × IQR to Q3 for the upper fence; values strictly above it are high outliers.
Worked example
Annual salaries in thousands: 32, 35, 36, 38, 40, 41, 43, 45, 48, 52, 120. The median is 41, Q1 = 36 (median of 32, 35, 36, 38, 40) and Q3 = 48 (median of 43, 45, 48, 52, 120). IQR = 12 and upper fence = 48 + 1.5 × 12 = 66, so the 120k salary is a high outlier. The lower fence is 36 − 18 = 18, below every value.
Interpreting high outliers
A high outlier is not necessarily an error. In salary data it may be a manager; in response times it may be a timeout. Report the median and IQR, which are robust, and mention the outlier explicitly rather than silently removing it.
Formula
Upper fence = Q3 + k × IQR = Q3 + k × (Q3 − Q1), with k = 1.5 by default
Worked example
Inputs — Data set: 32, 35, 36, 38, 40, 41, 43, 45, 48, 52, 120; Fence multiplier (k): 1.5; Quartile method: Median of halves, median excluded (TI-84, most textbooks).
Results — Upper fence (Q3 + k × IQR) = 66; High outliers (above the upper fence) = 120; Third quartile (Q3) = 48; Interquartile range (IQR = Q3 − Q1) = 12; First quartile (Q1) = 36; Lower fence (Q1 − k × IQR) = 18.
Key terms
- Upper fence
- Q3 + k × IQR.
- High outlier
- A value above the upper fence.
- Right-skewed
- A distribution with a long tail of large values.
Frequently asked questions
What is the upper fence?
Q3 + 1.5 × IQR. Values above it are high outliers.
How do I find the upper fence of 32, 35, 36, 38, 40, 41, 43, 45, 48, 52, 120?
Q1 = 36 and Q3 = 48, so IQR = 12 and the upper fence is 48 + 18 = 66. The value 120 is a high outlier.
What is the outer upper fence?
Q3 + 3 × IQR – here 48 + 36 = 84. 120 is beyond it, so it is an extreme outlier.
Do high outliers affect the mean?
Strongly. The example mean is 48.2 with 120 and 41 without it; the median moves only from 41 to 40.5.
Should salaries use the 1.5 rule?
Salary data are right-skewed, so 1.5 × IQR flags many genuine high earners. Consider k = 3 or analyse log-salaries.
Is a value equal to the upper fence an outlier?
No, only values strictly above it.