How this instrument works
A box plot's whiskers are bounded by two numbers, not one: a lower fence below which a value counts as an unusually low outlier, and an upper fence above which a value counts as an unusually high outlier. Both are built from the same interquartile range, IQR — the spread of the middle 50% of the data between the first quartile Q1 and third quartile Q3 — stepped out by 1.5 × IQR in each direction: lower fence = Q1 − 1.5 × IQR, upper fence = Q3 + 1.5 × IQR.
Computing them separately means finding Q1 and Q3 twice over, or copying the same IQR into two different calculations. This instrument does it once: sort the data, find Q1 and Q3, compute IQR, then return both fences from that single pass. It is built for exactly the moment you are constructing a full box plot by hand or screening a data set in both directions — checking whether anything sits either suspiciously high or suspiciously low.
The 1.5 multiplier behind both fences is a long-standing convention from exploratory data analysis, not a physical law — generous enough that ordinary extreme-but-real values in a roughly normal-shaped data set rarely get flagged, while genuinely separated values on either side usually do. Points beyond either fence are exactly what a standard box plot draws as individual dots past the whiskers.
- Paste or type your numbers into the Data set field — commas, spaces, or new lines all work as separators.
- The instrument sorts your data and finds Q1 and Q3 by linear interpolation, then computes the interquartile range between them once.
- Read Lower fence and Upper fence together — anything below the lower figure or above the upper figure counts as an outlier by the standard 1.5 × IQR rule.
- Use both numbers as the two whisker boundaries when hand-drawing a complete box plot, or as a two-sided screening range for new data.
- Needs at least 2 values to compute quartiles — for a specific yes/no check on one number, use this site's outlier calculator instead.
Worked example — the eight quiz scores
Paste the data set {2, 4, 4, 4, 5, 5, 7, 9} — n = 8 — into Data set. By linear interpolation, Q1 = 4 and Q3 = 5.5, so IQR = 5.5 − 4 = 1.5. Lower fence reads 1.75, computed as 4 − 1.5 × 1.5, and Upper fence reads 7.75, computed as 5.5 + 1.5 × 1.5.
Checking both bounds against the data at once: the minimum, 2, sits above the lower fence of 1.75, so nothing is flagged on the low side. The maximum, 9, sits above the upper fence of 7.75, so that value is flagged as a high-side outlier. One pass over the data settled both questions together, which is exactly the point of getting both fences from a single calculation.
Questions
Why get both fences at once instead of computing them separately?
Both numbers come from the same Q1, Q3, and IQR, computed once from your data — asking for them together avoids finding the quartiles twice or risking the two figures drifting out of sync if entered into two separate tools. It also matches how box plots actually work: you need both whiskers' boundaries together to draw one complete chart.
What if I only need one of the two fences?
This site also offers a standalone upper fence calculator and a standalone lower fence calculator, if you only care about one side — screening only for unusually high values, or only for unusually low ones. This combined tool exists for the more common case of building a complete box plot or a two-sided screen.
Why 1.5 times the IQR specifically, and not some other number?
It is a long-standing convention from exploratory data analysis, chosen to be generous enough that ordinary values at either end of a roughly normal-shaped distribution rarely get flagged by chance, while genuinely separated values usually do. Some fields use a stricter multiplier like 3 for 'extreme' outliers on both sides rather than 1.5's 'possible' outliers, but 1.5 is the standard default behind most box plots.
Can the lower fence come out negative?
Yes, whenever Q1 is smaller than 1.5 × IQR — that is expected, not an error. If your data physically cannot go below zero (counts, weights, durations), a negative lower fence just means there is effectively no low-side outlier boundary in practice for that data.
How is this different from the outlier checker on this site?
This tool reports both boundary values themselves, which is what you need to build a box plot or screen a whole data set by eye. The outlier checker instead takes one specific value you supply and returns a direct yes-or-no verdict for that value against both fences — use this tool to get the bounds, and the outlier checker when you already have a particular number you want tested.
Does a value beyond either fence have to be removed from my data?
No — the fences flag a value as statistically unusual relative to the rest of the data, not automatically wrong, erroneous, or safe to discard. Whether to investigate, correct, keep, or exclude a flagged value depends on subject-matter context the formula itself cannot supply, such as whether it reflects a measurement error or a genuine, valid extreme case.