Upper Fence Calculator

An upper fence is a single number above which a value is worth investigating. It is the top of the region a box plot draws before it stops and starts marking individual points, which is why the whisker ends where it does rather than at the maximum.

It is built from the upper quartile and the spread of the middle half of the data, never from the maximum itself. That detail is what separates a fence from a simple range check, and it is why one absurd reading cannot push the boundary out to meet itself.

Sort the values and find the first and third quartiles. Their difference is the interquartile range, which describes how spread out the middle half of the data is.

Multiply the interquartile range by the multiplier, 1.5 by convention, and add it to the third quartile to get the fence.

Count the values above the fence and note the largest one, which is usually several times further past the fence than the second largest.

Compare the count with what a normal sample of the same size would put there, about 0.35% of values at 1.5 IQR.

Recompute the mean with the flagged values removed, and again with them capped at the fence. The two adjusted means bracket what the flagged readings were doing to the average.

A fence of 34 with one of thirteen values flagged means the reading of 71 is 37 units past the fence, over twice the entire interquartile range.

The mean of 29.15 falls to 25.67 when the 71 is trimmed out, which is a shift of 3.5 units caused by a single value in a batch otherwise spanning 22 to 30.

The winsorized mean of 26.31 sits between the plain and trimmed means, because capping keeps the outlier present at a reduced value.

A flagged count of one in a list of thirteen is what chance produces, so on its own it is weak evidence of anything.

If no values are flagged, the fence is still worth noting: it records what the top of this batch looked like, which is what makes it useful for watching a later batch.

• The multiplier. At 1.5 a normal sample flags 0.35% of values, at 3 about 0.27%, so widening the threshold changes the count slowly.
• The quartile definition. Interpolated and hinge quartiles differ unless the list length is a multiple of four, and the difference grows with the multiplier.
• List length. The upper quartile of a short list is an average of very few values, so a fence from eight readings moves visibly when one changes.
• The basis. The interquartile range and the median deviation ignore the tail, while the standard deviation is built from it, so the same data gives different fences.
• Any extreme value. It barely moves a Tukey or MAD fence, but it inflates a standard deviation enough to widen that fence around itself.

• Screening a freshly exported data column before running any statistics on it.
• Checking whether a new batch of measurements is consistent with the earlier ones that defined the process.
• Before reporting an average: a plain mean alongside the trimmed and winsorized means shows how much of the result rests on one reading.
• Setting up a box plot or a run chart that needs to know where the whiskers stop.
• Comparing two data sets for contamination, where a fence from the clean one flags what is odd in the other.

• The expected share assumes normality, and real measurement data is often heavier-tailed, so actual counts run a little higher than the prediction.
• The 1.5 multiplier is a convention inherited from hand-drawn box plots, not an optimum, and it carries no sharp justification.
• Fences flag a difference without explaining it. Cause and intent have to come from outside the numbers.
• A trimmed mean discards real information, and a winsorized mean keeps an artificial value in the sample. Both are adjustments, not corrections.
• On a skewed distribution the upper fence will flag a legitimate long tail, so a flagged value is not evidence of an error.