The Same Value Is an Outlier by One Rule and Not by Another

Masking: the suspect hides behind its own effect

The three-sigma rule fails here for a specific and predictable reason: the standard deviation it divides by was computed from data that includes the point under test. One extreme value inflates that denominator enough to make itself look ordinary, and two or three of them can hide each other completely — the phenomenon has a name, masking, and it is why a rule built on the mean and the standard deviation is a poor detector of the very thing it is meant to detect. Rules built on ranks do not have this weakness, because a single extreme value cannot inflate a quartile or a median.

Even the quartiles are not one thing

Statistical software offers half a dozen definitions of a quartile, and they disagree on small samples. This calculator interpolates linearly between order statistics, which puts the quartiles of a seven-value set at 11.5 and 12.5 where the older method of splitting at the median would put them at 11 and 13 — an interquartile range of 1 against 2, and fences twice as far apart. On the seven readings 10, 11, 12, 12, 12, 13 and 40 both conventions flag the 40, but on a borderline case they will not agree. Any pipeline that removes flagged points is therefore sensitive to a choice nobody wrote down, and the honest fix is to record which convention was used alongside the results.

Frequently asked questions

Should flagged points be deleted?

Almost never automatically. A flag is a request to look, not a verdict: the value may be a transcription error, a genuine rare event, a different population accidentally included, or the most interesting thing in the dataset. Deleting on the strength of a rule removes real discoveries along with the mistakes, and it does so invisibly, because the deleted points leave no trace in the summary that follows. The defensible workflow is to keep the raw data, record which points were excluded and why, and be able to produce the analysis both ways when somebody asks.

Why 1.5 and not some other multiple?

Because it was chosen to be a useful compromise and then stuck. With normally distributed data the 1.5 × IQR fences leave roughly seven values in a thousand outside, which is rare enough to be worth investigating and common enough that the rule finds something in a real dataset. Three times the interquartile range is the conventional threshold for a far more extreme point. Neither number comes from a theorem, and both are adjustable inputs in serious work — a rule that flags a tenth of a large dataset is telling you about the distribution's shape rather than about individual points.