Outlier Lower Fence

John Tukey's fences turn outlier hunting into arithmetic. Take the interquartile range, multiply by 1.5, and subtract that from Q₁: anything below the result is a candidate outlier and gets plotted as an individual dot beyond the box plot's whisker. With Q₁ = 48 and IQR = 24, the lower fence sits at 48 − 36 = 12, so a score of 9 is flagged while a score of 15 is not. The 1.5 is not derived from anything deep: Tukey chose it because on normal data it flags roughly 0.7% of observations, often enough to be useful and rarely enough not to cry wolf.

The essential trap is treating a flagged point as bad data. A fence says "this value is unusual relative to the middle half", not "this value is wrong". Skewed distributions such as incomes or hospital lengths of stay throw points past the fences constantly and legitimately, which is why some analysts use 3.0 × IQR for a "far out" tier before considering exclusion. Delete points only when you can identify a cause (a transcription error, a broken instrument) never merely because a fence flagged them.