What the lower fence is
The lower fence is the cutoff below which a data value is treated as a potential low outlier. It comes from John Tukey's boxplot method and is built from two quartiles of your dataset. Anything smaller than the lower fence sits outside the "whisker" of a boxplot and is worth a second look.
The formula
The lower fence uses the first quartile (Q1), the third quartile (Q3), and the interquartile range (IQR):
- IQR = Q3 − Q1 — the spread of the middle 50% of the data.
- Lower Fence = Q1 − 1.5 × IQR — the standard "inner" fence for ordinary outlier screening.
The companion upper fence is Q3 + 1.5 × IQR, and together the two fences define which points a boxplot draws as separate outlier dots. Some analysts also use a 3 × IQR "outer" fence to flag extreme ("far out") values; you can enter 3 in the multiplier field to get that instead.
Why quartiles instead of the mean and standard deviation
Quartiles are order statistics, so a single very large or very small value shifts them only slightly. That makes the fence robust: it can flag an outlier without that outlier having already distorted the boundary, which is exactly what happens if you screen using mean ± k·standard deviation. This robustness is the main reason the IQR rule is the default in exploratory data analysis.
A worked example
Take the dataset 4, 7, 8, 9, 10, 11, 12, 13, 14, 40. Splitting the ten sorted values into two halves of five, the lower half is 4, 7, 8, 9, 10 (median Q1 = 8) and the upper half is 11, 12, 13, 14, 40 (median Q3 = 13). So IQR = 13 − 8 = 5 and the lower fence = 8 − 1.5 × 5 = 0.5. No value is below 0.5, so there are no low outliers here — though 40 exceeds the upper fence of 13 + 1.5 × 5 = 20.5 and is flagged as a high outlier.
How this tool finds the quartiles
This calculator sorts your numbers, splits them at the median, and takes the median of each half (Tukey's "median-of-halves" method). For an odd count the overall median is excluded from both halves. This matches the quartiles most textbooks and boxplots use. Note that spreadsheet functions such as QUARTILE.INC use linear interpolation and can return slightly different Q1/Q3 values on the same data.