Upper and Lower Fence Calculator — Box Plot Outliers
Enter Q1 and Q3 from your data set to compute both the lower fence (Q1 − k × IQR) and the upper fence (Q3 + k × IQR) — the outlier thresholds used in Tukey's rule and box-and-whisker plots.
Upper Fence = Q3 + k × IQR
- 1
IQR
Q3 − Q1 = 60 − 20 = 40 - 2
k × IQR
1.5 × 40 = 60 - 3
Lower Fence
Q1 − k × IQR = 20 − 60 = -40 - 4
Upper Fence
Q3 + k × IQR = 60 + 60 = 120Values outside the lower and upper fences are potential outliers.
How does this calculator work?
Lower Fence = Q1 − k × IQR; Upper Fence = Q3 + k × IQR; IQR = Q3 − Q1. Use k = 1.5 for mild outliers (box-plot standard) or k = 3 for extreme outliers. Values outside both fences are potential outliers; investigate before removing them.
Formula
How this is calculated
Tukey's fence method defines outliers using the interquartile range (IQR = Q3 − Q1), which measures the spread of the central 50% of data. The lower fence is placed at Q1 − k × IQR and the upper fence at Q3 + k × IQR. Observations below the lower fence or above the upper fence are potential outliers.
The conventional multiplier k = 1.5 gives the "mild" outlier threshold, which is the rule used in standard box-and-whisker plots: the whisker ends extend to the most extreme observed values that still fall within the fences, and any points beyond are drawn individually as dots. A second check at k = 3 identifies "extreme" outliers — values so far from the data centre that they are very unlikely to arise from the same population.
This calculator computes both fences simultaneously. Neither fence assumes normality; the method works on any ordinal or ratio data. The fences are descriptive tools — they do not prove a point is an error. Investigate each flagged observation: it may be a valid extreme, a recording error, or a genuine outlier that belongs in a separate sub-population.
Frequently asked questions
Sort your data, find the median, then split the sorted list at the median. Q1 is the median of the lower half and Q3 is the median of the upper half. The exact rule for even vs odd counts differs between methods (inclusive vs exclusive). Use the 5-number-summary calculator to compute Q1, Q3, and the fences directly from raw data.
Mild fences use k = 1.5 and are the box-plot standard. Values between the mild and extreme fences are "mild outliers"; values beyond the extreme fence (k = 3) are "extreme outliers". Under normality, the mild fence excludes roughly 0.7% per tail and the extreme fence fewer than 0.01%.
IQR fences make no distributional assumption — they work on any data shape. Z-score outlier detection assumes normality and is sensitive to the very outliers it tries to detect (since extreme values inflate the mean and standard deviation). Tukey fences are more robust for skewed or heavy-tailed data.
Also known as
TG we-Calculate Editorial Team. (2026). Upper and Lower Fence Calculator — Box Plot Outliers [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/upper-lower-fence-calculator
TG we-Calculate Editorial Team. "Upper and Lower Fence Calculator — Box Plot Outliers." TG we-Calculate. 2026. https://we-calculate.com/calculator/upper-lower-fence-calculator.
TG we-Calculate Editorial Team, "Upper and Lower Fence Calculator — Box Plot Outliers," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/upper-lower-fence-calculator
@misc{wecalculate_upper_lower_fence_calculator, title = {Upper and Lower Fence Calculator — Box Plot Outliers}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/upper-lower-fence-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?