Upper Lower Fence

Identify outliers in your dataset using the Interquartile Range method. Calculate Upper and Lower Fences instantly.

Data Set

Result

What are Upper and Lower Fences?

In statistics, Upper and Lower Fences are boundaries used to identify outliers in a dataset. These fences are calculated based on the Interquartile Range (IQR), which describes the middle 50% of your data.

Any data point that falls outside these fences is considered a potential outlier:

  • Lower Outlier: Any value less than the Lower Fence.
  • Upper Outlier: Any value greater than the Upper Fence.

How to Calculate Fences

1. Sort the Data

Arrange your dataset in ascending order from smallest to largest.

2. Find Quartiles (Q1 and Q3)

  • Q1 (First Quartile): The median of the lower half of the data (25th percentile).
  • Q3 (Third Quartile): The median of the upper half of the data (75th percentile).

3. Calculate IQR

The Interquartile Range is the distance between the third and first quartiles: IQR=Q3−Q1IQR = Q3 - Q1

4. Determine Fences

The standard formulas (Tukey's method) use a multiplier of 1.5:

  • Lower Fence=Q1−(1.5×IQR)\text{Lower Fence} = Q1 - (1.5 \times IQR)
  • Upper Fence=Q3+(1.5×IQR)\text{Upper Fence} = Q3 + (1.5 \times IQR)

For extreme outliers, some statisticians use a multiplier of 3.0 instead of 1.5.

Example Walkthrough

Dataset: {4, 8, 15, 16, 23, 42, 108}

  1. Sort: The data is already sorted.
  2. Find Median: The middle value is 16.
  3. Find Q1: Median of {4, 8, 15} is 8.
  4. Find Q3: Median of {23, 42, 108} is 42.
  5. Calculate IQR: 42−8=3442 - 8 = 34.
  6. Calculate Fences:
    • Lower: 8−(1.5×34)=8−51=−438 - (1.5 \times 34) = 8 - 51 = \mathbf{-43}
    • Upper: 42+(1.5×34)=42+51=9342 + (1.5 \times 34) = 42 + 51 = \mathbf{93}

Analysis: The value 108 is greater than the Upper Fence (93). Therefore, 108 is an outlier. All other values fall within the range [-43, 93].

Why Outlier Detection Matters

Outliers can significantly skew statistical results, especially the mean and standard deviation. Identifying them helps in:

  • Data Cleaning: Removing errors or bad data points.
  • Analysis: Deciding whether to use robust statistics (like median) instead of sensitive ones (like mean).
  • Insight: Sometimes the outlier is the most interesting part of the story (e.g., a credit card fraud transaction).

Frequently Asked Questions

Q: Why do we use 1.5 as the multiplier? A: John Tukey, who invented the box plot, chose 1.5 as a practical "rule of thumb." In a perfect normal distribution, this threshold captures about 99.3% of the data, marking only 0.7% as outliers, which is a reasonable standard for "unusual" observations.

Q: Can fences be negative? A: Yes, if the data range allows for it (as seen in the example calculation). If your data is strictly positive (like height or weight), a negative lower fence simply means there are no low outliers.