In statistics, the upper and lower fences represent the cut-off values for upper and lower outliers in a dataset. They are calculated as:
- Lower fence = Q1 – (1.5*IQR)
- Upper fence = Q3 + (1.5*IQR)
where IQR stands for “interquartile range” which represents the difference between the 75th percentile (Q3) and 25th percentile (Q1) in a dataset.
An observation that lies above the upper fence or below the lower fence is often considered to be an outlier.
Example: Calculating the Upper and Lower Fence
Suppose we have the following dataset:
Dataset: 11, 13, 14, 14, 15, 16, 18, 22, 24, 27, 34, 36, 38, 41, 45
We can use the following steps to calculate the upper and lower fence of the dataset:
Step 1: Find Q1 and Q3.
Q1 represents the 25th percentile of the dataset and Q3 represents the 75th percentile. According to the Interquartile Range Calculator, Q1 and Q3 for this dataset are as follows:
- Q1: 14
- Q3: 36
Step 2: Find the IQR (Interquartile Range).
The interquartile range represents the difference between Q3 and Q1, which is calculated as:
- Interquartile Range: Q3 – Q1 = 36 – 14 = 22
Step 3: Find the Upper and Lower Fence
We can use the following formulas to calculate the upper and lower fences:
- Lower fence: Q1 – (1.5*IQR) = 14 – (1.5*22) = -19
- Upper fence: Q3 + (1.5*IQR) = 36 + (1.5*22) = 69
Since none of the observations in our dataset lie below the lower fence or above the upper fence, none of the observations would be considered outliers.
We can also create a boxplot to visualize our distribution of data values along with the upper and lower fences:
Bonus: Upper and Lower Fence Calculator
Instead of calculating the upper and lower fence of a dataset by hand, feel free to use the Upper and Lower Fence Calculator:
You can find more helpful statistics calculators on this page.