*9*

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.