Upper and Lower Fence Calculator (2024)

Now that we know what the fences are, we'd love to know exactly how to find the upper and lower fence. There are a few steps involved, so let's get started!

Before we can get to the upper and lower fence formulas, we need two basic but important values: the dataset's quartiles. We denote the first quartile with $Q_1$Q1​ and the third quartile with $Q_3$Q3​. We can then calculate the interquartile range (IQR) as $\text{IQR} = Q_3 - Q_1$IQR=Q3​−Q1​.

Now that we have the building blocks, how do we find the lower fence, and how do we find the upper fence?

• The formula for the upper fence is $\text{Upper fence} = Q_3 + 1.5\times\text{IQR}$Upperfence=Q3​+1.5×IQR.

• The formula for the lower fence is $\text{Lower fence} = Q_1 - 1.5\times\text{IQR}$Lowerfence=Q1​−1.5×IQR.

Some select sources replace the $1.5$1.5 in our formulas with other values, like $2$2 or sometimes even $3$3. The choice of the multiplier is problem-specific and depends on the dataset distribution. An outlier in a dataset of children's heights would still be close to the rest of the dataset, so we'd choose a smaller multiplier like $1.5$1.5. A millionaire's salary in a dataset of salaries would be much more outlying, so we'd pick a bigger multiplier, like $3$3 or more. If you'd like to use a different multiplier for your dataset, you can enter the Advanced mode and change it from the default of $1.5$1.5.

So, there you have it! We've found the formula for the upper fence and the lower fence, and now we can find outliers in any dataset we want.

Let's use our upper and lower fence formula in a practical example. Suppose we have a dataset of each year's rainfall volumes for January from 2010 to 2021 in New York, and it looks like this:

$1.33$1.33, $1.96$1.96, $3.12$3.12, $2.20$2.20, $1.58$1.58, $2.04$2.04, $1.80$1.80, $6.32$6.32, $1.90$1.90, $3.84$3.84, $2.93$2.93, $2.34$2.34.

Can you already guess which value is going to be an outlier?
Let's back up that guess with some math.

1. We first have to sort the dataset in increasing order:

   $1.33$1.33, $1.58$1.58, $1.80$1.80, $1.90$1.90, $1.96$1.96, $2.04$2.04, $2.20$2.20, $2.34$2.34, $2.93$2.93, $3.12$3.12, $3.84$3.84, $6.32$6.32.

2. We can determine that the dataset's first and third quartiles are $Q_1= 1.85$Q1​=1.85 and $Q_3 = 3.025$Q3​=3.025.

3. Using $Q_1$Q1​ and $Q_3$Q3​, we can calculate the interquartile range with

   $\text{IQR} = Q_3 - Q_1 = 3.025 - 1.85 = 1.175$IQR=Q3​−Q1​=3.025−1.85=1.175

4. Finally, we can use the upper fence formula to get

   $\text{Upper fence} = Q_3 + 1.5 \times \text{IQR} = 3.025 + 1.5 \times 1.175 = 4.7875$Upperfence=Q3​+1.5×IQR=3.025+1.5×1.175=4.7875

5. …and we can use the lower fence formula to get

   $\text{Lower fence} = Q_1 - 1.5 \times \text{IQR} = 1.85 - 1.5 \times 1.175 = 0.0875$Lowerfence=Q1​−1.5×IQR=1.85−1.5×1.175=0.0875

We can look at our data with these upper and lower fences and see that 2017's rainfall of $6.32$6.32 is an outlier. Was your initial guess correct?

Our upper and lower fence calculator takes all these steps for you and gives you the fences in the blink of an eye so that you can get to find outliers in your dataset.

1. Enter your dataset's individual values in the rows. You can input up to 50 values.

2. Optionally, change the multiplier used in the fence formulas in the Advanced mode.

3. The calculator will determine the fences and display them at the bottom of the list of values, along with your dataset's outliers and the steps taken to calculate them. Happy outlier hunting!