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Lower Fence Calculator

Created by Luis Hoyos
Reviewed by Komal Rafay
Last updated: May 20, 2024

With this tool, you can calculate lower fences in statistics, which are crucial for determining outliers (what is an outlier?)

To use this calculator, input your data set, and, using the 1.5 IQR rule, the tool will provide the following:

  • The lower fence;
  • The upper fence;
  • The outliers, and
  • The upper and lower fence calculation procedure.

In the following article, we briefly discuss the role of fences in statistics and how to calculate them. For a more detailed explanation, visit our upper and lower fence calculator.

The upper and lower fences in statistics

In a dataset, the upper and lower fences are the numerical boundaries outside which any data point can be considered an outlier. To obtain these boundaries, we use the quartiles of the dataset.

Simple, right? Now, let's look at the upper and lower fence formulas.

How to calculate the lower fence?

To calculate the lower fence of a dataset:

  1. Obtain the first and third quartiles (Q and Q, respectively). Visit our quartile calculator to learn how to do it.

  2. Find the interquartile range (IQR), which is the difference between the third and first quartiles:

    IQR = Q₃ - Q

  3. Finally, use the lower fence formula:

    Lower fence = Q₁ ​− 1.5 × IQR

Finding the upper fence is almost the same; the only difference is a slight change in the formula used. Visit our upper fence calculator to learn how to do it.

💡 Using the 1.5 coefficient before IQR is the most common way to establish the fences. Even so, we can use other values (such as 2 or even 3), depending on how dispersed we expect data to be. You can modify the coefficient by enabling the calculator's advanced mode.


How to calculate the lower fence and upper fence of 1, 2, 3, 4, and 5?

The lower and upper fences of the {1, 2, 3, 4, 5} data set are -1 and 7, respectively.

To get to this answer:

  1. Obtain the first and third quartiles

    First quartile: Q1 = 2
    Third quartile: Q3 = 4.

  2. Find the interquartile range (IQR):

    IQR = Q3Q1 = 4 - 2 = 2

  3. Finally, calculate the upper and lower fences with the following formulas:

    Lower fence = Q1 ​− 1.5 × IQR = 2 - 1.5 × 2 = -1
    Upper fence = Q3 + 1.5 × IQR = 4 + 1.5 × 2 = 7

What is the 1.5 IQR rule?

The 1.5 IQR rule says a data point is an outlier if it is:

  • 1.5 × IQR below the first quartile (Q1); or
  • 1.5 × IQR above the third quartile (Q3).

In other words, if it's greater than Q1 + 1.5 × IQR or lower than Q1 - 1.5 × IQR, it is outside the lower and upper fences and, therefore, is an outlier.

Luis Hoyos
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