Lower Fence Calculator

Q: 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 ( Q 1 ); or 1.5 × IQR above the third quartile ( Q 3 ). In other words, if it's greater than Q 1 + 1.5 × IQR or lower than Q 1 - 1.5 × IQR , it is outside the lower and upper fences and, therefore, is an outlier.

Creators

Luis Hoyos

Luis is a mechanical engineer who works at Omni Calculator as a content creator and researcher specializing in math and physics calculators with a focus on thermodynamics and classical mechanics. With a strong foundation in mechanical engineering, Luis is passionate about creating tools that simplify complex concepts and improve everyday life. Outside of work, Luis likes weightlifting and applying his engineering knowledge to optimize the thermal conditions of his home, all while caring for four beloved cats. See full profile

Check our editorial policy

Reviewers

Komal Rafay

Komal has a master's in Bioinformatics, with 3+ years of experience as a project manager. She has a vast experience of 7+ years in teaching Math, Computer science, and General Sciences, not only professionally but also helping out local children. Komal is an enthusiastic reader with a thirst for knowledge. As a side hustle that satisfies her love for art, she is a body art and henna expert. In her free time, you will find her reading books, writing short stories, playing with her pets, or experimenting with food while listening to some bouncy music. See full profile

Check our editorial policy

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:

Obtain the first and third quartiles (Q₁ and Q₃, respectively). Visit our quartile calculator to learn how to do it.
Find the interquartile range (IQR), which is the difference between the third and first quartiles:

IQR = Q₃ - Q₁
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. Our tool offers you the option to modify the coefficient.

FAQs

How do I 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:

Obtain the first and third quartiles

First quartile: Q₁ = 2
Third quartile: Q₃ = 4.
Find the interquartile range (IQR):

IQR = Q₃ − Q₁ = 4 - 2 = 2
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.