Upper Fence Calculator

To calculate the upper and lower fences, you need three parameters:

• The first quartile (Q₁);
• The third quartile (Q₃); and
• The interquartile range (IQR)

🙋 The interquartile range is the difference between the third and the first quartile:

IQR = Q₃ - Q₁

🙋 In our quartile calculator, we explain how to find Q₁ and Q₃.

Once you know those parameters, the problem of how to find the upper fence and lower fence boils down to using two formulas:

Upper fence = Q₃ + 1.5 × IQR

The lower fence calculation is similar:

Lower fence = Q₂ ​− 1.5 × IQR

🙋 Using the 1.5 coefficient before IQR is the most common practice, but we can use other values (e.g., 2 or 3), depending on how dispersed we expect data to be. You can modify it in the calculator's Results section of our tool by unticking the Use the default multiplier 1.5 checkbox.

Now that we know how to find the upper fence and the lower fence, let's look at the tricky part: obtaining Q₁ and Q₃.

The tricky part about calculating the upper fence is finding the quartiles (Q₁ and Q₃). The procedure is as follows:

1. Order your numbers in ascending order (our order from least to greatest calculator explains some algorithms for large data sets).
2. Divide the ordered data set into two halves;
3. The first quartile is the median of the first half, and the third quartile is that of the second (our median calculator explains how to obtain it).

If the number of elements is odd, the middle number marks the end of the first and the beginning of the second half (although there are various ways to deal with uneven amounts. Wikipedia lists four methods).

Example

Let's use the following ordered set:

0, 7, 11, 12, 13, 14, 15, 16, 22, 30, 39, 49, 51, 56, 62, 74, 92, 97 98, 99

First, divide the data set into two halves:

• First half: 0, 7, 11, 12, 13, 14, 15, 16, 22, 30.
• Second half: 39, 49, 51, 56, 62, 74, 92, 97, 98, 99.

The medians of the first and second halves are 13.5 and 68, respectively. Therefore:

• Q₁ = 13.5; and
• Q₃ = 68.