How to Use the Lower and Upper Fence Calculator
This calculator helps you find the lower fence and upper fence of your data and identifies the possible outliers. To use the calculator:
- Enter your data values in the data input field. You can separate values using commas, spaces, tabs, or line breaks. The calculator also allows you to copy-paste values from Excel, text documents, and Google Sheets.
- Click Calculate.
The calculator will instantly return the lower fence, upper fence, and tell you the possible outliers, if any. It also provides a clear, step-by-step solution to help you learn how to find these values manually.
What is Lower Fence?
In statistics, the lower fence is a cutoff value used to identify unusually small data points in a dataset. Values below the lower fence are considered to be potential outliers and are flagged or excluded from the data when finding the summary statistics.
Lower Fence Formula
The lower fence formula is: Lower fence = Q1 − 1.5 × IQR
Where:
- Q1 is the lower quartile
- IQR is the interquartile range
How to Calculate Lower Fence
To calculate the lower fence:
- Find the lower quartile (Q1) and upper quartile (Q3).
- Find the interquartile range (IQR)
- Apply the lower fence formula
Example 1. Find the lower fence for the dataset: 4, 7, 8, 9, 10, 12, 15, 18, 40
Solution
To find the lower fence of the above dataset, follow these steps:
Step 1. Find the lower quartile (Q1) and upper quartile (Q3).
Using the quartile calculator:
- Lower quartile, Q1 = 7.5
- Upper Quartile, Q3 = 16.5
Step 2. Find the interquartile range
By definition, the interquartile range formula is IQR = Q3 – Q1
= 16.5-7.5
= 9
Alternatively, you can use the interquartile range calculator to get instant results.
Step 3. Apply the lower fence formula
By definition, Lower fence = Q1 − 1.5 × IQR
Substituting the values, we get:
Lower fence = 7.5 − 1.5 × 9
= 7.5 − 13.5
= −6
Therefore, the lower fence for the dataset is -6. Since there is no value in the dataset that is below -6, there is no possible lower outlier.
Want to verify the results using this lower and upper fence calculator? Follow these steps:
- Copy and paste the values into the data input field
- Click calculate
The calculator will instantly return the lower fence as -6.
What is Upper Fence?
The upper fence is a cutoff value used to identify unusually large data points in a dataset. Values above the upper fence are considered potential outliers and are often flagged or excluded from the data before analysis.
Upper Fence Formula
The upper fence formula is: Upper fence = Q3 + 1.5 × IQR
Where:
- Q3 is the third quartile.
- IQR is the interquartile range.
Recall. IQR = Q3 − Q1.
Therefore, to find the lower and upper fence for any dataset, you’ll need to have the lower quartile (Q1) and the upper quartile (Q3).
How to Calculate Upper Fence
To find the upper fence manually:
- Find the lower quartile (Q1) and upper quartile (Q3)
- Find the interquartile range (IQR)
- Apply the upper fence formula
Example 2. Find the upper fence of the following dataset: 55, 60, 62, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90, 110
Solution
To find the upper fence of the dataset by hand, follow these steps:
Step 1. Find the lower quartile and upper quartile
Using the quartile calculator:
- Lower quartile, Q1 = 65
- Upper Quartile, Q3 = 85
Step 2. Find the interquartile range
Using the interquartile range calculator, IQR = 20. Alternatively, with Q3 and Q1, you can find the interquartile range as follows:
IQR = Q3 – Q1
= 85- 65
= 20
Step 3. Apply the upper fence formula
By definition, Upper fence = Q3 + 1.5 × IQR
Substituting the values, we get:
Upper fence = 85 + 1.5 × 20
= 85 + 30
= 115
Therefore, the upper fence of the dataset is 115. Since no value in the dataset is above 115, there is no possible upper outlier.
You can also verify this result using this lower and upper fence calculator. Just follow these steps:
- Copy and paste the values into the data input field
- Click calculate
The calculator will return the upper fence as 115.
What Counts as an Outlier?
Using the lower and upper fence rule:
- A value is a possible lower outlier if it is less than the lower fence.
- A value is a possible upper outlier if it is greater than the upper fence.
- A value that is exactly equal to the lower fence or upper fence is not an outlier.
When Should You Be Careful?
A lower or upper fence can flag possible outliers, but it does not automatically prove that a value is wrong. In other words, a value may be unusual and still be valid. Therefore, you should always consider the context of the data before removing or changing any value.
For example, a very high income, very large order size, or unusually long completion time may be a real observation rather than a data error. This implies that you should use fences as a screening rule and then determine what value to exclude based on context.
Frequently Asked Questions
It calculates the lower fence and upper fence of a dataset and identifies possible outliers using the 1.5 × IQR rule. It also provides a clear, step-by-step solution to help you learn how to compute these fences manually.
The lower fence is the cutoff value used to identify possible unusually small values. It is calculated as Q1 − 1.5 × IQR.
The upper fence is the cutoff value used to identify possible unusually large values. It is calculated as Q3 + 1.5 × IQR.
Values below the lower fence and above the upper fence are possible outliers.
Not automatically. Values outside the fences are possible outliers. However, you should always review the context before removing or changing them.
