Excel ERF Function

last modified April 4, 2025

The ERF function calculates the error function integrated between two limits. It’s used in engineering and statistics for probability calculations. This tutorial provides a comprehensive guide to using the ERF function with detailed examples. You’ll learn basic syntax, practical applications, and advanced techniques.

ERF Function Basics

The ERF function returns the error function integrated between lower and upper limits. It’s related to the normal distribution in statistics. The syntax has optional arguments for flexibility.

Component
Description


Function Name
ERF


Syntax
=ERF(lower_limit, [upper_limit])


Arguments
1-2 limits for integration


Return Value
Error function result (0 to 1)

This table breaks down the essential components of the ERF function. It shows the function name, basic syntax format, argument options, and return value characteristics.

Basic ERF Example

This example demonstrates the simplest use of the ERF function with a single limit. The function integrates from 0 to the specified value.

Basic ERF formula

=ERF(1)

This formula calculates the error function from 0 to 1. The result is approximately 0.8427. This shows how ERF works with a single argument.

ERF with Two Limits

ERF can calculate the integral between any two points, not just from zero. Here’s an example with both lower and upper limits specified.

A
B


0.5



1.5




=ERF(A1, A2)

The table shows a simple spreadsheet with limits in cells A1 and A2. The ERF formula in B3 calculates the integral between these two points.

ERF with two limits

=ERF(0.5, 1.5)

This formula calculates the error function between 0.5 and 1.5. The result is approximately 0.3351. Using two arguments provides more flexibility.

ERF with Negative Values

ERF can handle negative input values, maintaining mathematical symmetry. This example shows ERF’s behavior with negative numbers.

A
B


-1




=ERF(A1)

This table demonstrates ERF’s calculation with a negative input value. The function maintains proper mathematical properties for negative inputs.

ERF with negative value

=ERF(-1)

This formula calculates the error function from 0 to -1. The result is approximately -0.8427. The negative input produces a negative result.

ERF in Probability Calculations

ERF is often used in probability calculations related to normal distributions. This example shows a practical statistical application.

A
B


1.96




=ERF(A1/SQRT(2))

The table shows how to use ERF to calculate probabilities for standard normal distributions. The formula converts a z-score to a probability value.

ERF for normal distribution

=ERF(1.96/SQRT(2))

This formula calculates the probability for z=1.96 in a standard normal distribution. The result is approximately 0.9500, matching statistical tables.

ERF with Cell References

For practical applications, ERF is often used with cell references rather than hard-coded values. This example demonstrates this approach.

A
B
C


0.2
0.8





=ERF(A1, B1)

The table illustrates using ERF with cell references for both limits. This approach makes the formula dynamic and easily adjustable.

ERF with cell references

=ERF(A1, B1)

This formula calculates the error function between values in A1 (0.2) and B1 (0.8). The result is approximately 0.5205. Cell references make the formula more flexible.

ERFC Complementary Function

Excel also provides ERFC, the complementary error function. This example shows the relationship between ERF and ERFC.

ERF and ERFC relationship

=1-ERF(1)

This formula demonstrates that ERFC(x) equals 1-ERF(x). For x=1, ERF(1) is 0.8427, so 1-ERF(1) equals 0.1573, which matches ERFC(1).

The ERF function is essential for statistical and engineering calculations in Excel. From basic error function evaluation to complex probability calculations, ERF handles it precisely. Mastering its applications will enhance your statistical analysis capabilities.

Author

My name is Jan Bodnar, and I am a passionate programmer with extensive programming experience. I have been writing programming articles since 2007. To date, I have authored over 1,400 articles and 8 e-books. I possess more than ten years of experience in teaching programming.

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