In mathematics, the error function (also called the Gauss error function or ERF) is a special, non-elementary function that occurs in probability, statistics and partial differential equations describing diffusion. It is defined as: 
In statistics, for nonnegative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and variance 1/2, ERF(x) describes the probability of Y falling in the range [−x, x]. This function returns the error function integrated between lower_limit and upper_limit.
Syntax: ERF(lower_limit,[upper_limit])
The ERF function syntax has the following arguments:
- Lower_limit Required. The lower bound for integrating ERF.
-
Upper_limit Optional. The upper bound for integrating ERF. If omitted, ERF integrates between zero and lower_limit.
Example: Let’s look at some Excel ERF function examples and explore how to use the ERF function as a worksheet function in Microsoft Excel:
Syntax: =ERF(A2,B2)
Result:
Based on the Excel spreadsheet above, the following ERF examples would return:
Syntax: =ERF(A3,B3)
Result: 0.842700793
Syntax: =ERF(A4,B4)
Result: 0.520499878
Syntax: =ERF(A5,B5)
Result: 0.742100965
Syntax: =ERF(A6,B6)
Result: 0.910313978
Syntax: =ERF(A7,B7)
Result: 0.976348383
Syntax: =ERF(A8,B8)
Result: 0.023651617
Syntax: =ERF(A9,B9)
Result: 0.966105146
Syntax: =ERF(A10,B10)
Result: 0.152621472
Syntax: =ERF(A11,B11)
Result: 0.004677735
Syntax: =ERF(A12,B12)
Result: 0
Syntax: =ERF(A13,B13)
Result: -1.842700793
Note:
- If lower_limit is nonnumeric, ERF returns the #VALUE! error value.
- If upper_limit is nonnumeric, ERF returns the #VALUE! error value
