New simple bounds for standard normal distribution function

This paper presents new simple lower and upper bounds for the cumulative normal distribution function, Φ(z). The accuracy and closeness of the proposed bounds to the exact Φ(z) are investigated based on the maximum absolute error and the mean absolute error. It is found that the maximum absolute error of the proposed lower bound is 8.55×10−3 and it is 4.1×10−4 for the upper bound. In addition, based on 5001 values between z=0 and z=5 with step 0.001, we found that the mean absolute error is 3.27×10−3 for the lower bound and it is 1.1×10−4 for the upper bound and these two values decrease with increasing the z value.