In the study of layered neural networks using the sigmoidal function as the characteristic function, certain statistical properties are still poorly understood. This study makes a theoretical and numerical comparison of this kind of network with the network using the Heaviside function as the characteristic function, in terms of the learning error. In the comparison, the Heaviside function with a ramp function is considered as a characteristic function which has properties intermediate between the two, and can be considered as a limit of the sigmoidal function. A similarity from the viewpoint of statistical properties is suggested. It is also shown that there is no significant difference in terms of learning error between the cases of the Heaviside function with and without a ramp function. This implies that the layered neural networks with the sigmoidal function and the Heaviside function in which the number of hidden layer units is 1 have similar properties, which are different from the conventional linear model. © 2005 Wiley Periodicals, Inc.