The usage of machine-learning techniques, such as neural networks, is common in a large variety of domains. Estimating the certainty of a predicted value is important when precise information is gained. Nevertheless, the forward propagation of uncertainty in machine-learning models is hardly understood. In general, providing error bars for measurements (measurement uncertainty) is crucial when high precision is needed for decision-making. The objective of this work is the development of an analytical method for aleatoric uncertainty forward propagation in neural networks, based on analytical uncertainty propagation well known from physics and engineering. With that, the method gives provable correct results. A benefit is that the method does not require a different training procedure, but only needs the weights and biases of the neural network and is computationally inexpensive. The analytical method is applied to real-world examples from the semiconductor industry (regression and image classification). Its usefulness is demonstrated by the provided examples, which show how meaningful error bars are when machine learning may be used for decision-making.