Directional data occur in version fields of applications. Directions of winds, daily times of events, and the directions of the birds’ flight are examples of directional observations. This paper deals with the problem of statistical hypotheses using an evidential approach based on directional data. It is assumed that a sample from the circular normal distribution is available. Hypotheses about the mean direction parameter are considered when the concentration parameter is either known or unknown. Evidential measures, including strong, misleading, and weak pieces of evidence are derived in explicit expressions. The evidential approach does not require the identification of a loss function, as is needed in the classical approach. Unlike the Bayesian method, which requires the specification of a prior, the evidential approach operates without the need for a prior. The evidential approach complements both Bayesian and classical methods by preventing the influence and inclusion of researchers’ personal biases and opinions. For big data sets, some approximations are also provided. These approximations may be used for fast computations when dealing with massive data sets. Finally, to assess the performance of the obtained results, a real data set on times of urban injury accidents is also examined.

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