In this work, we classify all templates according to their periods and for each template we evaluate the exact probabilities using generating functions. Afterwards, we propose a new family of statistical randomness tests, that is periodic template tests, for a collection of binary sequences. We apply these tests to the outputs of AES, SHA-3, SHA-2 family, SHA-1 and MD5 and the binary expansion of pi and root 2 and biased non-random data to test the power of new tests. Moreover, we give the probabilities for all templates for the overlapping template matching test in the NIST test suite. Afterwards, we analyse the power of templates and compare the periodic template tests with NIST overlapping template test. (C) 2019 Elsevier B.V. All rights reserved.