By analyzing the fluctuations of ability estimates in testing, we first obtain the purely probabilistic fluctuations of ability estimates in a onetime testing under the condition that the students' abilities can be estimated by using the item response theory, and next, by taking into account such the probabilistic fluctuations, we find students who reveal the discrepancies of observed abilities between two separated testings. When such discrepancies of abilities are observed, test results are considered to be affected by some factors such as the physical conditions of the examinees, the teacher's teaching skills, and students' study skill developments. To describe such a phenomenon, we proposed a basic formula. Using an example case took in a university mathematics testing, we have shown how we have extracted the purely probabilistic fluctuations and segregated fluctuations due to other factors. 




item response theory, purely probabilistic fluctuation, basic formula representing fluctuations, bootstrap method, Fisher information matrix, learning analytics.
%correct answer rate, evaluation bias, description type testing, multiple choice type testing, academic growth, ability equation. 

