In a celebrated paper in 1979, Gidas, Ni & Nirenberg proved a symmetry result for a rigidity problem. With minimal hypotheses, the authors showed that positive solutions of semilinear elliptic equations in the unitary ball are radial and radially decreasing. This result had a big impact on the PDE community and stemmed several generalizations. In a recent work in collaboration with Ciraolo, Cozzi & Perugini this problem was investigated from a quantitative viewpoint, starting with the following question: given that the rigidity condition implies symmetry, is it possible to prove that if said condition is "almost" satisfied the problem is "almost" symmetrical? With the employment of the method of moving planes and quantitative maximum principles we are able to give a positive answer to the question, proving approximate radial symmetry and almost monotonicity for positive solutions of the perturbed problem.