Perturbing sensitive data is one of the standard ways of protecting confidential data. The multiplicative noise method is one of these data perturbation methods and this method has attracted researchers' attention in the recent decade.
However, values close to zero in datasets cannot be well protected by using the multiplicative noise method directly.
This paper proposes a method for safeguarding the values close to zero through noise-multiplied shifted data. We demonstrate that those values can be reasonably protected through noise-multiplied data by following the approach proposed in this paper. This paper also indicates that the density function of the original data can be reasonably reconstructed from the noise-multiplied shifted data by using the software MaskDensity14 or MaskDensityBM.