In this paper we extend the notions of statistical limit points and statistical cluster points of double sequences of real numbers to $${\mathcal {I}}$$ -statistical limit points and $${\mathcal {I}}$$ -statistical cluster points of double sequences of real numbers. We study some basic properties of the set of all $${\mathcal {I}}$$ -statistical cluster points and the set of all $${\mathcal {I}}$$ -statistical limit points of a double sequence of real numbers including their interrelationship. Also introducing additive property of double $${\mathcal {I}}$$ -natural density zero sets we present $${\mathcal {I}}$$ -statistical analogue of some well known theorems concerning double sequences that are equivalent to the completeness of the real line.